Ground state solutions for a quasilinear elliptic equation with general critical nonlinearity

Abstract

In this article, we study the quasilinear elliptic equation: $-\mathrm{\Delta }u+V\left(x\right)u-\mathrm{\Delta }\left[\right(1+u^2{)}^{1/2}]\frac{u}{2(1+u^2{)}^{1/2}}=h(u),x\in {\mathbb{R}}^N,$ where $N\ge 3$, $V:{\mathbb{R}}^N\to \mathbb{R}$ satisfies suitable assumptions. Unlike $h\in C^1(\mathbb{R},\mathbb{R})$, we only need to assume that $h\in C(\mathbb{R},\mathbb{R})$. By using a change of variable, we obtain the existence of ground state solutions with general critical growth.

COMMUNICATED BY:

• D. Repov$\breve{s}$

Keywords:

• Quasilinear elliptic equation
• critical growth
• ground state solution

AMS SUBJECT CLASSIFICATIONS:

• 35J50
• 35J15

Disclosure statement

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

Additional information

Funding

This work is supported in part by National Natural Science Foundation of China [grant number 11001274].