Ground states for fractional Schrödinger equations involving critical or supercritical exponent

Abstract

In this paper, we study the following fractional Schrödinger equation involving critical or supercritical exponent $(-\mathrm{\Delta }{)}^{s}u+V(x)u=\lambda |u{|}^{p-2}u+f(x,u),x\in {\mathbb{R}}^N,$where 0<s<1, N>2s, $2_s^{\ast }=\frac{2N}{N-2s}$, $p\ge 2_s^{\ast }$, $\lambda >0$, $(-\mathrm{\Delta }{)}^s$ denotes the fractional Laplacian of order s and f is a continuous superlinear but subcritical nonlinearity. When V and f are asymptotically periodic in x, we prove that the equation has a ground state solution for small $\lambda >0$ by the Nehari method. Our main contribution is that we are able to deal with the supercritical case $p>2_s^{\ast }$.

Keywords:

• Fractional Schrödinger equation
• critical or supercritical exponent
• ground state solutions

1991 Mathematics Subject Classifications:

• 35J20
• 35J70
• 35P05
• 35P30
• 34B15
• 58E05
• 47H04

Disclosure statement

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

Additional information

Funding

This work is supported in part by the National Natural Science Foundation of China (11801153; 12026227; 12026228; 11901514 ) and the Yunnan Province Applied Basic Research for Youths (2018FD085) and the Yunnan Province Applied Basic Research for General Project (2019FB001) and Youth Outstanding-notch Talent Support Program in Yunnan Province and Technology Innovation Team of University in Yunnan Province.