Existence and concentration of positive solutions for a fractional Schrödinger logarithmic equation. (English) Zbl 1532.35198
Summary: In this paper, we study the existence and concentration of positive solutions for the following fractional Schrödinger logarithmic equation:
\[
\begin{cases}
\varepsilon^{2s} (-\Delta)^s u+V(x)u =u\log u^2,\:\; x\in \mathbb{R}^N,\\
u\in H^s(\mathbb{R}^N),
\end{cases}
\]
where \(\varepsilon > 0\) is a small parameter, \( N> 2s\), \(s \in ( 0 ,1)\), \((-\Delta )^s\) is the fractional Laplacian, the potential \(V\) is a continuous function having a global minimum. Using variational method to modify the nonlinearity with the sum of a \(C^1\) functional and a convex lower semicontinuous functional, we prove the existence of positive solutions and concentration around of a minimum point of \(V\) when \(\varepsilon\) tends to zero.
MSC:
| 35J61 | Semilinear elliptic equations |
| 35R11 | Fractional partial differential equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A15 | Variational methods applied to PDEs |
Keywords:
fractional Schrödinger logarithmic equation; existence of positive solutions; variational methodsReferences:
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