### Infinitely many solutions to quasilinear elliptic equation with concave and convex terms

#### Abstract

In this paper, we are concerned with the following quasilinear elliptic equation with concave and convex terms

$$

-\Delta u-{\frac12}u\Delta(|u|^2)=\alpha|u|^{p-2}u+\beta|u|^{q-2}u,\quad x\in

\Omega,

\leqno(\rom{P})

$$%

where $\Omega\subset\mathbb{R}^N$ is a bounded smooth domain,

$1< p< 2$, $4< q\leq 22^*$. The existence of infinitely many solutions

is obtained by the perturbation methods.

$$

-\Delta u-{\frac12}u\Delta(|u|^2)=\alpha|u|^{p-2}u+\beta|u|^{q-2}u,\quad x\in

\Omega,

\leqno(\rom{P})

$$%

where $\Omega\subset\mathbb{R}^N$ is a bounded smooth domain,

$1< p< 2$, $4< q\leq 22^*$. The existence of infinitely many solutions

is obtained by the perturbation methods.

#### Keywords

Concave and convex terms; fountain theorem; perturbation methods

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