Advances in Nonlinear Analysis

Open Access

ISSN: 2191-950X

First published: March 15, 2012

Editors-in-chief: Vicentiu Radulescu, Runzhang Xu

Impact Factor: 2.667

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Objective
Advances in Nonlinear Analysis (ANONA) aims to publish selected research contributions devoted to nonlinear problems coming from different areas, with particular reference to those introducing new techniques capable of solving a wide range of problems. The Journal focuses on papers that address significant problems in pure and applied nonlinear analysis. ANONA seeks to present the most significant advances in this field to a wide readership, including researchers and graduate students in mathematics, physics, and engineering.

Topics

calculus of variations and PDEs
variational and topological methods for ODEs and PDEs
non-smooth analysis and optimization
generalized differentiability
set-valued functions
bifurcation theory
convex analysis
variational inequalities

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Papers accepted for publication in Advances in Nonlinear Analysis are subject to Article Processing Charges (APCs). Currently the APC is 1200 Euro.

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History

Advanced Nonlinear Analysis started publishing in 2012. The origin of this new publication was due to the high demand of the understanding of many nonlinear phenomena in the applied sciences which are described by models involving partial differential equations.

Since 2019, the journal has become an Open Access publication and applied a continuous model of publication. All published articles are made freely and permanently accessible online without any restrictions.

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Volume 10 Issue 1

January 2021

Editorial

Editorial to Volume 10 of ANA

Vicenţiu D. Rădulescu May 4, 2020 Page range: 1-1

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Regular Articles

Convergence Results for Elliptic Variational-Hemivariational Inequalities

Dong-ling Cai, Mircea Sofonea, Yi-bin Xiao May 27, 2020 Page range: 2-23

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We consider an elliptic variational-hemivariational inequality 𝓟 in a reflexive Banach space, governed by a set of constraints K , a nonlinear operator A , and an element f . We associate to this inequality a sequence {𝓟 n } of variational-hemivariational inequalities such that, for each n ∈ ℕ, inequality 𝓟 n is obtained by perturbing the data K and A and, moreover, it contains an additional term governed by a small parameter ε n . The unique solvability of 𝓟 and, for each n ∈ ℕ, the solvability of its perturbed version 𝓟 n , are guaranteed by an existence and uniqueness result obtained in literature. Denote by u the solution of Problem 𝓟 and, for each n ∈ ℕ, let u n be a solution of Problem 𝓟 n . The main result of this paper states the strong convergence of u n → u in X , as n → ∞. We show that the main result extends a number of results previously obtained in the study of Problem 𝓟. Finally, we illustrate the use of our abstract results in the study of a mathematical model which describes the contact of an elastic body with a rigid-deformable foundation and provide the corresponding mechanical interpretations.

Weak and stationary solutions to a Cahn–Hilliard–Brinkman model with singular potentials and source terms

Matthias Ebenbeck, Kei Fong Lam May 27, 2020 Page range: 24-65

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We study a phase field model proposed recently in the context of tumour growth. The model couples a Cahn–Hilliard–Brinkman (CHB) system with an elliptic reaction-diffusion equation for a nutrient. The fluid velocity, governed by the Brinkman law, is not solenoidal, as its divergence is a function of the nutrient and the phase field variable, i.e., solution-dependent, and frictionless boundary conditions are prescribed for the velocity to avoid imposing unrealistic constraints on the divergence relation. In this paper we give a first result on the existence of weak and stationary solutions to the CHB model for tumour growth with singular potentials, specifically the double obstacle potential and the logarithmic potential, which ensures that the phase field variable stays in the physically relevant interval. New difficulties arise from the interplay between the singular potentials and the solution-dependent source terms, but can be overcome with several key estimates for the approximations of the singular potentials, which maybe of independent interest. As a consequence, included in our analysis is an existence result for a Darcy variant, and our work serves to generalise recent results on weak and stationary solutions to the Cahn–Hilliard inpainting model with singular potentials.

Single peaked traveling wave solutions to a generalized μ-Novikov Equation

Byungsoo Moon May 22, 2020 Page range: 66-75

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In this paper, we study the existence of peaked traveling wave solution of the generalized μ -Novikov equation with nonlocal cubic and quadratic nonlinearities. The equation is a μ -version of a linear combination of the Novikov equation and Camassa-Hom equation. It is found that the equation admits single peaked traveling wave solutions.

Constant sign and nodal solutions for superlinear (p, q)–equations with indefinite potential and a concave boundary term

Nikolaos S. Papageorgiou, Youpei Zhang May 30, 2020 Page range: 76-101

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We consider a nonlinear elliptic equation driven by the ( p , q )–Laplacian plus an indefinite potential. The reaction is ( p − 1)–superlinear and the boundary term is parametric and concave. Using variational tools from the critical point theory together with truncation, perturbation and comparison techniques and critical groups, we show that for all small values of the parameter, the problem has at least five nontrivial smooth solutions, all with sign information and which are linearly ordered.

On isolated singularities of Kirchhoff equations

Huyuan Chen, Mouhamed Moustapha Fall, Binling Zhang May 27, 2020 Page range: 102-120

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In this note, we study isolated singular positive solutions of Kirchhoff equation Mθ (u)(− Δ )u=upinΩ ∖ {0},u=0on∂ Ω , $$\begin{array}{} \displaystyle M_\theta(u)(-{\it\Delta}) u =u^p \quad{\rm in}\quad {\it\Omega}\setminus \{0\},\qquad u=0\quad {\rm on}\quad \partial {\it\Omega}, \end{array}$$ where p > 1, θ ∈ ℝ, M θ ( u ) = θ + ∫ Ω |∇ u | dx , Ω is a bounded smooth domain containing the origin in ℝ N with N ≥ 2. In the subcritical case: 1 < p < NN− 2 $\begin{array}{} \displaystyle \frac{N}{N-2} \end{array}$ if N ≥ 3, 1 < p < + ∞ if N = 2, we employee the Schauder fixed point theorem to derive a sequence of positive isolated singular solutions for the above equation such that M θ ( u ) > 0. To estimate M θ ( u ), we make use of the rearrangement argument. Furthermore, we obtain a sequence of isolated singular solutions such that M θ ( u ) < 0, by analyzing relationship between the parameter λ and the unique solution u λ of − Δ u+λ up=kδ 0inB1(0),u=0on∂ B1(0). $$\begin{array}{} \displaystyle -{\it\Delta} u+\lambda u^p=k\delta_0\quad{\rm in}\quad B_1(0),\qquad u=0\quad {\rm on}\quad \partial B_1(0). \end{array}$$ In the supercritical case: NN− 2 $\begin{array}{} \displaystyle \frac{N}{N-2} \end{array}$ ≤ p < N+2N− 2 $\begin{array}{} \displaystyle \frac{N+2}{N-2} \end{array}$ with N ≥ 3, we obtain two isolated singular solutions u i with i = 1, 2 such that M θ ( u i ) > 0 under other assumptions.

On the existence of periodic oscillations for pendulum-type equations

J. Ángel Cid May 27, 2020 Page range: 121-130

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We provide new sufficient conditions for the existence of T -periodic solutions for the ϕ -laplacian pendulum equation ( ϕ ( x ′))′ + k x ′ + a sin x = e ( t ), where e ∈ C͠ T . Our main tool is a continuation theorem due to Capietto, Mawhin and Zanolin and we improve or complement previous results in the literature obtained in the framework of the classical, the relativistic and the curvature pendulum equations.

Multiplicity of concentrating solutions for a class of magnetic Schrödinger-Poisson type equation

Yueli Liu, Xu Li, Chao Ji May 17, 2020 Page range: 131-151

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In this paper, we study the following nonlinear magnetic Schrödinger-Poisson type equation (ε i∇ − A(x))2u+V(x)u+ϵ − 2(|x|− 1∗ |u|2)u=f(|u|2)uin R3,u∈ H1(R3,C), $$\begin{array}{} \displaystyle \left\{\!\begin{aligned}&\Big(\frac{\varepsilon}{i}\nabla-A(x)\Big)^{2}u+V(x)u+\epsilon^{-2}(\vert x\vert^{-1}\ast \vert u\vert^{2})u = f(|u|^{2})u\quad\hbox{in }\mathbb{R}^3,\\&u\in H^{1}(\mathbb{R}^{3}, \mathbb{C}),\end{aligned}\right. \end{array}$$ where ϵ > 0, V : ℝ 3 → ℝ and A : ℝ 3 → ℝ 3 are continuous potentials. Under a local assumption on the potential V , by variational methods, penalization technique, and Ljusternick-Schnirelmann theory, we prove multiplicity and concentration properties of nontrivial solutions for ε > 0 small. In this problem, the function f is only continuous, which allow to consider larger classes of nonlinearities in the reaction.

Nehari-type ground state solutions for a Choquard equation with doubly critical exponents

Sitong Chen, Xianhua Tang, Jiuyang Wei May 30, 2020 Page range: 152-171

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This paper deals with the following Choquard equation with a local nonlinear perturbation: − Δ u+u=Iα ∗ |u|α 2+1|u|α 2− 1u+f(u),x∈ R2;u∈ H1(R2), $$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} - {\it\Delta} u+u=\left(I_{\alpha}*|u|^{\frac{\alpha}{2}+1}\right)|u|^{\frac{\alpha}{2}-1}u +f(u), & x\in \mathbb{R}^2; \\ u\in H^1(\mathbb{R}^2), \end{array} \right. \end{array}$$ where α ∈ (0, 2), I α : ℝ 2 → ℝ is the Riesz potential and f ∈ 𝓒(ℝ, ℝ) is of critical exponential growth in the sense of Trudinger-Moser. The exponent α 2+1 $\begin{array}{} \displaystyle \frac{\alpha}{2}+1 \end{array}$ is critical with respect to the Hardy-Littlewood-Sobolev inequality. We obtain the existence of a nontrivial solution or a Nehari-type ground state solution for the above equation in the doubly critical case, i.e. the appearance of both the lower critical exponent α 2+1 $\begin{array}{} \displaystyle \frac{\alpha}{2}+1 \end{array}$ and the critical exponential growth of f ( u ).

Gradient estimate of a variable power for nonlinear elliptic equations with Orlicz growth

Shuang Liang, Shenzhou Zheng July 2, 2020 Page range: 172-193

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In this paper, we prove a global Calderón-Zygmund type estimate in the framework of Lorentz spaces for a variable power of the gradients to the zero-Dirichlet problem of general nonlinear elliptic equations with the nonlinearities satisfying Orlicz growth. It is mainly assumed that the variable exponents p ( x ) satisfy the log-Hölder continuity, while the nonlinearity and underlying domain ( A , Ω ) is ( δ , R 0 )-vanishing in x ∈ Ω .

The structure of 𝓐-free measures revisited

D. Mitrovic, Dj. Vujadinović June 10, 2020 Page range: 194-201

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We refine a recent result on the structure of measures satisfying a linear partial differential equation 𝓐 μ = σ , μ , σ are Radon measures, considering the measure μ ( x ) = g ( x ) d x + μ us ( x̃ )( μ s ( x̄ ) + d x̄ ) where x = ( x̃ , x̄ ) ∈ ℝ k × ℝ d − k , μ us is a uniformly singular measure in x̃ 0 and the measure μ s is a singular measure. We proved that for μ us -a.e. x̃ 0 the range of the Radon-Nykodim derivative f~ (x~ 0)=dμ usd|μ us|(x~ 0) $\begin{array}{} \tilde{f}(\tilde{{\bf x}}_0) = \frac{d \mu_{us}}{d | \mu_{us}|}(\tilde{{\bf x}}_0) \end{array}$ is in the set ∩ ξ̃ ∈ P̃ 𝓚 er A P̃ ( ξ ) and, if μ s is different to zero, for μ s -a.e. x̄ 0 the range of the Radon-Nykodim derivative f¯ (x¯ 0)=dμ sd|μ s|(x¯ 0) $\begin{array}{} \bar{f}(\bar{{\bf x}}_0) = \frac{d \mu_{s}}{d | \mu_{s}|}(\bar{{\bf x}}_0) \end{array}$ is in the set ∪ ξ̄ ∈ P̄ 𝓚 er A P̄ ( ξ ) where P̃ × P̄ = P is a manifold determined by the main symbol A P = A P̃ ⋅ A P̄ of the operator 𝓐.

Solvability of an infinite system of integral equations on the real half-axis

Józef Banaś, Weronika Woś June 19, 2020 Page range: 202-216

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The aim of the paper is to investigate the solvability of an infinite system of nonlinear integral equations on the real half-axis. The considerations will be located in the space of function sequences which are bounded at every point of the half-axis. The main tool used in the investigations is the technique associated with measures of noncompactness in the space of functions defined, continuous and bounded on the real half-axis with values in the space l ∞ consisting of real bounded sequences endowed with the standard supremum norm. The essential role in our considerations is played by the fact that we will use a measure of noncompactness constructed on the basis of a measure of noncompactness in the mentioned sequence space l ∞ . An example illustrating our result will be included.

Positive Solutions for Resonant (p, q)-equations with convection

Zhenhai Liu, Nikolaos S. Papageorgiou July 17, 2020 Page range: 217-232

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We consider a nonlinear parametric Dirichlet problem driven by the (p, q)-Laplacian (double phase problem) with a reaction exhibiting the competing effects of three different terms. A parametric one consisting of the sum of a singular term and of a drift term (convection) and of a nonparametric perturbation which is resonant. Using the frozen variable method and eventually a fixed point argument based on an iterative asymptotic process, we show that the problem has a positive smooth solution.

Concentration behavior of semiclassical solutions for Hamiltonian elliptic system

Jian Zhang, Jianhua Chen, Quanqing Li, Wen Zhang July 17, 2020 Page range: 233-260

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In this paper, we study the following nonlinear Hamiltonian elliptic system with gradient term − ϵ 2Δ ψ +ϵ b→ ⋅ ∇ ψ +ψ +V(x)φ =f(|η |)φ in RN,− ϵ 2Δ φ − ϵ b→ ⋅ ∇ φ +φ +V(x)ψ =f(|η |)ψ in RN, $$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} -\epsilon^{2}{\it\Delta} \psi +\epsilon \vec{b}\cdot \nabla \psi +\psi+V(x)\varphi=f(|\eta|)\varphi~~\hbox{in}~\mathbb{R}^{N},\\ -\epsilon^{2}{\it\Delta} \varphi -\epsilon \vec{b}\cdot \nabla \varphi +\varphi+V(x)\psi=f(|\eta|)\psi~~\hbox{in}~\mathbb{R}^{N},\\ \end{array} \right. \end{array}$$ where η = ( ψ , φ ) : ℝ N → ℝ 2 , ϵ is a small positive parameter and b⃗ is a constant vector. We require that the potential V only satisfies certain local condition. Combining this with other suitable assumptions on f , we construct a family of semiclassical solutions. Moreover, the concentration phenomena around local minimum of V , convergence and exponential decay of semiclassical solutions are also explored. In the proofs we apply penalization method, linking argument and some analytical techniques since the local property of the potential and the strongly indefinite character of the energy functional.

Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation

Xingchang Wang, Runzhang Xu July 28, 2020 Page range: 261-288

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In this paper, the initial boundary value problem for a nonlocal semilinear pseudo-parabolic equation is investigated, which was introduced to model phenomena in population dynamics and biological sciences where the total mass of a chemical or an organism is conserved. The existence, uniqueness and asymptotic behavior of the global solution and the blowup phenomena of solution with subcritical initial energy are established. Then these results are extended parallelly to the critical initial energy. Further the blowup phenomena of solution with supercritical initial energy is proved, but the existence, uniqueness and asymptotic behavior of the global solution with supercritical initial energy are still open.

On variational nonlinear equations with monotone operators

Marek Galewski August 2, 2020 Page range: 289-300

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Using monotonicity methods and some variational argument we consider nonlinear problems which involve monotone potential mappings satisfying condition (S) and their strongly continuous perturbations. We investigate when functional whose minimum is obtained by a direct method of the calculus of variations satisfies the Palais-Smale condition, relate minimizing sequence and Galerkin approximaitons when both exist, then provide structure conditions on the derivative of the action functional under which bounded Palais-Smale sequences are convergent. Finally, we make some comment concerning the convergence of Palais-Smale sequence obtained in the mountain pass theorem due to Rabier.

Existence results for nonlinear degenerate elliptic equations with lower order terms

Weilin Zou, Xinxin Li August 2, 2020 Page range: 301-310

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In this paper, we prove the existence and regularity of solutions of the homogeneous Dirichlet initial-boundary value problem for a class of degenerate elliptic equations with lower order terms. The results we obtained here, extend some existing ones of [2, 9, 11] in some sense.

Blow-up criteria and instability of normalized standing waves for the fractional Schrödinger-Choquard equation

Feng Binhua, Ruipeng Chen, Jiayin Liu July 28, 2020 Page range: 311-330

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In this paper, we study blow-up criteria and instability of normalized standing waves for the fractional Schrödinger-Choquard equation i∂ tψ − (− Δ )sψ +(Iα ∗ |ψ |p)|ψ |p− 2ψ =0. $$\begin{array}{} \displaystyle i\partial_t\psi- (-{\it\Delta})^s \psi+(I_\alpha \ast |\psi|^{p})|\psi|^{p-2}\psi=0. \end{array}$$ By using localized virial estimates, we firstly establish general blow-up criteria for non-radial solutions in both L 2 -critical and L 2 -supercritical cases. Then, we show existence of normalized standing waves by using the profile decomposition theory in H s . Combining these results, we study the strong instability of normalized standing waves. Our obtained results greatly improve earlier results.

Ground states and multiple solutions for Hamiltonian elliptic system with gradient term

Wen Zhang, Jian Zhang, Heilong Mi July 30, 2020 Page range: 331-352

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This paper is concerned with the following nonlinear Hamiltonian elliptic system with gradient term − Δ u+b→ (x)⋅ ∇ u+V(x)u=Hv(x,u,v)inRN,− Δ v− b→ (x)⋅ ∇ v+V(x)v=Hu(x,u,v)inRN. $$\begin{array}{} \displaystyle \left\{\,\, \begin{array}{ll} -{\it\Delta} u +\vec{b}(x)\cdot \nabla u+V(x)u = H_{v}(x,u,v)\,\,\hbox{in}\,\mathbb{R}^{N},\\[-0.3em] -{\it\Delta} v -\vec{b}(x)\cdot \nabla v +V(x)v = H_{u}(x,u,v)\,\,\hbox{in}\,\mathbb{R}^{N}.\\ \end{array} \right. \end{array}$$ Compared with some existing issues, the most interesting feature of this paper is that we assume that the nonlinearity satisfies a local super-quadratic condition, which is weaker than the usual global super-quadratic condition. This case allows the nonlinearity to be super-quadratic on some domains and asymptotically quadratic on other domains. Furthermore, by using variational method, we obtain new existence results of ground state solutions and infinitely many geometrically distinct solutions under local super-quadratic condition. Since we are without more global information on the nonlinearity, in the proofs we apply a perturbation approach and some special techniques.

Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations

Hans-Christoph Grunau, Nobuhito Miyake, Shinya Okabe August 2, 2020 Page range: 353-370

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This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. One has eventual local positivity for positive initial data, but on short time scales, one will in general have also regions of negativity. The first goal of this paper is to find sufficient conditions on initial data which ensure the existence of solutions to the Cauchy problem for the linear biharmonic heat equation which are positive for all times and in the whole space . The second goal is to apply these results to show existence of globally positive solutions to the Cauchy problem for a semilinear biharmonic parabolic equation.

Convex solutions of Monge-Ampère equations and systems: Existence, uniqueness and asymptotic behavior

Meiqiang Feng August 25, 2020 Page range: 371-399

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In this paper, the equations and systems of Monge-Ampère with parameters are considered. We first show the uniqueness of of nontrivial radial convex solution of Monge-Ampère equations by using sharp estimates. Then we analyze the existence and nonexistence of nontrivial radial convex solutions to Monge-Ampère systems, which includes some new ingredients in the arguments. Furthermore, the asymptotic behavior of nontrivial radial convex solutions for Monge-Ampère systems is also considered. Finally, as an application, we obtain sufficient conditions for the existence of nontrivial radial convex solutions of the power-type system of Monge-Ampère equations.

Multiple solutions for critical Choquard-Kirchhoff type equations

Sihua Liang, Patrizia Pucci, Binlin Zhang August 22, 2020 Page range: 400-419

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In this article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents, − a+b∫ RN|∇ u|2dxΔ u=α k(x)|u|q− 2u+β ∫ RN|u(y)|2μ ∗ |x− y|μ dy|u|2μ ∗ − 2u,x∈ RN, $$\begin{array}{} \displaystyle -\left(a + b\int\limits_{\mathbb{R}^N} |\nabla u|^2 dx\right){\it\Delta} u = \alpha k(x)|u|^{q-2}u + \beta\left(\,\,\displaystyle\int\limits_{\mathbb{R}^N}\frac{|u(y)|^{2^*_{\mu}}}{|x-y|^{\mu}}dy\right)|u|^{2^*_{\mu}-2}u, \quad x \in \mathbb{R}^N, \end{array}$$ where a > 0, b ≥ 0, 0 < μ < N , N ≥ 3, α and β are positive real parameters, 2μ ∗ =(2N− μ )/(N− 2) $\begin{array}{} 2^*_{\mu} = (2N-\mu)/(N-2) \end{array}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, k ∈ L r (ℝ N ), with r = 2 ∗ /(2 ∗ − q ) if 1 < q < 2 * and r = ∞ if q ≥ 2 ∗ . According to the different range of q , we discuss the multiplicity of solutions to the above equation, using variational methods under suitable conditions. In order to overcome the lack of compactness, we appeal to the concentration compactness principle in the Choquard-type setting.

Regularity for sub-elliptic systems with VMO-coefficients in the Heisenberg group: the sub-quadratic structure case

Jialin Wang, Maochun Zhu, Shujin Gao, Dongni Liao August 7, 2020 Page range: 420-449

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We consider nonlinear sub-elliptic systems with VMO-coefficients for the case 1 < p < 2 under controllable growth conditions, as well as natural growth conditions, respectively, in the Heisenberg group. On the basis of a generalization of the technique of 𝓐-harmonic approximation introduced by Duzaar-Grotowski-Kronz, and an appropriate Sobolev-Poincaré type inequality established in the Heisenberg group, we prove partial Hölder continuity results for vector-valued solutions of discontinuous sub-elliptic problems. The primary model covered by our analysis is the non-degenerate sub-elliptic p -Laplacian system with VMO-coefficients, involving sub-quadratic growth terms.

Inducing strong convergence of trajectories in dynamical systems associated to monotone inclusions with composite structure

Radu Ioan Boţ, Sorin-Mihai Grad, Dennis Meier, Mathias Staudigl August 25, 2020 Page range: 450-476

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In this work we investigate dynamical systems designed to approach the solution sets of inclusion problems involving the sum of two maximally monotone operators. Our aim is to design methods which guarantee strong convergence of trajectories towards the minimum norm solution of the underlying monotone inclusion problem. To that end, we investigate in detail the asymptotic behavior of dynamical systems perturbed by a Tikhonov regularization where either the maximally monotone operators themselves, or the vector field of the dynamical system is regularized. In both cases we prove strong convergence of the trajectories towards minimum norm solutions to an underlying monotone inclusion problem, and we illustrate numerically qualitative differences between these two complementary regularization strategies. The so-constructed dynamical systems are either of Krasnoselskiĭ-Mann, of forward-backward type or of forward-backward-forward type, and with the help of injected regularization we demonstrate seminal results on the strong convergence of Hilbert space valued evolutions designed to solve monotone inclusion and equilibrium problems.

A posteriori analysis of the spectral element discretization of a non linear heat equation

Mohamed Abdelwahed, Nejmeddine Chorfi August 7, 2020 Page range: 477-493

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The paper deals with a posteriori analysis of the spectral element discretization of a non linear heat equation. The discretization is based on Euler’s backward scheme in time and spectral discretization in space. Residual error indicators related to the discretization in time and in space are defined. We prove that those indicators are upper and lower bounded by the error estimation.

Liouville property of fractional Lane-Emden equation in general unbounded domain

Ying Wang, Yuanhong Wei August 22, 2020 Page range: 494-500

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Our purpose of this paper is to consider Liouville property for the fractional Lane-Emden equation (− Δ )α u=upinΩ ,u=0inRN∖ Ω , $$\begin{array}{} \displaystyle (-{\it\Delta})^\alpha u = u^p\quad {\rm in}\quad {\it\Omega},\qquad u = 0\quad {\rm in}\quad \mathbb{R}^N\setminus {\it\Omega}, \end{array}$$ where α ∈ (0, 1), N ≥ 1, p > 0 and Ω ⊂ ℝ N –1 × [0, +∞) is an unbounded domain satisfying that Ω t := { x ′ ∈ ℝ N –1 : ( x ′, t ) ∈ Ω } with t ≥ 0 has increasing monotonicity, that is, Ω t ⊂ Ω t ′ for t ′ ≥ t . The shape of Ω ∞ := lim t →∞ Ω t in ℝ N –1 plays an important role to obtain the nonexistence of positive solutions for the fractional Lane-Emden equation.

Large data existence theory for three-dimensional unsteady flows of rate-type viscoelastic fluids with stress diffusion

Michal Bathory, Miroslav Bulíček, Josef Málek September 2, 2020 Page range: 501-521

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We prove that there exists a weak solution to a system governing an unsteady flow of a viscoelastic fluid in three dimensions, for arbitrarily large time interval and data. The fluid is described by the incompressible Navier-Stokes equations for the velocity v , coupled with a diffusive variant of a combination of the Oldroyd-B and the Giesekus models for a tensor 𝔹. By a proper choice of the constitutive relations for the Helmholtz free energy (which, however, is non-standard in the current literature, despite the fact that this choice is well motivated from the point of view of physics) and for the energy dissipation, we are able to prove that 𝔹 enjoys the same regularity as v in the classical three-dimensional Navier-Stokes equations. This enables us to handle any kind of objective derivative of 𝔹, thus obtaining existence results for the class of diffusive Johnson-Segalman models as well. Moreover, using a suitable approximation scheme, we are able to show that 𝔹 remains positive definite if the initial datum was a positive definite matrix (in a pointwise sense). We also show how the model we are considering can be derived from basic balance equations and thermodynamical principles in a natural way.

On some classes of generalized Schrödinger equations

Amanda S. S. Correa Leão, Joelma Morbach, Andrelino V. Santos, João R. Santos Júnior August 22, 2020 Page range: 522-533

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Some classes of generalized Schrödinger stationary problems are studied. Under appropriated conditions is proved the existence of at least 1 + ∑ i=2m $\begin{array}{} \sum_{i=2}^{m} \end{array}$ dim V λ i pairs of nontrivial solutions if a parameter involved in the equation is large enough, where V λ i denotes the eigenspace associated to the i-th eigenvalue λ i of laplacian operator with homogeneous Dirichlet boundary condition.

Variational formulations of steady rotational equatorial waves

Jifeng Chu, Joachim Escher August 22, 2020 Page range: 534-547

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When the vorticity is monotone with depth, we present a variational formulation for steady periodic water waves of the equatorial flow in the f -plane approximation, and show that the governing equations for this motion can be obtained by studying variations of a suitable energy functional 𝓗 in terms of the stream function and the thermocline. We also compute the second variation of the constrained energy functional, which is related to the linear stability of steady water waves.

On a class of critical elliptic systems in ℝ⁴

Xin Zhao, Wenming Zou September 13, 2020 Page range: 548-568

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In the present paper, we consider the following classes of elliptic systems with Sobolev critical growth: − Δ u+λ 1u=μ 1u3+β uv2+2qpyu2qp− 1v2inΩ ,− Δ v+λ 2v=μ 2v3+β u2v+2yu2qpvinΩ ,u,v> -->0inΩ ,u,v=0on∂ Ω , $$\begin{array}{} \displaystyle \begin{cases} -{\it\Delta} u+\lambda_1u=\mu_1 u^3+\beta uv^2+\frac{2q}{p} y u^{\frac{2q}{p}-1}v^2\quad &\hbox{in}\;{\it\Omega}, \\ -{\it\Delta} v+\lambda_2v=\mu_2 v^3+\beta u^2v+2 y u^{\frac{2q}{p}}v\quad&\hbox{in}\;{\it\Omega}, \\ u,v \gt 0&\hbox{in}\;{\it\Omega}, \\ u,v=0&\hbox{on}\;\partial{\it\Omega}, \end{cases} \end{array}$$ where Ω ⊂ ℝ 4 is a smooth bounded domain with smooth boundary ∂Ω ; p , q are positive coprime integers with 1 < 2qp $\begin{array}{} \displaystyle \frac{2q}{p} \end{array}$ < 2; μ i > 0 and λ i ∈ ℝ are fixed constants, i = 1, 2; β > 0, y > 0 are two parameters. We prove a nonexistence result and the existence of the ground state solution to the above system under proper assumptions on the parameters. It seems that this system has not been explored directly before.

Exponential stability of the nonlinear Schrödinger equation with locally distributed damping on compact Riemannian manifold

Fengyan Yang, Zhen-Hu Ning, Liangbiao Chen October 18, 2020 Page range: 569-583

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In this paper, we consider the following nonlinear Schrödinger equation: iut+Δ gu+ia(x)u− |u|p− 1u=0(x,t)∈ M× (0,+∞ ),u(x,0)=u0(x)x∈ M, $$\begin{array}{} \displaystyle \begin{cases}iu_t+{\it\Delta}_g u+ia(x)u-|u|^{p-1}u=0\qquad (x,t)\in \mathcal{M} \times (0,+\infty), \cr u(x,0)=u_0(x)\qquad x\in \mathcal{M},\end{cases} \end{array}$$(0.1) where (𝓜, g ) is a smooth complete compact Riemannian manifold of dimension n ( n = 2, 3) without boundary. For the damping terms − a ( x )(1 − Δ ) −1 a( x ) u t and ia(x)(− Δ )12a(x)u, $\begin{array}{} \displaystyle ia(x)(-{\it\Delta})^{\frac12}a(x)u, \end{array}$ the exponential stability results of system (0.1) have been proved by Dehman et al. (Math Z 254(4): 729-749, 2006), Laurent. (SIAM J. Math. Anal. 42(2): 785-832, 2010) and Cavalcanti et al. (Math Phys 69(4): 100, 2018). However, from the physical point of view, it would be more important to consider the stability of system (0.1) with the damping term ia ( x ) u , which is still an open problem. In this paper, we obtain the exponential stability of system (0.1) by Morawetz multipliers in non Euclidean geometries and compactness-uniqueness arguments.

On a degenerate hyperbolic problem for the 3-D steady full Euler equations with axial-symmetry

Yanbo Hu, Fengyan Li October 18, 2020 Page range: 584-615

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The transonic channel flow problem is one of the most important problems in mathematical fluid dynamics. The structure of solutions near the sonic curve is a key part of the whole transonic flow problem. This paper constructs a local classical hyperbolic solution for the 3-D axisymmetric steady compressible full Euler equations with boundary data given on the degenerate hyperbolic curve. By introducing a novel set of dependent and independent variables, we use the idea of characteristic decomposition to transform the axisymmetric Euler equations as a new system which has explicitly singularity-regularity structures. We first establish a local classical solution for the new system in a weighted metric space and then convert the solution in terms of the original variables.

Existence, multiplicity and nonexistence results for Kirchhoff type equations

Wei He, Dongdong Qin, Qingfang Wu October 30, 2020 Page range: 616-635

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In this paper, we study following Kirchhoff type equation: − a+b∫ Ω |∇ u|2dxΔ u=f(u)+h in Ω ,u=0 on ∂ Ω . $$\begin{array}{} \left\{ \begin{array}{lll} -\left(a+b\int_{{\it\Omega}}|\nabla u|^2 \mathrm{d}x \right){\it\Delta} u=f(u)+h~~&\mbox{in}~~{\it\Omega}, \\ u=0~~&\mbox{on}~~ \partial{\it\Omega}. \end{array} \right. \end{array}$$ We consider first the case that Ω ⊂ ℝ 3 is a bounded domain. Existence of at least one or two positive solutions for above equation is obtained by using the monotonicity trick. Nonexistence criterion is also established by virtue of the corresponding Pohožaev identity. In particular, we show nonexistence properties for the 3-sublinear case as well as the critical case. Under general assumption on the nonlinearity, existence result is also established for the whole space case that Ω = ℝ 3 by using property of the Pohožaev identity and some delicate analysis.

Combined effects of Choquard and singular nonlinearities in fractional Kirchhoff problems

Fuliang Wang, Die Hu, Mingqi Xiang December 3, 2020 Page range: 636-658

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The aim of this paper is to study the existence and multiplicity of solutions for a class of fractional Kirchho problems involving Choquard type nonlinearity and singular nonlinearity. Under suitable assumptions, two nonnegative and nontrivial solutions are obtained by using the Nehari manifold approach combined with the Hardy-Littlehood-Sobolev inequality.

Convergence analysis for double phase obstacle problems with multivalued convection term

Shengda Zeng, Yunru Bai, Leszek Gasiński, Patrick Winkert November 26, 2020 Page range: 659-672

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In the present paper, we introduce a family of the approximating problems corresponding to an elliptic obstacle problem with a double phase phenomena and a multivalued reaction convection term. Denoting by 𝓢 the solution set of the obstacle problem and by 𝓢 n the solution sets of approximating problems, we prove the following convergence relation ∅ ≠ w-lim supn→ ∞ Sn=s-lim supn→ ∞ Sn⊂ S, $$\begin{array}{} \displaystyle \emptyset\neq w\text{-}\limsup\limits_{n\to\infty}{\mathcal S}_n=s\text{-}\limsup\limits_{n\to\infty}{\mathcal S}_n\subset \mathcal S, \end{array}$$ where w -lim sup n →∞ 𝓢 n and s -lim sup n →∞ 𝓢 n denote the weak and the strong Kuratowski upper limit of 𝓢 n , respectively.

Multiple solutions for weighted Kirchhoff equations involving critical Hardy-Sobolev exponent

Zupei Shen, Jianshe Yu November 7, 2020 Page range: 673-683

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In this article, we consider a class of Kirchhoff equations with critical Hardy-Sobolev exponent and indefinite nonlinearity, which has not been studied in the literature. We prove very nicely that this equation has at least two solutions in ℝ 3 . And some known results in the literature are improved.

Boundary value problems associated with singular strongly nonlinear equations with functional terms

Stefano Biagi, Alessandro Calamai, Cristina Marcelli, Francesca Papalini November 12, 2020 Page range: 684-706

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We study boundary value problems associated with singular, strongly nonlinear differential equations with functional terms of type ( Φ ( k ( t ) x ′ ( t ) ) ) ′ + f ( t , G x ( t ) ) ρ ( t , x ′ ( t ) ) = 0 , $$\big({\it \Phi}(k(t)\,x'(t))\big)' + f(t,{{\mathcal{G}}}_x(t))\,\rho(t, x'(t)) = 0,$$ on a compact interval [ a , b ]. These equations are quite general due to the presence of a strictly increasing homeomorphism Φ , the so-called Φ -Laplace operator, of a non-negative function k , which may vanish on a set of null measure, and moreover of a functional term G x . We look for solutions, in a suitable weak sense, which belong to the Sobolev space W 1,1 ([ a , b ]). Under the assumptions of the existence of a well-ordered pair of upper and lower solutions and of a suitable Nagumo-type growth condition, we prove an existence result by means of fixed point arguments.

Global solvability in a three-dimensional Keller-Segel-Stokes system involving arbitrary superlinear logistic degradation

Yulan Wang, Michael Winkler, Zhaoyin Xiang December 1, 2020 Page range: 707-731

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The Keller-Segel-Stokes system (*) n t + u ⋅ ∇ n = Δ n − ∇ ⋅ ( n ∇ c ) + ρ n − μ n α , c t + u ⋅ ∇ c = Δ c − c + n , u t = Δ u + ∇ P − n ∇ Λ , ∇ ⋅ u = 0 , $$\begin{eqnarray*} \left\{ \begin{array}{lcll} n_t + u\cdot\nabla n &=& \it\Delta n - \nabla \cdot (n\nabla c) + \rho n - \mu n^\alpha, \\[1mm] c_t + u\cdot\nabla c &=& \it\Delta c-c+n, \\[1mm] u_t &=& \it\Delta u + \nabla P - n\nabla \it\Lambda, \qquad \nabla\cdot u =0, \end{array} \right. \end{eqnarray*}$$ is considered in a bounded domain Ω ⊂ ℝ 3 with smooth boundary, with parameters ρ ≥ 0, μ > 0 and α > 1, and with a given gravitational potential Λ ∈ W 2,∞ ( Ω ). It is shown that in this general setting, when posed under no-flux boundary conditions for n and c and homogeneous Dirichlet boundary conditions for u , and for any suitably regular initial data, an associated initial value problem possesses at least one globally defined solution in an appropriate generalized sense. Since it is well-known that in the absence of absorption, already the corresponding fluid-free subsystem with u ≡ 0 and μ = 0 admits some solutions blowing up in finite time, this particularly indicates that any power-type superlinear degradation of the form in (*) goes along with some significant regularizing effect.

Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth

Zhipeng Yang, Fukun Zhao December 1, 2020 Page range: 732-774

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In this paper, we study the singularly perturbed fractional Choquard equation ε 2s(− Δ )su+V(x)u=ε μ − 3(∫R3|u(y)|2μ ,s∗ +F(u(y))|x− y|μ dy)(|u|2μ ,s∗ − 2u+12μ ,s∗ f(u))inR3, $$\begin{equation*}\varepsilon^{2s}(-{\it\Delta})^su+V(x)u=\varepsilon^{\mu-3}(\int\limits_{\mathbb{R}^3}\frac{|u(y)|^{2^*_{\mu,s}}+F(u(y))}{|x-y|^\mu}dy)(|u|^{2^*_{\mu,s}-2}u+\frac{1}{2^*_{\mu,s}}f(u)) \, \text{in}\, \mathbb{R}^3, \end{equation*}$$ where ε > 0 is a small parameter, (− △ ) s denotes the fractional Laplacian of order s ∈ (0, 1), 0 < μ < 3, 2μ,s⋆=6−μ3−2s $2_{\mu ,s}^{\star }=\frac{6-\mu }{3-2s}$is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and fractional Laplace operator. F is the primitive of f which is a continuous subcritical term. Under a local condition imposed on the potential V , we investigate the relation between the number of positive solutions and the topology of the set where the potential attains its minimum values. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory.

Concentration results for a magnetic Schrödinger-Poisson system with critical growth

Jingjing Liu, Chao Ji November 27, 2020 Page range: 775-798

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This paper is concerned with the following nonlinear magnetic Schrödinger-Poisson type equation (ϵ i∇ − A(x))2u+V(x)u+ϵ − 2(|x|− 1∗ |u|2)u=f(|u|2)u+|u|4uin R3,u∈ H1(R3,C), $$\begin{array}{} \displaystyle \left\{ \begin{aligned} &\Big(\frac{\epsilon}{i}\nabla-A(x)\Big)^{2}u+V(x)u+\epsilon^{-2}(\vert x\vert^{-1}\ast \vert u\vert^{2})u=f(|u|^{2})u+\vert u\vert^{4}u \quad \hbox{in }\mathbb{R}^3,\\ &u\in H^{1}(\mathbb{R}^{3}, \mathbb{C}), \end{aligned} \right. \end{array}$$ where ϵ > 0, V : ℝ 3 → ℝ and A : ℝ 3 → ℝ 3 are continuous potentials, f : ℝ → ℝ is a subcritical nonlinear term and is only continuous. Under a local assumption on the potential V , we use variational methods, penalization technique and Ljusternick-Schnirelmann theory to prove multiplicity and concentration of nontrivial solutions for ϵ > 0 small.

Periodic solutions for a differential inclusion problem involving the p(t)-Laplacian

Peng Chen, Xianhua Tang December 19, 2020 Page range: 799-815

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In the present paper, we consider the nonlinear periodic systems involving variable exponent driven by p ( t )-Laplacian with a locally Lipschitz nonlinearity. Our arguments combine the variational principle for locally Lipschitz functions with the properties of the generalized Lebesgue-Sobolev space. Applying the non-smooth critical point theory, we establish some new existence results.

The concentration-compactness principles for W^s,p(·,·)(ℝ^N) and application

Ky Ho, Yun-Ho Kim December 8, 2020 Page range: 816-848

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We obtain a critical imbedding and then, concentration-compactness principles for fractional Sobolev spaces with variable exponents. As an application of these results, we obtain the existence of many solutions for a class of critical nonlocal problems with variable exponents, which is even new for constant exponent case.

Regularity for commutators of the local multilinear fractional maximal operators

Xiao Zhang, Feng Liu December 8, 2020 Page range: 849-876

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In this paper we introduce and study the commutators of the local multilinear fractional maximal operators and a vector-valued function b⃗ = ( b 1 , …, b m ). Under the condition that each b i belongs to the first order Sobolev spaces, the bounds for the above commutators are established on the first order Sobolev spaces.

An improvement to the John-Nirenberg inequality for functions in critical Sobolev spaces

Ángel D. Martínez, Daniel Spector December 20, 2020 Page range: 877-894

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It is known that functions in a Sobolev space with critical exponent embed into the space of functions of bounded mean oscillation, and therefore satisfy the John-Nirenberg inequality and a corresponding exponential integrability estimate. While these inequalities are optimal for general functions of bounded mean oscillation, the main result of this paper is an improvement for functions in a class of critical Sobolev spaces. Precisely, we prove the inequality H∞β({x∈Ω:|Iαf(x)|>t})≤Ce−ctq′ $$\mathcal{H}^{\beta}_{\infty}(\{x\in \Omega:|I_\alpha f(x)|>t\})\leq Ce^{-ct^{q'}}$$ for all ∥f∥LN/α,q(Ω)≤1 $\|f\|_{L^{N/\alpha,q}(\Omega)}\leq 1$ and any β∈(0,N],whereΩ⊂RN,H∞β $\beta \in (0,N], \; {\text{where}} \; \Omega \subset \mathbb{R}^N, \mathcal{H}^{\beta}_{\infty}$ is the Hausdorff content, L N / α , q ( Ω ) is a Lorentz space with q ∈ (1,∞], q' = q /( q − 1) is the Hölder conjugate to q , and I α f denotes the Riesz potential of f of order α ∈ (0, N ).

Local versus nonlocal elliptic equations: short-long range field interactions

Daniele Cassani, Luca Vilasi, Youjun Wang December 31, 2020 Page range: 895-921

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In this paper we study a class of one-parameter family of elliptic equations which combines local and nonlocal operators, namely the Laplacian and the fractional Laplacian. We analyze spectral properties, establish the validity of the maximum principle, prove existence, nonexistence, symmetry and regularity results for weak solutions. The asymptotic behavior of weak solutions as the coupling parameter vanishes (which turns the problem into a purely nonlocal one) or goes to infinity (reducing the problem to the classical semilinear Laplace equation) is also investigated.

Analysis of a diffusive host-pathogen model with standard incidence and distinct dispersal rates

Jinliang Wang, Renhao Cui December 31, 2020 Page range: 922-951

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This paper concerns with detailed analysis of a reaction-diffusion host-pathogen model with space-dependent parameters in a bounded domain. By considering the fact the mobility of host individuals playing a crucial role in disease transmission, we formulate the model by a system of degenerate reaction-diffusion equations, where host individuals disperse at distinct rates and the mobility of pathogen is ignored in the environment.We first establish the well-posedness of the model, including the global existence of solution and the existence of the global compact attractor. The basic reproduction number is identified, and also characterized by some equivalent principal spectral conditions, which establishes the threshold dynamical result for pathogen extinction and persistence. When the positive steady state is confirmed, we investigate the asymptotic profiles of positive steady state as host individuals disperse at small and large rates. Our result suggests that small and large diffusion rate of hosts have a great impacts in formulating the spatial distribution of the pathogen.

Blowing-up solutions of the time-fractional dispersive equations

Ahmed Alsaedi, Bashir Ahmad, Mokhtar Kirane, Berikbol T. Torebek January 12, 2021 Page range: 952-971

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This paper is devoted to the study of initial-boundary value problems for time-fractional analogues of Korteweg-de Vries, Benjamin-Bona-Mahony, Burgers, Rosenau, Camassa-Holm, Degasperis-Procesi, Ostrovsky and time-fractional modified Korteweg-de Vries-Burgers equations on a bounded domain. Sufficient conditions for the blowing-up of solutions in finite time of aforementioned equations are presented. We also discuss the maximum principle and influence of gradient non-linearity on the global solvability of initial-boundary value problems for the time-fractional Burgers equation. The main tool of our study is the Pohozhaev nonlinear capacity method. We also provide some illustrative examples.

Fixed point of some Markov operator of Frobenius-Perron type generated by a random family of point-transformations in ℝ^d

Peter Bugiel, Stanisław Wędrychowicz, Beata Rzepka January 16, 2021 Page range: 972-981

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Existence of fixed point of a Frobenius-Perron type operator P : L 1 ⟶ L 1 generated by a family { φ y } y ∈ Y of nonsingular Markov maps defined on a σ -finite measure space ( I , Σ , m ) is studied. Two fairly general conditions are established and it is proved that they imply for any g ∈ G = { f ∈ L 1 : f ≥ 0, and ∥ f ∥ = 1}, the convergence (in the norm of L 1 ) of the sequence {Pjg}j=1∞ $\begin{array}{} \{P^{j}g\}_{j = 1}^{\infty} \end{array} $ to a unique fixed point g 0 . The general result is applied to a family of C 1+ α -smooth Markov maps in ℝ d .

Non-stationary Navier–Stokes equations in 2D power cusp domain

Konstantin Pileckas, Alicija Raciene February 28, 2021 Page range: 982-1010

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The initial boundary value problem for the non-stationary Navier-Stokes equations is studied in 2D bounded domain with a power cusp singular point O on the boundary. The case of the boundary value with a nonzero flow rate is considered. In this case there is a source/sink in O and the solution necessary has infinite energy integral. In the first part of the paper the formal asymptotic expansion of the solution near the singular point is constructed. The justification of the asymptotic expansion and the existence of a solution are proved in the second part of the paper.

Non-stationary Navier–Stokes equations in 2D power cusp domain

Konstantin Pileckas, Alicija Raciene February 28, 2021 Page range: 1011-1038

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The initial boundary value problem for the non-stationary Navier-Stokes equations is studied in 2D bounded domain with a power cusp singular point O on the boundary. We consider the case where the boundary value has a nonzero flux over the boundary. In this case there is a source/sink in O and the solution necessary has infinite energy integral. In the first part of the paper the formal asymptotic expansion of the solution near the singular point was constructed. In this, second part, the constructed asymptotic decomposition is justified, i.e., existence of the solution which is represented as the sum of the constructed asymptotic expansion and a term with finite energy norm is proved. Moreover, it is proved that the solution represented in this form is unique.

Nontrivial solutions to the p-harmonic equation with nonlinearity asymptotic to |t|^p–2t at infinity

Qihan He, Juntao Lv, Zongyan Lv March 5, 2021 Page range: 1039-1060

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We consider the following p -harmonic problem Δ (|Δ u|p− 2Δ u)+m|u|p− 2u=f(x,u),x∈ RN,u∈ W2,p(RN), $$\begin{array}{} \displaystyle \left\{ \displaystyle\begin{array}{ll} \displaystyle {\it\Delta} (|{\it\Delta} u|^{p-2}{\it\Delta} u)+m|u|^{p-2}u=f(x,u), \ \ x\in {\mathbb R}^N, \\ u \in W^{2,p}({\mathbb R}^N), \end{array} \right. \end{array}$$ where m > 0 is a constant, N > 2 p ≥ 4 and limt→ ∞ f(x,t)|t|p− 2t=l $\begin{array}{} \displaystyle \lim\limits_{t\rightarrow \infty}\frac{f(x,t)}{|t|^{p-2}t}=l \end{array}$ uniformly in x , which implies that f ( x , t ) does not satisfy the Ambrosetti-Rabinowitz type condition. By showing the Pohozaev identity for weak solutions to the limited problem of the above p-harmonic equation and using a variant version of Mountain Pass Theorem, we prove the existence and nonexistence of nontrivial solutions to the above equation. Moreover, if f ( x , u ) ≡ f ( u ), the existence of a ground state solution and the nonexistence of nontrivial solutions to the above problem is also proved by using artificial constraint method and the Pohozaev identity.

Iterative methods for monotone nonexpansive mappings in uniformly convex spaces

Rahul Shukla, Andrzej Wiśnicki March 5, 2021 Page range: 1061-1070

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We show the nonlinear ergodic theorem for monotone 1-Lipschitz mappings in uniformly convex spaces: if C is a bounded closed convex subset of an ordered uniformly convex space ( X , ∣·∣, ⪯), T : C → C a monotone 1-Lipschitz mapping and x ⪯ T ( x ), then the sequence of averages 1n∑i=0n−1Ti(x) $ \frac{1}{n}\sum\nolimits_{i=0}^{n-1}T^{i}(x) $ converges weakly to a fixed point of T . As a consequence, it is shown that the sequence of Picard’s iteration { T n ( x )} also converges weakly to a fixed point of T . The results are new even in a Hilbert space. The Krasnosel’skiĭ-Mann and the Halpern iteration schemes are studied as well.

Optimality of Serrin type extension criteria to the Navier-Stokes equations

Reinhard Farwig, Ryo Kanamaru March 5, 2021 Page range: 1071-1085

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We prove that a strong solution u to the Navier-Stokes equations on (0, T ) can be extended if either u ∈ L θ (0, T ; U˙ ∞ ,1/θ ,∞ − α $\begin{array}{} \displaystyle \dot{U}^{-\alpha}_{\infty,1/\theta,\infty} \end{array}$) for 2/ θ + α = 1, 0 < α < 1 or u ∈ L 2 (0, T ; V˙ ∞ ,∞ ,20 $\begin{array}{} \displaystyle \dot{V}^{0}_{\infty,\infty,2} \end{array}$), where U˙ p,β ,σ s $\begin{array}{} \displaystyle \dot{U}^{s}_{p,\beta,\sigma} \end{array}$ and V˙ p,q,θ s $\begin{array}{} \displaystyle \dot{V}^{s}_{p,q,\theta} \end{array}$ are Banach spaces that may be larger than the homogeneous Besov space B˙ p,qs $\begin{array}{} \displaystyle \dot{B}^{s}_{p,q} \end{array}$. Our method is based on a bilinear estimate and a logarithmic interpolation inequality.

Fractional Hardy-Sobolev equations with nonhomogeneous terms

Mousomi Bhakta, Souptik Chakraborty, Patrizia Pucci March 25, 2021 Page range: 1086-1116

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This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity: (− Δ )su− γ u|x|2s=K(x)|u|2s∗ (t)− 2u|x|t+f(x)inRN,u∈ H˙ s(RN), $$\begin{array}{} \displaystyle \begin{cases} (-{\it\Delta})^s u -\gamma\dfrac{u}{|x|^{2s}}=K(x)\dfrac{|u|^{2^*_s(t)-2}u}{|x|^t}+f(x) \quad\mbox{in}\quad\mathbb R^N,\\ \qquad\qquad\qquad\quad u\in \dot{H}^s(\mathbb R^N), \end{cases} \end{array}$$ where N > 2 s , s ∈ (0, 1), 0 ≤ t < 2 s < N and 2s∗ (t):=2(N− t)N− 2s $\begin{array}{} \displaystyle 2^*_s(t):=\frac{2(N-t)}{N-2s} \end{array}$. Here 0 < γ < γ N , s and γ N , s is the best Hardy constant in the fractional Hardy inequality. The coefficient K is a positive continuous function on ℝ N , with K (0) = 1 = lim | x |→∞ K ( x ). The perturbation f is a nonnegative nontrivial functional in the dual space Ḣ s (ℝ N )′ of Ḣ s (ℝ N ). We establish the profile decomposition of the Palais-Smale sequence associated with the functional. Further, if K ≥ 1 and ∥ f ∥ ( Ḣ s )′ is small enough (but f ≢ 0), we establish existence of at least two positive solutions to the above equation.

New class of sixth-order nonhomogeneous p(x)-Kirchhoff problems with sign-changing weight functions

Mohamed Karim Hamdani, Nguyen Thanh Chung, Dušan D. Repovš March 24, 2021 Page range: 1117-1131

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In this paper, we prove the existence of multiple solutions for the following sixth-order p ( x )-Kirchhoff-type problem − M∫ Ω 1p(x)|∇ Δ u|p(x)dxΔ p(x)3u=λ f(x)|u|q(x)− 2u+g(x)|u|r(x)− 2u+h(x)inΩ ,u=Δ u=Δ 2u=0,on∂ Ω , $$\begin{array}{} \displaystyle \begin{cases} -M\left( \int\limits_{\it\Omega} \frac{1}{p(x)}|\nabla {\it\Delta} u|^{p(x)}dx\right){\it\Delta}^3_{p(x)} u = \lambda f(x)|u|^{q(x)-2}u + g(x)|u|^{r(x)-2}u + h(x) &\mbox{in}\quad {\it\Omega}, \\[0.3em] u = {\it\Delta} u = {\it\Delta}^2 u = 0, \quad &\mbox{on}\quad \partial{\it\Omega}, \end{cases} \end{array}$$ where Ω ⊂ ℝ N is a smooth bounded domain, N> 3,Δ p(x)3u:=div⁡ (Δ (|∇ Δ u|p(x)− 2∇ Δ u)) $\begin{array}{} N \,\,\gt\,\, 3, {\it\Delta}_{p(x)}^3u\,\, : =\,\, \operatorname{div}\Big({\it\Delta}(|\nabla {\it\Delta} u|^{p(x)-2}\nabla {\it\Delta} u)\Big) \end{array}$ is the p ( x )-triharmonic operator, p , q , r ∈ C ( Ω ), 1 < p ( x ) < N3 $\begin{array}{} \displaystyle \frac N3 \end{array}$ for all x ∈ Ω , M ( s ) = a − bs γ , a , b , γ > 0, λ > 0, g : Ω × ℝ → ℝ is a nonnegative continuous function while f , h : Ω × ℝ → ℝ are sign-changing continuous functions in Ω . To the best of our knowledge, this paper is one of the first contributions to the study of the sixth-order p ( x )-Kirchhoff type problems with sign changing Kirchhoff functions.

On the set of positive solutions for resonant Robin (p, q)-equations

Nikolaos S. Papageorgiou, Youpei Zhang March 25, 2021 Page range: 1132-1153

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We consider a nonlinear Robin problem driven by the ( p , q )-Laplacian and a parametric reaction exhibiting the competition of a concave term and of a resonant perturbation. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter λ moves on ℝ̊ + = (0, +∞). Also, we determine the continuity properties of the solution multifunction.

Solving Composite Fixed Point Problems with Block Updates

Patrick L. Combettes, Lilian E. Glaudin March 25, 2021 Page range: 1154-1177

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Various strategies are available to construct iteratively a common fixed point of nonexpansive operators by activating only a block of operators at each iteration. In the more challenging class of composite fixed point problems involving operators that do not share common fixed points, current methods require the activation of all the operators at each iteration, and the question of maintaining convergence while updating only blocks of operators is open. We propose a method that achieves this goal and analyze its asymptotic behavior. Weak, strong, and linear convergence results are established by exploiting a connection with the theory of concentrating arrays. Applications to several nonlinear and nonsmooth analysis problems are presented, ranging from monotone inclusions and inconsistent feasibility problems, to variational inequalities and minimization problems arising in data science.

Lions-type theorem of the p-Laplacian and applications

Yu Su, Zhaosheng Feng March 25, 2021 Page range: 1178-1200

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Abstract

In this article, our aim is to establish a generalized version of Lions-type theorem for the p -Laplacian. As an application of this theorem, we consider the existence of ground state solution for the quasilinear elliptic equation with the critical growth.

Half-space Gaussian symmetrization: applications to semilinear elliptic problems

J. I. Díaz, F. Feo, M. R. Posteraro April 14, 2021 Page range: 1201-1221

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Abstract

We consider a class of semilinear equations with an absorption nonlinear zero order term of power type, where elliptic condition is given in terms of Gauss measure. In the case of the superlinear equation we introduce a suitable definition of solutions in order to prove the existence and uniqueness of a solution in ℝ N without growth restrictions at infinity. A comparison result in terms of the half-space Gaussian symmetrized problem is also proved. As an application, we give some estimates in measure of the growth of the solution near the boundary of its support for sublinear equations. Finally we generalize our results to problems with a nonlinear zero order term not necessary of power type.

Positive radial symmetric solutions for a class of elliptic problems with critical exponent and -1 growth

Chun-Yu Lei, Jia-Feng Liao April 14, 2021 Page range: 1222-1234

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Abstract

In this paper, we consider a class of semilinear elliptic equation with critical exponent and -1 growth. By using the critical point theory for nonsmooth functionals, two positive solutions are obtained. Moreover, the symmetry and monotonicity properties of the solutions are proved by the moving plane method. Our results improve the corresponding results in the literature.

Global well-posedness of the full compressible Hall-MHD equations

Qiang Tao, Canze Zhu April 24, 2021 Page range: 1235-1254

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Abstract

This paper deals with a Cauchy problem of the full compressible Hall-magnetohydrodynamic flows. We establish the existence and uniqueness of global solution, provided that the initial energy is suitably small but the initial temperature allows large oscillations. In addition, the large time behavior of the global solution is obtained.

Σ-Shaped Bifurcation Curves

A. Acharya, N. Fonseka, J. Quiroa, R. Shivaji May 4, 2021 Page range: 1255-1266

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Abstract

We study positive solutions to the steady state reaction diffusion equation of the form: − Δ u=λ f(u); Ω ∂ u∂ η +λ u=0; ∂ Ω $$\begin{array}{} \displaystyle \left\lbrace \begin{matrix} -{\it\Delta} u =\lambda f(u);~ {\it\Omega} \\ \frac{\partial u}{\partial \eta}+ \sqrt{\lambda} u=0;~\partial {\it\Omega}\end{matrix} \right. \end{array}$$ where λ > 0 is a positive parameter, Ω is a bounded domain in ℝ N when N > 1 (with smooth boundary ∂ Ω ) or Ω = (0, 1), and ∂ u∂ η $\begin{array}{} \displaystyle \frac{\partial u}{\partial \eta} \end{array}$ is the outward normal derivative of u . Here f ( s ) = ms + g ( s ) where m ≥ 0 (constant) and g ∈ C 2 [0, r ) ∩ C [0, ∞) for some r > 0. Further, we assume that g is increasing, sublinear at infinity, g (0) = 0, g ′(0) = 1 and g ″(0) > 0. In particular, we discuss the existence of multiple positive solutions for certain ranges of λ leading to the occurrence of Σ -shaped bifurcation diagrams. We establish our multiplicity results via the method of sub-supersolutions.

On the critical behavior for inhomogeneous wave inequalities with Hardy potential in an exterior domain

Mohamed Jleli, Bessem Samet, Calogero Vetro May 4, 2021 Page range: 1267-1283

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Abstract

We study the wave inequality with a Hardy potential ∂ ttu− Δ u+λ |x|2u≥ |u|pin (0,∞ )× Ω , $$\begin{array}{} \displaystyle \partial_{tt}u-{\it\Delta} u+\frac{\lambda}{|x|^2}u\geq |u|^p\quad \mbox{in } (0,\infty)\times {\it\Omega}, \end{array}$$ where Ω is the exterior of the unit ball in ℝ N , N ≥ 2, p > 1, and λ ≥ − N− 222 $\begin{array}{} \displaystyle \left(\frac{N-2}{2}\right)^2 \end{array}$, under the inhomogeneous boundary condition α ∂ u∂ ν (t,x)+β u(t,x)≥ w(x)on (0,∞ )× ∂ Ω , $$\begin{array}{} \displaystyle \alpha \frac{\partial u}{\partial \nu}(t,x)+\beta u(t,x)\geq w(x)\quad\mbox{on } (0,\infty)\times \partial{\it\Omega}, \end{array}$$ where α , β ≥ 0 and ( α , β ) ≠ (0, 0). Namely, we show that there exists a critical exponent p c ( N , λ ) ∈ (1, ∞] for which, if 1 < p < p c ( N , λ ), the above problem admits no global weak solution for any w ∈ L 1 ( ∂ Ω ) with ∫ ∂Ω w ( x ) dσ > 0, while if p > p c ( N , λ ), the problem admits global solutions for some w > 0. To the best of our knowledge, the study of the critical behavior for wave inequalities with a Hardy potential in an exterior domain was not considered in previous works. Some open questions are also mentioned in this paper.

On singular quasilinear elliptic equations with data measures

Nour Eddine Alaa, Fatima Aqel, Laila Taourirte May 16, 2021 Page range: 1284-1300

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Abstract

The aim of this work is to study a quasilinear elliptic equation with singular nonlinearity and data measure. Existence and non-existence results are obtained under necessary or sufficient conditions on the data, where the main ingredient is the isoperimetric inequality. Finally, uniqueness results for weak solutions are given.

On the sub–diffusion fractional initial value problem with time variable order

Eduardo Cuesta, Mokhtar Kirane, Ahmed Alsaedi, Bashir Ahmad June 4, 2021 Page range: 1301-1315

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Abstract

We consider a fractional derivative with order varying in time. Then, we derive for it a Leibniz' inequality and an integration by parts formula. We also study an initial value problem with our time variable order fractional derivative and present a regularity result for it, and a study on the asymptotic behavior.

Partial regularity of stable solutions to the fractional Geľfand-Liouville equation

Ali Hyder, Wen Yang May 22, 2021 Page range: 1316-1327

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Abstract

We analyze stable weak solutions to the fractional Geľfand problem (− Δ )su=euinΩ ⊂ Rn. $$\begin{array}{} \displaystyle (-{\it\Delta})^su = e^u\quad\mathrm{in}\quad {\it\Omega}\subset\mathbb{R}^n. \end{array}$$ We prove that the dimension of the singular set is at most n − 10 s .

Ground state solutions for a class of fractional Schrodinger-Poisson system with critical growth and vanishing potentials

Yuxi Meng, Xinrui Zhang, Xiaoming He June 10, 2021 Page range: 1328-1355

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Abstract

In this paper, we study the fractional Schrödinger-Poisson system (−Δ)su+V(x)u+K(x)ϕ|u|q−2u=h(x)f(u)+|u|2s∗−2u,in R3,(−Δ)tϕ=K(x)|u|q,in R3, $$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} (-{\it\Delta})^{s}u+V(x)u+ K(x) \phi|u|^{q-2}u=h(x)f(u)+|u|^{2^{\ast}_{s}-2}u,&\mbox{in}~ {\mathbb R^{3}},\\ (-{\it\Delta})^{t}\phi=K(x)|u|^{q},&\mbox{in}~ {\mathbb R^{3}}, \end{array}\right. \end{array}$$ where s , t ∈ (0, 1), 3 < 4 s < 3 + 2 t , q ∈ (1, 2s∗ $\begin{array}{} \displaystyle 2^*_s \end{array}$/2) are real numbers, (− Δ ) s stands for the fractional Laplacian operator, 2s∗ :=63− 2s $\begin{array}{} \displaystyle 2^{*}_{s}:=\frac{6}{3-2s} \end{array}$ is the fractional critical Sobolev exponent, K , V and h are non-negative potentials and V , h may be vanish at infinity. f is a C 1 -function satisfying suitable growth assumptions. We show that the above fractional Schrödinger-Poisson system has a positive and a sign-changing least energy solution via variational methods.

Initial boundary value problems for the three-dimensional compressible elastic Navier-Stokes-Poisson equations

Yong Wang, Wenpei Wu June 4, 2021 Page range: 1356-1383

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Abstract

We study the initial-boundary value problems of the three-dimensional compressible elastic Navier-Stokes-Poisson equations under the Dirichlet or Neumann boundary condition for the electrostatic potential. The unique global solution near a constant equilibrium state in H 2 space is obtained. Moreover, we prove that the solution decays to the equilibrium state at an exponential rate as time tends to infinity. This is the first result for the three-dimensional elastic Navier-Stokes-Poisson equations under various boundary conditions for the electrostatic potential.

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5-year Impact Factor	2.567
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Vicentiu Radulescu
Mathematics Institute "Simion Stoilow" of the Romanian Academy, Bucharest / Department of Mathematics, University of Craiova, Romania
vicentiu.radulescu@math.cnrs.fr
Runzhang Xu
College of Mathematical Sciences, Harbin Engineering University, China
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Viorel Barbu, Institute of Mathematics "Octav Mayer", Romania
Haim Brezis, Department of Mathematics, Rutgers University, United States / Department of Mathematics, Technion, Israel
Philippe G. Ciarlet, Department of Mathematics, City University of Hong Kong, Hong Kong
Yoshikazu Giga, University of Tokyo, Japan
Jean Mawhin, Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, Belgium
Tatsien Li, Fudan University, Shanghai, China
Dušan Repovš, Faculty of Mathematics and Physics, University of Ljubljana, Slovenia
Roger Temam, The Institute for Scientific Computing & Applied Mathematics, Indiana University, United States
Michel Willem, Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, Belgium

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Nicola Bellomo, Polytechnic University of Torino, Italy
Luigi Carlo Berselli, Dipartimento di Matematica, Universit degli Studi di Pisa, Italy
José A. Carrillo, Oxford University, United Kingdom
Gui-Qiang G. Chen, University of Oxford, UK
Manuel del Pino, Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Chile
Jesus-Ildefonso Diaz, Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, Spain
Frank Duzaar, Department Mathematics, University of Erlangen-Nuremberg, Germany
Joachim Escher, University of Hannover, Germany
Olivier Goubet, Université de Picardie Jules Verne, France
Xianpeng Hu, City University of Hong Kong, Hong Kong, China
Louis Jeanjean, Laboratoire de Mathématiques, Université de Franche-Comté, France
Irena Lasiecka, Department of Mathematical Sciences, The University of Memphis
Giuseppe Mingione, Dipartimento di Matematica, Università degli Studi di Parma, Italy
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Wei-Ming Ni, University of Minnesota, USA & The Chinese University of Hong Kong, Shenzhen, China
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Panagiotis Souganidis, The University of Chicago, USA
Takis Souganidis, Department of Mathematics, University of Chicago, USA
Catherine Sulem, University of Toronto, Canada
Xianhua Tang, School of Mathematics and Statistics, Central South University, China
Dehua Wang, University of Pittsburgh, USA
Juncheng Wei, University of British Columbia, Canada
Michael Winkler, Universität Paderborn, Germany
Chao Zhang, School of Mathematics, Harbin Institute of Technology, China
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