## Advances in Differential Equations

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

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

Francisco Júlio S.A. Corrêa and Giovany M. Figueiredo

#### 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

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1363266258

**Mathematical Reviews number (MathSciNet)**

MR3086466

**Zentralblatt MATH identifier**

1277.45016

**Subjects**

Primary: 45M20: Positive solutions 35J25: Boundary value problems for second-order elliptic equations 34B18: Positive solutions of nonlinear boundary value problems 34C11: Growth, boundedness 34K12: Growth, boundedness, comparison of solutions

#### Citation

Corrêa, Francisco Júlio S.A.; Figueiredo, Giovany M. Existence and multiplicity of nontrivial solutions for a bi-nonlocal equation. Adv. Differential Equations 18 (2013), no. 5/6, 587--608. https://projecteuclid.org/euclid.ade/1363266258