Existence of solutions for a class of quasilinear elliptic equations involving the p-Laplacian

Abstract

This paper is concerned with the existence of solutions for the quasilinear elliptic equations $-{\mathrm{\Delta }}_{p}u-{\mathrm{\Delta }}_p\left(|u{|}^{2\alpha }\right)|u{|}^{2\alpha -2}u+V\left(x\right)|u{|}^{p-2}u=|u{|}^{q-2}u,x\in {\mathbb{R}}^N,$ where $\alpha \ge 1$, 1<p<N, $p^{\ast }=\mathrm{Np}/(N-p)$, ${\mathrm{\Delta }}_p$ is the p-Laplace operator and the potential $V\left(x\right)>0$ is a continuous function. In this work, we mainly focus on nontrivial solutions. When $2\mathrm{\alpha p}<q<p^{\ast }$, we establish the existence of nontrivial solutions by using Mountain-Pass lemma; when $q\ge 2\alpha p^{\ast }$, by using a Pohozaev type variational identity, we prove that the equation has no nontrivial solutions.

Keywords:

• Quasilinear elliptic equations
• existence and non-existence
• variational methods
• cerami sequence
• critical points

AMS Subject Classifications:

• 35J20
• 35J62
• 35J92
• 35Q55

COMMUNICATED BY:

• D. Repov$\breve{\mathrm{s}}$

Disclosure statement

No potential conflict of interest was reported by the author(s).