Existence and concentration of positive solutions for a fractional Schrödinger logarithmic equation

ABSTRACT

In this paper, we study the existence and concentration of positive solutions for the following fractional Schrödinger logarithmic equation: $\{\begin{array}{l}{\epsilon }^{2s}(-\mathrm{\Delta }{)}^{s}u+V(x)u=u\log u^2,x\in {\mathbb{R}}^N,\\ u\in H^s({\mathbb{R}}^N),\end{array}$ where $\epsilon >0$ is a small parameter, $N>2s,$ $s\in (0,1),(-\mathrm{\Delta }{)}^s$ 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 ε tends to zero.

KEYWORDS:

• Fractional Schrödinger logarithmic problem
• variational methods
• positive solutions

AMS SUBJECT CLASSIFICATIONS:

• 35J10
• 35R11
• 35A15
• 35B09

Acknowledgements

The authors would like to express their sincere gratitude to one anonymous referee for his/her constructive comments for improving the quality of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported by National Natural Science Foundation of China [No. 12161038], Natural Science Foundation program of Jiangxi Provincial [20202BABL211005].