Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells

Abstract

The purpose of this paper is to investigate the ground state solutions for the following nonlinear Schrödinger equations involving the fractional p-Laplacian (−Δ)psu(x)+λV(x)u(x)p−1=u(x)q−1,u(x)≥0,x∈RN,{\left(-\Delta )}_{p}^{s}u\left(x)+\lambda V\left(x)u{\left(x)}^{p-1}=u{\left(x)}^{q-1},\hspace{1em}u\left(x)\ge 0,\hspace{0.33em}x\in {{\mathbb{R}}}^{N}, where λ>0\lambda \gt 0 is a parameter, 1<p<q<NpN−sp1\lt p\lt q\lt \frac{Np}{N-sp}, N≥2N\ge 2, and V(x)V\left(x) is a real continuous function on RN{{\mathbb{R}}}^{N}. For λ\lambda large enough, the existence of ground state solutions are obtained, and they localize near the potential well int(V−1(0))\left({V}^{-1}\left(0)).

Keywords

• nonlinear schrödinger equation
• ground state solution
• fractional p-laplacian
• variational methods
• 35j60
• 35b33