Abstract

This paper is concerned with the radial symmetry weak positive solutions for a class of singular fractional Laplacian. The main results in the paper demonstrate the existence and multiplicity of radial symmetry weak positive solutions by Schwarz spherical rearrangement, constrained minimization, and Ekeland’s variational principle. It is worth pointing out that our results extend the previous works of T. Mukherjee and K. Sreenadh to a setting in which the testing functions need not have a compact support. Moreover, we weakened one of the conditions used in their papers. Our results improve on existing studies on radial symmetry solutions of nonlocal boundary value problems.

Highlights

  • We focus on radial symmetry positive solutions to a singular elliptic problem involving a nonlocal operator: the fractional powers of the Laplacian in a bounded sphere domain in BR (0) ⊆ R N ( N ≥ 3)

  • We are interested in the existence of radial symmetry weak positive solutions that satisfy the singular fractional Laplacian boundary value problem

  • This paper is concerned with the radial symmetry weak positive solutions for a class of singular fractional Laplacian

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Summary

Introduction

We focus on radial symmetry positive solutions to a singular elliptic problem involving a nonlocal operator: the fractional powers of the Laplacian in a bounded sphere domain in BR (0) ⊆ R N ( N ≥ 3). We are interested in the existence of radial symmetry weak positive solutions that satisfy the singular fractional Laplacian boundary value problem,. The study of elliptic problems with a singular nonlinearity has attracted many researchers of partial differential equations ([23,24,25] and the references therein). In [24], the authors studied the existence, regularity, and multiplicity of weak solutions for fractional p-Laplacian equations with singular nonlinearities via fibering maps. Sire [25] provided new results with respect to the existence and regularity of radial extremal solutions for some nonlocal problems with smooth nonlinearity by following the s-harmonic extension approach, as in [3].

Preliminaries and Main Results
Conclusions

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