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Finite Difference Methods

## Abstract

In this paper, we study the Cauchy problem of the parabolic Monge–Ampère equation

$-ut⁢det⁡D2⁢u=f⁢(x,t)$

and obtain the existence and uniqueness of viscosity solutions with asymptotic behavior by using the Perron method.

## Abstract

In this paper we prove an existence result of multiple positive solutions for the following quasilinear problem:

where Ω is a smooth and bounded domain in $ℝN,N≥3$. More specifically we prove that, for p near the critical exponent $22*=4⁢N/(N-2)$, the number of positive solutions is estimated below by topological invariants of the domain Ω: the Ljusternick–Schnirelmann category and the Poincaré polynomial. With respect to the case involving semilinear equations, many difficulties appear here and the classical procedure does not apply immediately. We obtain also en passant some new results concerning the critical case.

## Abstract

We discuss, by topological methods, the solvability of systems of second-order elliptic differential equations subject to functional boundary conditions under the presence of gradient terms in the nonlinearities. We prove the existence of nonnegative solutions and provide a non-existence result. We present some examples to illustrate the applicability of the existence and non-existence results.

## Abstract

We study the self-similar blow-up profiles associated to the following second-order reaction-diffusion equation with strong weighted reaction and unbounded weight:

$∂t⁡u=∂x⁢x⁡(um)+|x|σ⁢up,$

posed for $x∈ℝ$, $t≥0$, where $m>1$, $0 and $σ>2⁢(1-p)m-1$. As a first outcome, we show that finite time blow-up solutions in self-similar form exist for $m+p>2$ and σ in the considered range, a fact that is completely new: in the already studied reaction-diffusion equation without weights there is no finite time blow-up when $p<1$. We moreover prove that, if the condition $m+p>2$ is fulfilled, all the self-similar blow-up profiles are compactly supported and there exist two different interface behaviors for solutions of the equation, corresponding to two different interface equations. We classify the self-similar blow-up profiles having both types of interfaces and show that in some cases global blow-up occurs, and in some other cases finite time blow-up occurs only at space infinity. We also show that there is no self-similar solution if $m+p<2$, while the critical range $m+p=2$ with $σ>2$ is postponed to a different work due to significant technical differences.

## Abstract

We study the existence of sign-changing solutions for a non-local version of the sinh-Poisson equation on a bounded one-dimensional interval I, under Dirichlet conditions in the exterior of I. This model is strictly related to the mathematical description of galvanic corrosion phenomena for simple electrochemical systems. By means of the finite-dimensional Lyapunov–Schmidt reduction method, we construct bubbling families of solutions developing an arbitrarily prescribed number sign-alternating peaks. With a careful analysis of the limit profile of the solutions, we also show that the number of nodal regions coincides with the number of blow-up points.

## Abstract

In this paper we establish a new critical point theorem for a class of perturbed differentiable functionals without satisfying the Palais–Smale condition. We prove the existence of at least one critical point to such functionals, provided that the perturbation is sufficiently small. The main abstract result of this paper is applied both to perturbed nonhomogeneous equations in Orlicz–Sobolev spaces and to nonlocal problems in fractional Sobolev spaces.

## Abstract

By means of a suitable degree theory, we prove persistence of eigenvalues and eigenvectors for set-valued perturbations of a Fredholm linear operator. As a consequence, we prove existence of a bifurcation point for a non-linear inclusion problem in abstract Banach spaces. Finally, we provide applications to differential inclusions.

## Abstract

In this paper, we investigate the existence of nontrivial solutions to the following fractional p-Laplacian system with homogeneous nonlinearities of critical Sobolev growth:

where $(-Δp)s$ denotes the fractional p-Laplacian operator, $p>1$, $s∈(0,1)$, $p⁢s, $ps*=N⁢pN-p⁢s$ is the critical Sobolev exponent, Ω is a bounded domain in $ℝN$ with Lipschitz boundary, and Q and H are homogeneous functions of degrees p and q with $p and $Qu$ and $Qv$ are the partial derivatives with respect to u and v, respectively. To establish our existence result, we need to prove a concentration-compactness principle associated with the fractional p-Laplacian system for the fractional order Sobolev spaces in bounded domains which is significantly more difficult to prove than in the case of single fractional p-Laplacian equation and is of its independent interest (see Lemma ). Our existence results can be regarded as an extension and improvement of those corresponding ones both for the nonlinear system of classical p-Laplacian operators (i.e., $s=1$) and for the single fractional p-Laplacian operator in the literature. Even a special case of our main results on systems of fractional Laplacian $(-Δ)s$ (i.e., $p=2$ and $0) has not been studied in the literature before.