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Comments on behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities

Mariusz Bodzioch
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• Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, ul. Sloneczna 54, 10-710 Olsztyn, Poland
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/ Damian Wiśniewski
• Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, ul. Sloneczna 54, 10-710 Olsztyn, Poland
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• Other articles by this author:
• De Gruyter OnlineGoogle Scholar
/ Krzysztof Żyjewski
• Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, ul. Sloneczna 54, 10-710 Olsztyn, Poland
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• Other articles by this author:
• De Gruyter OnlineGoogle Scholar
Published Online: 2017-05-23 | DOI: https://doi.org/10.1515/math-2017-0033

Abstract

We have investigated the behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities in bounded and unbounded domains. We found exponents of the solution’s decreasing rate near the boundary singularities.

MSC 2010: 35J20; 35J25; 35J60; 35J70

1 Introduction

This is a brief description of results of [13] with comments and improvements, which were presented at the International conference on differential equations dedicated to the 110th anniversary of Ya. B. Lopatynsky. In these articles we have investigated the behaviour of solutions of quasi-linear elliptic problems in a neighborhood of boundary singularities in bounded and unbounded domains. We have found exponents of the solution’s decreasing rate of the type |u(x)| ≤ O(|x|α), near the boundary singularities.

Let G ⊂ ℝn be an unbounded domain (see Fig. 1) with boundary ∂G that is a smooth surface everywhere except at the origin 𝓞 and near 𝓞 it is a conical surface, n ≥ 2. We assume that $\begin{array}{}G={G}_{0}^{R}\cup {G}_{R},\text{\hspace{0.17em}where\hspace{0.17em}}{G}_{0}^{R}={G}_{0}^{d}\cup {G}_{d}^{R}\text{\hspace{0.17em}and\hspace{0.17em}}{G}_{0}^{d}\end{array}$ is a rotational cone with the vertex at 𝓞 and the aperture ω0 ∈ (0,π), d ≪ 1, GR = {x = (r, ω) ∈ ℝn|r ∈ (R, ∞), ω ∈ Ω ⊂ Sn − 1, n ≥ 2}, R ≫ 1, Sn − 1 is the unit sphere.

Fig. 1

an unbounded cone-like domain.

We introduce the following notations for a domain G which has a boundary conical point:

• Ω : = GSn − 1, where Sn − 1 denotes the unit sphere in ℝn;

• Ω: the boundary of Ω;

• $\begin{array}{}{G}_{a}^{b}\end{array}$ := G ∩{(r, ω) : 0 ≤ a < r < b, ω ∈ Ω}: a layer in ℝn;

• $\begin{array}{}{\mathrm{\Gamma }}_{a}^{b}\end{array}$ := ∂G ∩{(r, ω) : 0 ≤ a < r < b, ωΩ}: the lateral surface of $\begin{array}{}{G}_{a}^{b}\end{array}$;

• $\begin{array}{}{G}_{d}:=G\mathrm{\setminus }{G}_{0}^{d},\phantom{\rule{1em}{0ex}}{\mathrm{\Gamma }}_{d}:=\mathrm{\partial }G\mathrm{\setminus }{\mathrm{\Gamma }}_{0}^{d}.\end{array}$

We use standard function spaces: $\begin{array}{}{C}^{k}\left(\overline{G}\right)\end{array}$ with the norm $\begin{array}{}|u{|}_{k,\overline{G}};{L}_{p}\left(G\right)\end{array}$ with the norm ∥up,G, p ≥ 1; the Sobolev space Wk,p(G) with the norm ∥uWk,p(G) for integer k ≥ 0; the weighted Sobolev-Kondratiev space $\begin{array}{}{V}_{p,\alpha }^{k}\left(G\right)\end{array}$ for integer k ≥ 0, 1 < p < ∞ and α ∈ ℝ with the finite norm $\begin{array}{}\parallel u{\parallel }_{{V}_{p,\alpha }^{k}\left(G\right)}=\left(\underset{G}{\int }\sum _{|\beta |\le k}{r}^{\alpha +p\left(|\beta |-k\right)}|{D}^{\beta }u{|}^{p}dx{\right)}^{\frac{1}{p}}\end{array}$ and the space $\begin{array}{}{V}_{p,\alpha }^{k,-\frac{1}{p}}\left(\mathrm{\Gamma }\right)\end{array}$ of functions φ, given on ∂G, with the norm $\begin{array}{}\parallel g{\parallel }_{{V}_{p,\alpha }^{k,-\frac{1}{p}}\left(\mathrm{\partial }G\right)}=\text{inf}\parallel \mathcal{G}{\parallel }_{{V}_{p,\alpha }^{k}\left(G\right)},\end{array}$ where the infimum is taken over all functions g such that 𝓖|∂G = g in the sense of traces.

2 Oblique derivative problem

In [1] we have investigated the behaviour of strong solutions to the oblique derivative problem for the general second-order quasi-linear elliptic equation in a neighbourhood of a conical boundary point of an n-dimensional bounded domain, n ≥ 2. In the case of the linear equation we refer to [4, 5].

Local maximum principle for strong solutions to the elliptic quasi-linear oblique derivative problem in convex rotational cones has been obtained by Lieberman in [6]. He [7] and Trudinger [8] have obtained local gradient bound estimate and local Hölder gradient estimate of strong solutions in any sub-domain with a C2 boundary portion of the domain.

The results obtained in [1] are a generalization and improvement of results of [9] on the case of the oblique boundary condition. We consider the oblique derivative problem for the elliptic second-order linear equation: $aij(x,u,ux)uxixj+a(x,u,ux)=0,aij=aji,x∈G0R,∂u∂n→+χ(ω)∂u∂r+1|x|γ(ω)u=g(x),x∈∂G0R∖O,$(QL) where $\begin{array}{}\stackrel{\to }{n}\end{array}$ denotes the unite exterior normal vector to $\begin{array}{}\mathrm{\partial }{G}_{0}^{R}\mathrm{\setminus }\mathcal{O},\left(r,\omega \right)\end{array}$ are spherical coordinates in ℝn with pole 𝓞; repeated indices are understood as summation from 1 to n.

Definition 2.1

A function u is called a strong solution of problem (QL) if $\begin{array}{}u\in {W}_{loc}^{2,q}\left({G}_{0}^{R}\mathrm{\setminus }\mathcal{O}\right)\cap {W}^{1}\left({G}_{0}^{R}\right)\cap {C}^{0}\left(\overline{{G}_{0}^{R}}\right),q>n\end{array}$ and satisfies the equation for almost all $\begin{array}{}x\in {G}_{\epsilon }^{R}\end{array}$ for all ε > 0 as well as the boundary condition in the sense of traces on $\begin{array}{}\mathrm{\partial }{G}_{0}^{R}\mathrm{\setminus }\mathcal{O}.\end{array}$ We assume that $\begin{array}{}{M}_{0}=x\underset{x\in \overline{{G}_{0}^{R}}}{max}|u\left(x\right)|\end{array}$ is known (see e.g. Theorem 13.1 [7]).

2.1 Assumptions

Let $\begin{array}{}\mathfrak{M}=\left\{\left(x,u,z\right)\phantom{\rule{thinmathspace}{0ex}}:\phantom{\rule{thinmathspace}{0ex}}x\in {G}_{0}^{R},u\in \mathbb{R},z\phantom{\rule{thinmathspace}{0ex}}\in {\mathbb{R}}^{n}\right\}.\end{array}$ With regard to problem (QL) we assume that the following conditions are satisfied on 𝔐:

1. $\begin{array}{}{a}^{ij}\left(x,u,z\right)\in {W}^{1,q}\left(\mathfrak{M}\right),q>n;a\left(x,u,z\right),\frac{\mathrm{\partial }a\left(x,u,z\right)}{\mathrm{\partial }u}\end{array}$ are Caratheodory functions;

2. the uniform elipticity $v|ξ|2<_∑i,j=1naij(x,u,z)ξiξj<_μ|ξ|2,∀ξ∈Rn,$ with the ellipticity constants $\begin{array}{}\mu \ge \nu >0;\phantom{\rule{1em}{0ex}}{a}^{ij}\left(0,0,0\right)={\delta }_{i}^{j},i,j=1,\dots ,n\end{array}$ -the Kronecker symbol;

3. $\begin{array}{}\frac{\mathrm{\partial }a\left(x,u,z\right)}{\mathrm{\partial }u}\le 0;\end{array}$

4. $\begin{array}{}\gamma \left(\omega \right),\chi \left(\omega \right)\in {C}^{1}\left(\overline{\mathrm{\Omega }}\right)\mathit{;}\end{array}$ there exist numbers $\begin{array}{}s>1,\phantom{\rule{thinmathspace}{0ex}}{\chi }_{0}\ge 0,\phantom{\rule{thinmathspace}{0ex}}{\gamma }_{0}>\mathrm{tan}\frac{{\omega }_{\text{O}}}{2}\ge 0\end{array}$ and γ1γ(ω) ≥ γ0 > 0, 0 ≤ χ(ω) ≤ χ0 as well as nonnegative constants μ1, k1, go, g1 and functions $\begin{array}{}b\left(x\right),f\left(x\right)\in {L}_{loc}^{q}\left(G\right),q\ge n\end{array}$ such that the inequalities $|a(x,u,z)|+|∂a(x,u,z)∂u|≤μ1|z|2+b(x)|z|+f(x),0≤b(x),f(x)≤k1|x|s−2,|g(x)|≤g0|x|s−1,|∇g|≤g1|x|s−2,$ hold;

5. coefficients of problem (QL) satisfy such conditions that guarantee $\begin{array}{}u\in {C}^{1+\stackrel{~}{\phantom{\rule{thinmathspace}{0ex}}x\phantom{\rule{thinmathspace}{0ex}}}}\left({G}^{\prime }\right)\end{array}$ and the existence of the local a priori estimate $|u|1+x~,G′≤M1,x~∈(0,1),$ for any smooth $\begin{array}{}{G}^{\prime }\subset \subset \overline{{G}_{0}^{R}}\mathrm{\setminus }\mathcal{O}\end{array}$ (see Theorems 13.13 and 13.14 [7]).

2.2 The main result

Theorem 2.2

([1]). Let u be a strong solution of problem (QL) and λ > 1 be the smallest positive eigenvalue of the following eigenvalue problem for the Laplace - Beltrami operator Δω on the unit sphere $Δωψ+λ(λ+n−2)ψ(ω)=0,ω∈Ω,∂ψ∂ν→+〈λχ(ω)+γ(ω)〉ψ(ω)=0,ω∈∂Ω,$(EVP) where $\begin{array}{}\stackrel{\to }{\nu }\end{array}$ is the unit exterior normal to $\begin{array}{}\mathrm{\partial }{G}_{0}^{1}\end{array}$ at the points of Ω. Suppose that assumptions (A) - (E) are satisfied. Then there exist numbers d > 0, c0, c1 independent of u such that

• $|u(x)|≤c0|x|λifs>λ,|x|λln32⁡1|x|ifs=λ,x∈G0d;|x|Sifs<λ,$(1) $|∇u(x)|≤c1|x|λ−1ifs>λ,|x|λ−1ln32⁡1|x|ifs=λ,x∈G0d.|x|s−1ifs<λ,$(2)

• If 4 + q(λ − 2) > 0 then $\begin{array}{}u\left(x\right)\in {V}_{q,4-n}^{2}\left({G}_{0}^{Q}\right)\end{array}$ and there exist numbers d > 0, c2, independent of u such that $∥u(x)∥Vq,4−n2(G0ϱ)≤C~2ϱλ−2+4qifs>λ,ϱλ−2+4qln3⁡1Qifs=λ,,0<ϱ(3) and

• if $\begin{array}{}1<\lambda <2,q>\frac{n}{2-\lambda }\end{array}$ then $u(x)∈Cλ(G0d¯)ifs≥λ,u(x)∈Cs(G0d¯)ifs<λ.$(4)

Theorem 2.3

([4, 5]). There exists the smallest positive eigenvalue λ of problem (EVP), which satisfies the following inequalities $0<λ<(πω0)2+(n−22)2−n−22$ for n ≥ 3.

3 Quasi-linear nonlocal Robin problem

In [3] we have investigated the behaviour of weak solutions for the nonlocal Robin problem with quasi-linear elliptic divergence second-order equations in a plain domain in a neighbourhood of the boundary corner point 𝓞. In the case of the linear equation we refer to [10, 11].

Here, we consider a different eigenvalue problem (see (EVP2)) and derive a new Friedrichs-Wirtinger type inequality adapted to the quasi-linear elliptic problem considered in [3]. It allows us to improve the main result of [3] (see Theorem 3.7): the exponent of weak solution behavior (see inequality (20)) in a neighborhood of an angular point 𝓞 in the case 𝓑 < 0 is better than that one obtained in [3].

We consider the type of nonlocal problems, where the support of nonlocal terms intersects the boundary. Namely, the situation in which a part Γ of domain boundary ∂G is mapped by transformation γ on γ(Γ) and $\begin{array}{}\overline{\gamma \left(\mathrm{\Gamma }\right)}\cap \mathrm{\partial }G\ne \mathrm{\varnothing }.\end{array}$

Let us consider the domain $\begin{array}{}{G}_{0}^{R}\end{array}$ ⊂ ℝ2. Moreover, let Γ+ and Γ be the part of boundary $\begin{array}{}{G}_{0}^{R}\end{array}$ for which x2 > 0 and x2 < 0 respectively. We assume, that $\begin{array}{}\mathrm{\partial }{G}_{0}^{R}={\overline{\mathrm{\Gamma }}}_{+}\cup \phantom{\rule{thinmathspace}{0ex}}{\overline{\mathrm{\Gamma }}}_{-}\end{array}$ is a smooth curve everywhere except at the origin 𝓞 ∈ $\begin{array}{}{G}_{0}^{R}\end{array}$ and near the point 𝓞 curves Γ± are lateral sides of an angle with the measure ω0 ∈ [0, π) and the vertex at 𝓞. Let Σ0 = $\begin{array}{}{G}_{0}^{R}\end{array}$ ∩{x2 = 0}, where $\begin{array}{}\mathcal{O}\in \overline{{\mathrm{\Sigma }}_{0}}.\end{array}$ Furthermore, let γ be a diffeomorphism mapping of Γ+ onto Σ0. Additionally, we suppose that there exists d > 0 such that in the neighbourhood of $\begin{array}{}{\mathrm{\Gamma }}_{0+}^{d}\end{array}$ the mapping γ is the rotation about the origin 𝓞 and $\begin{array}{}-\frac{{\omega }_{0}}{2}\end{array}$ is the angle of rotation. It means $\begin{array}{}\gamma \left(\overline{{\mathrm{\Gamma }}_{0+}^{d}}\right)=\overline{{\mathrm{\Sigma }}_{0}^{d}}=\overline{{G}_{0}^{d}}\cap {\mathrm{\Sigma }}_{0}.\end{array}$

We have considered a quasi-linear elliptic equation with the nonlocal boundary condition connecting the values of the unknown function u on the boudary part Γ+ with its values of u on Σ0 : $−ddxi(|u|q|∇u|m−2uxi)+a0r−mu|u|q+m−2−μu|u|q−2|∇u|m=f(x),x∈G0R;|u|q|∇u|m−2∂u∂n→+β+r1−mu|u|q+m−2+br1−mu(γ(x))|u(γ(x))|q+m−2=g(x,u),x∈Γ+;|u|q|∇u|m−2∂u∂n→+β−r1−mu|u|q+m−2=h(x,u),x∈Γ−$(QL2) here q ≥ 0, m > 1, μ ≥ 0, a0 ≥ 0, β± > 0, b ≥ 0 are given numbers and $\begin{array}{}\stackrel{\to }{n}\end{array}$denotes the unit outward with respect to $\begin{array}{}{G}_{0}^{R}\text{\hspace{0.17em}normal to\hspace{0.17em}}\mathrm{\partial }{G}_{0}^{R}\phantom{\rule{thinmathspace}{0ex}}\mathrm{\setminus }\phantom{\rule{thinmathspace}{0ex}}\mathcal{O}.\end{array}$

Recall that we are dealing with the nonlocal problem and the boundary $\begin{array}{}{G}_{0}^{R}\end{array}$ is non smooth, so the formulation of problem (QL2) does not make sense in general. To make our formulation precise, we give the definition of the weak solution of (QL2).

Definition 3.1

A function u is called a weak solution of problem (QL2) provided that $\begin{array}{}u\in {C}^{0}\left(\overline{{G}_{0}^{R}}\right)\cap {V}_{m,0}^{1}\left({G}_{0}^{R}\right)\end{array}$ and for all functions $\begin{array}{}\eta \in {C}^{0}\left(\overline{{G}_{0}^{R}}\right)\cap {V}_{m,0}^{1}\left({G}_{0}^{R}\right)\end{array}$ the following integral identity $∫G|u|q|∇u|m−2uxiηxidx+∫G{a01rmu|u|q+m−2−μ|u|q−1|∇u|msgnu}η(x)dx+β+∫Γ+1rm−1u|u|q+m−2η(x)ds+b∫Γ+1rm−1u(γ(x))|u(γ(x))|q+m−2η(x)ds+β−∫Γ−1rm−1u|u|q+m−2η(x)ds=∫Gf(x)η(x)dx+∫Γ+g(x,u)η(x)ds+∫Γ−h(x,u)η(x)ds$ is satisfied.

3.1 The Friedrichs-Wirtinger type inequality

We consider the following eigenvalue problem: $(|ψ′(ω)|m−2ψ′(ω))′+ϑ|ψ(ω)|m−2ψ(ω)=0,ω∈Ω|ψ′(ω02)|m−2ψ′(ω02)+β+|ψ(ω02)|m−2ψ(ω02)+b|ψ(0)|m−2ψ(0)=0−|ψ′(−ω02)|m−2ψ′(−ω02)+β−|ψ(−ω02)|m−2ψ(−ω02)=0,$(EVP2) with m ≥ 2, β+, β > 0, b ≥ 0, which consists in determining all values ϑ (eigenvalues) for which (EVP2) has nonzero weak solutions (eigenfunctions) ψ(ω).

Definition 3.2

A function ψ is called a weak solution of problem (EVP2) provided that ψW1,m(Ω) ∩ C0( $\begin{array}{}\overline{\mathrm{\Omega }}\end{array}$) and satisfies the integral identity $∫Ω(|ψ′(ω)|m−2ψ′(ω)η′(ω)−ϑ|ψ(ω)|m−2ψ(ω)η(ω))dω+β+|ψ(ω02)|m−2ψ(ω02)η(ω02)+β−|ψ(−ω02)|m−2ψ(−ω02)η(−ω02)+b|ψ(0)|m−2ψ(0)η(ω02)=0$ for all η(ω) ∈ W1,m(Ω) ∩ C0( $\begin{array}{}\overline{\mathrm{\Omega }}\end{array}$i).

Theorem 3.3

(Friedrichs-Wirtinger’s type inequality). Let Ω ⊂ S be an arc, m ≥ 2, ϑ be the eigenvalue ofproblem (EVP2) and ψW1,m(Ω) ∩ C0( $\begin{array}{}\overline{\mathrm{\Omega }}\end{array}$) be the corresponding eigenfunction. Then for any uW1,m(Ω) ∩ C0( $\begin{array}{}\overline{\mathrm{\Omega }}\end{array}$), uconst ≠ 0 the inequality $ϑ∫Ω|u(ω)|mdω≤∫Ω|u′(ω)|mdω+B|u(ω02)|m+β−|u(−ω02)|m$(F − W) holds, where $B=bψ(0)|ψ(0)|m−2ψ(ω02)|ψ(ω02)|−m+β+.$(5)

Proof

Let us first derive the estimate (FW) for functions uC0( $\begin{array}{}\overline{\mathrm{\Omega }}\end{array}$) ∩ C1(Ω) and ψC1( $\begin{array}{}\overline{\mathrm{\Omega }}\end{array}$) ∩ C2(Ω). Setting u(ω) = ψ(ω)v(ω) we obtain $|u′(ω)|m=|(ψ(ω)v(ω))′|m=[(v′(ω)ψ(ω)+v(ω)ψ′(ω))2]m2=[ψ′2(ω)v2(ω)+2ψ′(ω)ψ(ω)v′(ω)v(ω)+ψ2(ω)v′2(ω)]m2=|v(ω)|m|ψ′(ω)|m[1+2ψ(ω)ψ(ω)v′(ω)v(ω)+ψ2(ω)ψ′2(ω)v′2(ω)v2(ω)]m2.$(6)

By Bernoulli inequality (1 + x)α ≥ 1 + αx, with α ≥ 1 and x > − 1 from (6) we have $|u′(ω)|m≥|v(ω)|m|ψ′(ω)|m(1+m2[2v′(ω)v(ω)ψ(ω)ψ′(ω)+ψ2(ω)ψ′2(ω)v′2(ω)v2(ω)])≥|v(ω)|m|ψ′(ω)|m+m|v(ω)|m−2v(ω)v′(ω)ψ(ω)ψ′(ω)|ψ′(ω)|m−2.$(7)

Next, in virtue of (|v(ω)|m)′ = m |v(ω)|m−2v(ω)v′(ω) and from (EVP2) it follows that: $(|v(ω)|mψ(ω)ψ′(ω)|ψ′(ω)|m−2)′=(|v(ω)|m)′ψ(ω)ψ′(ω)|ψ′(ω)|m−2+|v(ω)|m|ψ′(ω)|m+|v(ω)|mψ(ω)(|ψ′(ω)|m−2ψ′(ω))′=m|v(ω)|m−2v(ω)v′(ω)ψ(ω)ψ′(ω)|ψ′(ω)|m−2+|v(ω)|m|ψ′(ω)|m+|v(ω)|mψ(ω)(|ψ′(ω)|m−2ψ′(ω))′=|v(ω)|m|ψ′(ω)|mm|v(ω)|m−2v(ω)v′(ω)ψ(ω)ψ′(ω)|ψ′(ω)|m−2−θ|v(ω)|m|ψ(ω)|m.$(8) Now, from (7), (8) and definition of function u(ω) we get the inequality $ϑ|u(ω)|m≤|u′(ω)|m−(|v(ω)|mψ(ω)ψ′(ω)|ψ′(ω)|m−2)′.$(9)

Integrating (9) over ω and since ψ(ω) is a solution of (EVP2) we obtain $ϑ∫Ω|u(ω)|mdω≤∫Ω|u′(ω)|mdω−|v(ω)|mψ(ω)ψ′(ω)|ψ′(ω)|m−2|ω=−ω02ω=ω02=∫Ω|u′(ω)|mdω+β+|v(ω02)|m|ψ(ω02)|m+β−|v(−ω02)|m|ψ(−ω02)|m+b|v(ω02)|mψ(ω02)ψ(0)|ψ(0)|m−2=∫Ω|u′(ω)|mdω+β++bψ(0)|ψ(0)|m−2ψ(ω02)|ψ(ω02)|−m|u(ω02)|m+β−|u(−ω02)|m.$ Thus, we get (FW) for smooths functions. The extension to arbitrary functions in $\begin{array}{}{W}^{1,m}\left(\mathrm{\Omega }\right)\cap {C}^{0}\left(\overline{\mathrm{\Omega }}\right)\end{array}$ follows by straight-forward approximation argument. □

Remark 3.4

In virtue of inequality (FW), Corollary 2.2 [3] for any $\begin{array}{}v\in {V}_{m,0}^{1}\left({G}_{0}^{d}\right)\cap {C}^{0}\left({\overline{G}}_{0}^{d}\right)\end{array}$ and ϱ ∈ (0, d), we have $∫G0ϱ|v|mdx≤ϱmϑ{∫G0ϱ|∇v|mdx+B∫Γ0+ϱ|v|mrm−1ds+β−∫Γ0−ϱ|v|mrm−1ds}$(H − W) where ϑ is the least positive eigenvalue of (EVP2) problem.

3.2 Assumptions

With regard to problem (QL2) we assume that the following conditions are satisfied:

1. let p > m > 1, $\begin{array}{}0\phantom{\rule{thinmathspace}{0ex}}\le \phantom{\rule{thinmathspace}{0ex}}\mu \phantom{\rule{thinmathspace}{0ex}}<\phantom{\rule{thinmathspace}{0ex}}\frac{q+m-1}{m-1}\end{array}$ be given numbers; g(x, u) be the Caratheodory and continuously differentiable with respect to variable u function Γ+ × ℝ → ℝ, h(x, u) be the Caratheodory and continuously differentiable with respect to variable u function Γ × ℝ → ℝ;

2. $\begin{array}{}\frac{\mathrm{\partial }h\left(x,u\right)}{\mathrm{\partial }u}\phantom{\rule{thinmathspace}{0ex}}\le \phantom{\rule{thinmathspace}{0ex}}0,\phantom{\rule{thinmathspace}{0ex}}\frac{\mathrm{\partial }g\left(x,u\right)}{\mathrm{\partial }u}\phantom{\rule{thinmathspace}{0ex}}\le \phantom{\rule{thinmathspace}{0ex}}0;\end{array}$

3. $\begin{array}{}f\left(x\right)\in {L}_{\frac{p}{m}}\left({G}_{0}^{R}\right);g\left(x,\phantom{\rule{thinmathspace}{0ex}}0\right),h\left(x,0\right)\in {W}^{1,\frac{p}{m-1}}\left({G}_{0}^{R}\right)\end{array}$ and there exist positive constants g1, h1 such that $|∂g(x,u)∂u|≤g1|u|m−1,x∈Γ+;|∂h(x,u)∂u|≤h1|u|m−1,x∈Γ−;$

4. $\begin{array}{}0\le b

For the following we will use the numbers: $K−:=min{ςm−1(1−μς)−bm−1mω0m−1;1} with ς=m−1q+m−1,$(10) $K+:=min{K−;1B(β+−b1+(m−1)2m−1m)},B>0,$(11) $Ξ(m)=mm2⋅(2m+2)m+22m,$(12) $kσ=:supϱ>0ϱ−mσ{∫G0ϱ|f(x)|mm−1dx+∫Γ0+ϱ|g(x,0)|mm−1ds+∫Γ0−ϱ|h(x,0)|mm−1ds},σ>1;$(13) and $ϑ~±:=K+ϑ1mΞ(m), if B>0K−ϑ1mΞ(m), if B≤0,$ where 𝓑 is defined by (5) and ϑ is the smallest positive eigenvalue of problem (EVP2).

3.3 Integral estimates

Lemma 3.5

Let 𝓑 be defined by (5) and $\begin{array}{}{G}_{0}^{d}\end{array}$ be a sector. Let $\begin{array}{}v\in {C}^{0}\left(\overline{{G}_{0}^{d}}\right)\cap {V}_{m,0}^{1}\left({G}_{0}^{d}\right)\end{array}$ for almost all ϱ ∈ (0, d) and $V±(ϱ)=∫G0ϱ|∇v|mdx+α±B∫Γ0+ϱ|v(x)|mrm−1ds+β−∫Γ0−ϱ|v(x)|mrm−1ds,$(14) where $α±=1,ifB>00,ifB≤0.$ Let ϑ be the smallest positive eigenvalue of problem (EVP2). Then for almost all ϱ ∈ (0, d) $∫Ωϱv∂v∂r|∇v|m−2|r=ϱdω≤Ξ(m)⋅ϱmϑ1mV±′(ϱ),m>_2,$ where Ξ is defined by (12).

Proof

We repeat the proof of Lemma 2.3 [3] applying inequality (FW) instead of (W)m and taking β+ = α± 𝓑. □

Theorem 3.6

Let 𝓑 be defined by (5) and u be a weak solution of problem (QL), ϑ be the smallest positive eigenvalue of (EVP2). Let us assume that assumptions (a)-(d) are satisfied. In addition, assume that there exists real number kσ ≥ 0 defined by (13). Then there exist d ∈ (0,1) and a constant c > 0 independent of u such that for any ϱ ∈ (0, d) $∫G0ϱ|u|qmm−1|∇u|mdx+∫Γ0+ϱ1rm−1|u|mm−1(q+m−1)ds+∫Γ0+ϱ1rm−1|u|mm−1(q+m−1)ds≤cΘm(ϱ),$ where $Θ(ϱ)=ϱϑ~±,σ>ϑ~±ϱϑ~±ln1m⁡(1ϱ),σ=ϑ~±ϱσ,σ<ϑ~±.$(15)

Proof

The proof we starting from estimates the integrals on the right hand side of the inequality (5.8) of Theorem 4.2[3]. By the Young inequality with parametrs $\begin{array}{}m,\frac{m}{m-1},\end{array}$ we get $∫Γ0+ϱ|v||g(x,0)|ds≤1mδm∫Γ0+ϱ|v|mds+m−1mδm1−m∫Γ0+ϱ|g(x,0)|mm−1dS,∀δ>0.$ Similarly, with δ = 1, we have $∫Γ0−ϱ|v||h(x,0)|ds≤1m∫Γ0−ϱ|v|mds+m−1m∫Γ0−ϱ|h(x,0)|mm−1ds$ and $∫G0ϱvf(x)dx≤1m∫G0ϱ|v|mdx+m−1m∫G0ϱ|f(x)|mm−1dx≤ϱmmϑV±(ϱ)+m−1m∫G0ϱ|f(x)|mm−1dx,$ in virtue of inequality (HW). Thus, by Lemma 3.5 and with regard to definition (14), from inequality (5.8) of Theorem 4.2[3], we have $ςm−1(1−μς)−bm−1mω0m−1∫G0ϱ|∇v|mdx+[β+−b1+(m−1)2m−1m]∫Γ0+ϱ1rm−1|v|mds+β−∫Γ0−ϱ1rm−1|v|mds≤ςm−1Ξ(m)mϑ1mϱV±′(ϱ)+m−1m(∫G0ϱ|f(x)|mm−1dx+δm1−m∫Γ0+ϱ|g(x,0)|mm−1ds+∫Γ0−ϱ|h(x,0)|mm−1dS)+1m(ϱmϑV±(ϱ)+δm∫Γ0+ϱ|v|mds+∫Γ0−ϱ|v|mds),∀δ>0.$(16)

Now, there are two cases to consider: 𝓑 > 0 and 𝓑 ≤ 0. Firstly, let 𝓑 > 0. Therefore $∫Γ0+ϱ|v|mds+∫Γ0−ϱ|v|mds≤ϰϱm−1V+(ϱ),$ where $\begin{array}{}\varkappa =max\left\{\frac{1}{\mathcal{B}};\frac{1}{\mathcal{B}-}\right\}.\end{array}$ Hence, setting δ = 1, by assumption (i v), Lemma 3.5 with regard to definition (14) and (13), from (16) it follows that $(K+−δ(ϱ))V+(ϱ)≤Ξ(m)mϑ1mςm−1ϱV+′(ϱ)+m−1mkσϱmσ,$(17) where δ(ϱ) = const (m, q, ϑ, β, 𝓑)⋅ϱm−1 and 𝓚+ is defined by (11).

Next, let 𝓑 ≤ 0. Setting in (16) $δ=[m2dm−1〈β+−b1+(m−1)2m−1m〉]1m,$ by assumption (d) and $\begin{array}{}\frac{1}{{d}^{m-1}}\le \frac{1}{{r}^{m-1}},\end{array}$ we have $ςm−1(1−μς)−bm−1mω0m−1∫G0ϱ|∇v|mdx+12dm−1[β;+−b1+(m−1)2m−1m]∫Γ0+ϱ|v|mds+β;−∫Γ0−ϱ1rm−1|v|mds≤ςm−1Ξ(m)mϑ1mϱV−′(ϱ)+m−1m(∫G0ϱ|f(x)|mm−1dx+c1(m,b,β;+)∫Γ0+ϱ|g(x,0)|mm−1ds+∫Γ0−ϱ|h(x,0)|mm−1ds)+1m(ϱmϑV−(ϱ)+∫Γ0−ϱ|v|mds).$(18)

Finally, by $\begin{array}{}\underset{{\mathrm{\Gamma }}_{0-}^{\varrho }}{\int }|v{|}^{m}ds\le {\varrho }^{m-1}V-\left(\varrho \right),\end{array}$ Lemma 3.5 and with regard to definition (14) and (13), from (18) it follows that $(K−−δ(ϱ))V−(ϱ)≤Ξ(m)mϑ1mςm−1ϱV−′(ϱ)+m−1mkσϱmσ,$(19) where δ(ϱ) = const(m, q, ϑ, β, 𝓑)⋅ϱm−1, and 𝓚 is defined by (10).

Thus, (17) and (19) we can write as follows $(K±−δ(ϱ))V±(ϱ)≤Ξ(m)mϑ1mςm−1ϱV±′(ϱ)+m−1mkσϱmσ.$ The rest of the proof follows verbatim the proof of Theorem 4.2 [3], starting from the paragraph following inequality (5.14). □

3.4 The main result

Main result in this part of work is the following statement:

Theorem 3.7

Let u be a weak solution of problem (QL2) and assumptions (a)-(d) are fulfilled. Let us assume that $\begin{array}{}{M}_{0}=\underset{x\in \overline{G}}{max}|u\left(x\right)|\end{array}$ is known (see e.g. Theorem 3.1 [3]). In addition, suppose that there exist a real number kσ ≥ 0 defined by (13) and K ≥ 0 such that $K=:supϱ>0ϱ2m−1Θ(ϱ){ϱm(p−2)p(m−1)∥f(x)∥pm,G0ϱ1m−1+ϱm−1m−2p(||g(x,0)||pm−1,G0ϱ1m−1+||h(x,0)||pm−1,G0ϱ1m−1)+ϱm2−m+1m(m−1)−2p(||∇g(x,0)||pm−1,G0ϱ1m−1+||∇h(x,0)||pm−1,G0ϱ1m−1)},$ where Θ(ϱ) is defined by (15) Then there exist d ∈ (0, 1) and a constant C > 0 independent of u such that $|u(x)|≤C(|x|1−2mΘ(|x|))m−1q+m−1,∀x∈G0d.$(20)

Proof

The estimate (20) we derive analogously to (6.1.7)[12] in virtue of Theorem 3.2 [3] and above proved Theorem 3.6 and the inequality (HW). □

4 Boundary value problems near the infinity

In [2] we consider the following boundary value problems for quasi-linear elliptic divergence equations: $−ddxiai(x,u,∇u)+b(x,u,∇u)=0,x∈Gd;α(x)∂u∂v+1|x|m−1γ(x|x|)u|u|q+m−2=g(x,u),x∈∂Gd;lim|x|→∞u(x)=0,$(QL3) where: d > 0, ai : Gd × ℝ × ℝn → ℝ, b : Gd × ℝ × ℝn → ℝ, m > 1, q ≥ 0, $\begin{array}{}\alpha \left(x\right)=\left\{\begin{array}{}0,\phantom{\rule{thinmathspace}{0ex}}if\phantom{\rule{thinmathspace}{0ex}}x\in \mathcal{D};\\ 1,\phantom{\rule{thinmathspace}{0ex}}if\phantom{\rule{thinmathspace}{0ex}}x\notin \mathcal{D},\end{array}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathcal{D}\subseteq \mathrm{\partial }{G}_{d}\end{array}$ is the part of the boundary ∂Gd, where the Dirichlet boundary condition is posed, γ : (0,+∞) → (0,+∞) and $\begin{array}{}\frac{\mathrm{\partial }u}{\mathrm{\partial }\nu }={a}_{i}\left(x,u,\mathrm{\nabla }u\right)\mathrm{cos}\left(\stackrel{\to }{n},{x}_{i}\right),\stackrel{\to }{n}\end{array}$ denotes the unit outward with respect to Gd normal to Gd.

Our aim was to find an exponent of (QL3) weak solutions’ decreasing rate at the infinity (in the case of the linear equation we refer to [13, 14]).

Definition 4.1

A function u is called a weak solution of problem (QL3) provided that $\begin{array}{}u\in {C}^{0}\left(\overline{{G}_{d}}\right)\cap {V}_{m,0}^{1}\left({G}_{d}\right),\underset{|x|\to \mathrm{\infty }}{lim}u\left(x\right)=0\end{array}$ and satisfies the integral identity $∫Gd{ai(x,u,ux)ηxi+b(x,u,ux)η(x)}dx+∫∂Gdα(x)γ(ω)rm−1u|u|q+m−2η(x)ds=∫∂Gdα(x)g(x,u)η(x)ds$ for all functions $\begin{array}{}\eta \in {C}^{0}\left(\overline{{G}_{d}}\right)\cap {V}_{m,0}^{1}\left({G}_{d}\right),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}such\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}that\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\underset{|x|\to \mathrm{\infty }}{lim}\eta \left(x\right)=0.\end{array}$

4.1 Assumptions

Let m < n < p, $\begin{array}{}0\le \mu <\frac{q+m-1}{m-1}\end{array}$ be given numbers; a0(x), α1(x), b0(x) be non-negative measurable functions. We assume that:

1. $\begin{array}{}\sqrt{\sum _{i=1}^{n}{a}_{i}^{2}\left(x,u,\xi \right)}+\sqrt{\sum _{i=1}^{n}|\frac{\mathrm{\partial }{a}_{i}\left(x,u,\xi \right)}{\mathrm{\partial }{x}_{i}}{|}^{2}}\le |u{|}^{q}|\xi {|}^{m-1}+{\alpha }_{1}\left(x\right);\phantom{\rule{2em}{0ex}}{\alpha }_{1}\left(x\right)\in {L}_{\frac{p}{m-1}}\left({G}_{d}\right)\cap {L}_{\frac{m}{m-1}}\left({G}_{d}\right);\end{array}$

2. $\begin{array}{}{a}_{i}\left(x,u,\xi \right){\xi }_{i}\ge |u{|}^{q}|\xi {|}^{m}-{a}_{0}\left(x\right);\phantom{\rule{1em}{0ex}}{a}_{0}\left(x\right)\in {L}_{\frac{p}{m}}\left({G}_{d}\right);\end{array}$

3. $\begin{array}{}|b\left(x,u,\xi \right)|\le \mu |u{|}^{q-1}|\xi {|}^{m}+{b}_{0}\left(x\right);\\ b\left(x,u,\xi \right)=\mathrm{\beta ;}\left(x,u\right)+\stackrel{~}{b}\left(x,u,\xi \right),\phantom{\rule{2em}{0ex}}u\cdot \mathrm{\beta ;}\left(x,u\right)\ge |u{|}^{q+m};\\ |\stackrel{~}{b}\left(x,u,\xi \right)|\le \mu |u{|}^{q-1}|\xi {|}^{m}+{b}_{0}\left(x\right);\phantom{\rule{2em}{0ex}}{b}_{0}\left(x\right)\in {L}_{\frac{p}{m}}\left({G}_{d}\right)\cap {L}_{1}\left({G}_{d}\right);\end{array}$

4. $\begin{array}{}\frac{\mathrm{\partial }g\left(x,u\right)}{\mathrm{\partial }u}\le 0;\end{array}$

5. γ(ω) ≥ γ0 > 0 on ∂ Gd.

In addition, suppose that the functions ai(x, u, ξ) are continuously differentiable with respect to u, ξ variables in $\begin{array}{}{\mathfrak{M}}_{\varrho ,{M}_{0}}=\overline{{G}_{\varrho }}×\left[-{M}_{0},{M}_{0}\right]×{\mathbb{R}}^{n},\varrho >R\end{array}$ and satisfy in 𝔐ϱ, M0 the following conditions

6. $\begin{array}{}\left(m-1\right)u\frac{\mathrm{\partial }{a}_{i}\left(x,u,\xi \right)}{\mathrm{\partial }u}=q\frac{\mathrm{\partial }{a}_{i}\left(x,u,\xi \right)}{\mathrm{\partial }{\xi }_{j}}{\xi }_{j};\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}i=1,\dots ,n;\end{array}$

7. $\begin{array}{}\sqrt{\sum _{i=1}^{n}|{a}_{i}\left(x,u,{u}_{x}\right)-|u{|}^{q}|\mathrm{\nabla }u{|}^{m-2}{u}_{{x}_{i}}{|}^{2}}\le \mathcal{A}\left(\frac{1}{|x|}\right)|u{|}^{q}|\mathrm{\nabla }u{|}^{m-1},\end{array}$

where 𝓐(t) is a monotonically increasing and Dini-continuous at zero function.

4.2 The main result

Suppose that there are finite numbers ks and K such that $ks=supϱ>Rϱms{∫Gϱrq(m+1)(q+m)(m−1)(a0(x))m(q+m−1)(m−1)(q+m)dx+∫Gϱrm+1m−1(b0(x))mm−1dx+∫Γϱα(x)(r|g(x,0)|)mm−1ds},s>0;K=supϱ>Rϱnm−1Θ(ϱ){ϱm(1−np)q+m−1(m−1)(q+m)∥a0(x)∥pm,Gϱ2ϱq+m−1(m−1)(q+m)+ϱ(1−np)mm−1∥b0(x)∥pm,Gϱ2ϱ1m−1+ϱ1−np∥α1(x)∥pm−1,Gϱ2ϱ1m−1+ϱ∥g(x,0)∥∞,Γϱ2ϱ1m−1},$ where $Θ(ϱ)=ϱ−ϑ1m(m)Ξ(m)⋅q+(m−1)(1−μ)q+m−1,s>ϑ1m(m)Ξ(m)⋅q+(m−1)(1−μ)q+m−1;ϱ−ϑ1m(m)Ξ(m)⋅q+(m−1)(1−μ)q+m−1ln1m⁡ϱ,s=ϑ1m(m)Ξ(m)⋅q+(m−1)(1−μ)q+m−1;ϱ−s,0 and ϑ is the smallest positive eigenvalue of the eigenvalue problem for the m-Laplace-Beltrami operator on the unit sphere: $divω(|∇ωψ|m−2∇ωψ)+ϑ(m)|ψ|m−2ψ=0,ω∈Ω;α(ω)|∇ωψ|m−2∂ψ∂ν→+γ(ω)|ψ|m−2ψ(ω)=0,ω∈∂Ω.$

Theorem 4.2

([2]). Let u be a weak solution ofproblem (QL3) and assumptions (1)(7) are satisfied. Suppose, in addition, that $\begin{array}{}g\left(x,0\right)\in {L}_{\frac{j}{j-1}}\left(\mathrm{\partial }{G}_{d}\right),10.\end{array}$ Then there exist $\begin{array}{}\stackrel{~}{R}>R\gg 1\end{array}$ and a constant C0 > 0 such that for all $\begin{array}{}x\in {G}_{\stackrel{~}{R}}\end{array}$ $|u(x)|≤C0(|x|1−nmΘ(|x|))m−1q+m−1.$

5 The ideas of proofs

The ideas of proofs of Theorem 2.2, Theorem 3.7 and Theorem 4.2 are based on the deduction of new inequalities of Friedrichs-Wirtinger type with exact constants as well as some integral-differential inequalities adapted to our problems. The precise exponents of the solution’s decrease rate depend on these exact constants. For details we refer to [13].

The existence of the smallest positive eigenvalue of problem (EVP) for n = 3 was proved in [4]. The ideas of proof of this theorem are based on the Legendre spherical harmonics (see [4]) and the Gegenbauer functions.

Acknowledgement

The authors would like to thank Prof. Mikhail Borsuk for suggesting the problems and numerous helpful discussions during these studies.

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Accepted: 2017-02-20

Published Online: 2017-05-23

Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 667–678, ISSN (Online) 2391-5455,

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