### Existence and multiplicity of nontrivial solutions for a bi-nonlocal equation

#### 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 $p-1 < q <p(\alpha+1)$.

#### Article information

Source
Adv. Differential Equations Volume 18, Number 5/6 (2013), 587-608.

Dates
First available in Project Euclid: 14 March 2013