Positive solutions for a weighted critical problem with mixed boundary conditions

TL;DR

This paper investigates the existence and multiplicity of positive solutions for a nonlocal elliptic problem with weighted critical nonlinearities, using variational methods and boundary conditions, extending classical results to the fractional Laplacian setting.

Contribution

It extends classical Dirichlet problem results to nonlocal fractional Laplacian problems with mixed boundary conditions and weighted critical nonlinearities.

Findings

Multiple positive solutions exist under certain weight function behaviors.

Explicit conditions relate the problem's parameters to solution existence.

The results generalize and improve previous classical Laplacian findings.

Abstract

We analyze the existence and multiplicity of positive solutions to a nonlocal elliptic problem involving the spectral fractional Laplace operator endowed with homogeneous mixed Dirichlet-Neumann boundary conditions and weighted critical nonlinearities. By means of variational methods and the Nehari manifold approach, we deduce the existence of multiple positive solutions under some assumptions on the behavior of the weight function around its maximum points. Such a behavior, formulated in terms of some rate growth, is explicitly determined and depends on the relation between the dimension, the order of the operator and the subcritical perturbation. In this way we extend and improve the results in "J.F. Liao, J. Liu, P. Zhang, C.L. Tang, Existence and multiplicity of positive solutions for a class of elliptic equations involving critical Sobolev exponents, RACSAM 110 (2016) 483--501", dealing with the Dirichlet problem for the classical Laplace operator, to the nonlocal setting involving mixed boundary conditions.