Multiple solutions for a class of fractional logarithmic Schrödinger equations

Siyuan He () and Xiaochun Liu ()
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Siyuan He: Wuhan University
Xiaochun Liu: Wuhan University

Partial Differential Equations and Applications, 2021, vol. 2, issue 6, 1-30

Abstract: Abstract In this paper, we study the following fractional logarithmic Schrödinger equation: $$\begin{aligned} (-\Delta )^s u+V(x)u=u \log u^2,\quad x \in {\mathbb {R}}^{N}, \end{aligned}$$ ( - Δ ) s u + V ( x ) u = u log u 2 , x ∈ R N , where $$s\in (0,1)$$ s ∈ ( 0 , 1 ) , $$N>2s$$ N > 2 s , and the external potential V(x) belongs to $$C({\mathbb {R}}^N)$$ C ( R N ) and is bounded from below. By the direction derivative and constrained minimization method, we obtain the existence of nonnegative and sign-changing weak solutions in $$H^s(R^N)$$ H s ( R N ) with various potentials. Moreover, we also construct a radial nodal solution, which changes sign exactly k-time (for any $$k \in {\mathbb {N}}$$ k ∈ N ) when the potential is radial symmetric.

Keywords: Fractional logarithmic Schrödinger equations; Nonlocal operator; Constrained minimization method; Nodal solutions; 35J75; 35J61 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s42985-021-00124-5

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