Quasilinear Schrödinger equations with concave and convex nonlinearities

Abstract

In this paper, we consider the following quasilinear Schrödinger equation \begin{align*} -Δu-uΔ(u^{2})=k(x)\left\vert u\right\vert ^{q-2}u-h(x)\left\vert u\right\vert ^{s-2}u\text{, }u\in D^{1,2}(\mathbb{R}^{N})\text{,} \end{align*} where $1<q<2<s<+\infty$. Unlike most results in the literature, the exponent $s$ here is allowed to be supercritical $s>2\cdot 2^{\ast }$. By taking advantage of geometric properties of a nonlinear transformation $f$ and a variant of Clark's theorem, we get a sequence of solutions with negative energy in a space smaller than $D^{1,2}({\mathbb{R}}^N)$. Nonnegative solution at negative energy level is also obtained.