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May/June 2013 Existence and multiplicity of nontrivial solutions for a bi-nonlocal equation
Francisco Júlio S.A. Corrêa, Giovany M. Figueiredo
Adv. Differential Equations 18(5/6): 587-608 (May/June 2013).

Abstract

In this work will use the Genus theory, introduced by Krasnolselskii, the mountain-pass theorem, introduced by Ambrosetti and Rabinowitz, and the concentration-compactness principle, due to Lions, to show results of existence and multiplicity of solutions for the bi-nonlocal equation $$ -M \Big (\int_{\Omega}|\nabla u|^{p} \, dx\ \Big )\Delta_{p} u = \lambda |u|^{q-2}u+ \mu g(x)|u|^{\gamma-2}u \Big [\frac{1}{\gamma}\int_{\Omega}g(x)|u|^{\gamma} \, dx \Big ]^{2r} \ \mbox{in} \ \ \Omega, $$ with Dirichlet boundary condition, where $\Omega$ is a bounded smooth domain of $\mathbb{R}^{N}$, $1<p<N$, $1<\gamma< p^{*}$, and $M:[0,+\infty)\rightarrow \mathbb{R}$ and $g:\Omega\rightarrow \mathbb{R}$ are continuous functions. We consider $r$ a positive parameter, and we study several cases: $\lambda=0$, $\lambda=1$, $\mu=1$, $p(\alpha+1)<q\leq p^{*}$, and $p-1 < q <p(\alpha+1)$.

Citation

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Francisco Júlio S.A. Corrêa. Giovany M. Figueiredo. "Existence and multiplicity of nontrivial solutions for a bi-nonlocal equation." Adv. Differential Equations 18 (5/6) 587 - 608, May/June 2013.

Information

Published: May/June 2013
First available in Project Euclid: 14 March 2013

zbMATH: 1277.45016
MathSciNet: MR3086466

Subjects:
Primary: 34B18 , 34C11 , 34K12 , 35J25 , 45M20

Rights: Copyright © 2013 Khayyam Publishing, Inc.

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Vol.18 • No. 5/6 • May/June 2013
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