## Topological Methods in Nonlinear Analysis

### Solutions for quasilinear elliptic systems with vanishing potentials

Billel Gheraibia

#### Abstract

In this paper, we study the following strongly coupled quasilinear elliptic system: $$\begin{cases} -\Delta_{p} u+\lambda a(x)|u|^{p-2}u=\dfrac{\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta}, & x\in {\mathbb R}^{N}, \\ -\Delta_{p} v+\lambda b(x)|v|^{p-2}v=\dfrac{\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v, & x\in {\mathbb R}^{N}, \\ u,v\in D^{1,p}(\mathbb R^{N}), \end{cases}$$ where $N\geq 3$, $\lambda> 0$ is a parameter, $p< \alpha+\beta< p^{*}:={Np}/({N-p})$. Under some suitable conditions which are given in section 1, we use variational methods to obtain both the existence and multiplicity of solutions for the system on an appropriated space when the parameter $\lambda$ is sufficiently large. Moreover, we study the asymptotic behavior of these solutions when $\lambda\rightarrow\infty$.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 54, Number 1 (2019), 153-175.

Dates
First available in Project Euclid: 16 July 2019

https://projecteuclid.org/euclid.tmna/1563242557

Digital Object Identifier
doi:10.12775/TMNA.2019.032

#### Citation

Gheraibia, Billel. Solutions for quasilinear elliptic systems with vanishing potentials. Topol. Methods Nonlinear Anal. 54 (2019), no. 1, 153--175. doi:10.12775/TMNA.2019.032. https://projecteuclid.org/euclid.tmna/1563242557

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