Concave-convex critical problems for the spectral fractional laplacian with mixed boundary conditions

In this work we study the existence of solutions to the following critical fractional problem with concave-convex nonlinearities, {(-delta)(s)u = lambda u(q) + u(2s & lowast; -1), u > 0 in omega,u = 0 on sigma(D),& part;u/& part;nu = 0 on sigma(N), with omega subset of R-N, N > 2s, a smooth bounded domain, 1/2 < s < 1, 0 < q < 2(s)* - 1, q &NOTEQUexpressionL; 1, being 2(s)* = 2N/n-2s the critical fractional Sobolev exponent, lambda > 0, nu is the outwards normal to & part;omega; sigma(D), sigma(N) are smooth (N - 1)-dimensional submanifolds of & part;omega such that sigma(D )boolean OR sigma(N )= & part;omega, sigma(D )& cap; sigma(N) = empty set , and sigma(D) & cap; (sigma)over bar(N) = gamma is a smooth (N - 2)-dimensional submanifold of & part;omega. In particular, we will prove that, for the sublinear case 0 < q < 1, there exists at least two solutions for every 0 < lambda < lambda for certain lambda is an element of R while, for the superlinear case 1 < q < 2(s)& lowast; - 1, we will prove that there exists at least one solution for every lambda > 0. We will also prove that solutions are bounded.