Abstract
Abstract 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.
Highlights
In this article, we study the following fractional logarithmic Schrödinger equation:(−Δ)su + V (x)u = u logu2, x ∈ N, (1.1)where N ≥ 1, 0 < s < 1, V (x) ∈ ( N) and V0 ≔ infxV (x) > 0
We study the following fractional logarithmic Schrödinger equation: (−Δ)su + V (x)u = u logu2, x ∈ N, (1.1)
∂t which is a generalization of the following logarithmic Schrödinger equation: i ∂Φ = −ΔΦ + (V (x) + λ)Φ − Φ log∣Φ∣2, (t, x) ∈ + × N
Summary
We study the following fractional logarithmic Schrödinger equation:. ∣, up to translations, which is the unique positive solution of the logarithmic equation They proved that the same result holds for bound state solutions. Let u1 ∈ Hs( N) be the solution of the equation (−Δ)su1 + u1 = η1h(x), x ∈ N It follows from Lemma 2.4 that η1h(x) ∈ Lq( N) for all q > 2, u1 ∈ W2s,q( N) and u1 ∈ some σ0 ∈ (0, σ). (iii) Suppose u ∈ such that I(u) = m, u+, u− ≠ 0, similar to the proof of Lemmas 2.1–2.2, we conclude that there exists a unique αu+ ∈ (0, 1) such that αu+u+ ∈ , and a unique βu− ∈ (0, 1) such that βu−u− ∈.
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