Positive solutions for a weighted critical problem with mixed boundary conditions

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.