Quasilinear Schr\"{o}dinger equations with concave and convex nonlinearities

TL;DR

This paper studies a quasilinear Schrödinger equation with mixed nonlinearities, allowing supercritical growth, and establishes the existence of multiple solutions with negative energy using geometric and variational methods.

Contribution

It introduces a novel approach to handle supercritical nonlinearities in quasilinear Schrödinger equations and proves the existence of multiple solutions with negative energy levels.

Findings

Existence of solutions with negative energy

Solutions include nonnegative solutions at negative energy

Handling supercritical nonlinearities beyond previous literature

Abstract

In this paper, we consider the following quasilinear Schr\"{o}dinger equation \begin{align*} -\Delta u-u\Delta(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 $s>2\cdot2^{\ast}$. 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.