On fractional logarithmic Schrödinger equations

Abstract

We study the following fractional logarithmic Schrödinger equation: (−Δ)su+V(x)u=ulogu2,x∈RN,{\left(-\Delta )}^{s}u+V\left(x)u=u\log {u}^{2},\hspace{1em}x\in {{\mathbb{R}}}^{N}, where N≥1N\ge 1, (−Δ)s{\left(-\Delta )}^{s} denotes the fractional Laplace operator, 0<s<10\lt s\lt 1 and V(x)∈C(RN)V\left(x)\in {\mathcal{C}}\left({{\mathbb{R}}}^{N}). Under different assumptions on the potential V(x)V\left(x), we prove the existence of positive ground state solution and least energy sign-changing solution for the equation. It is known that the corresponding variational functional is not well defined in Hs(RN){H}^{s}\left({{\mathbb{R}}}^{N}), and inspired by Cazenave (Stable solutions of the logarithmic Schrödinger equation, Nonlinear Anal. 7 (1983), 1127–1140), we first prove that the variational functional is well defined in a subspace of Hs(RN){H}^{s}\left({{\mathbb{R}}}^{N}). Then, by using minimization method and Lions’ concentration-compactness principle, we prove that the existence results.

Keywords

• fractional logarithmic schrödinger equations
• positive ground state solutions
• sign-changing solutions
• variational methods
• 35r11 (35b38 35q55 47j30)