| Popis: |
We study the following fractional logarithmic Schrödinger equation: ( − Δ ) s u + V ( x ) u = u log u 2 , x ∈ R N , {\left(-\Delta )}^{s}u+V\left(x)u=u\log {u}^{2},\hspace{1em}x\in {{\mathbb{R}}}^{N}, where N ≥ 1 N\ge 1 , ( − Δ ) s {\left(-\Delta )}^{s} denotes the fractional Laplace operator, 0 < s < 1 0\lt s\lt 1 and V ( x ) ∈ C ( R N ) 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 H s ( R N ) {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 H s ( R N ) {H}^{s}\left({{\mathbb{R}}}^{N}) . Then, by using minimization method and Lions’ concentration-compactness principle, we prove that the existence results. |
