Nonlocal critical exponent singular problems under mixed Dirichlet-Neumann boundary conditions

In this paper, we study the following singular problem, under mixed Dirichlet-Neumann boundary conditions, and involving the fractional Laplacian {(-Delta)(s)u = lambda u(-q) + u(2s)*(-1), u > 0 in Omega, (P-lambda) A(u) = 0 on partial derivative Omega = Sigma(D) boolean OR Sigma(N), where Omega R-N is a bounded domain with smooth boundary partial derivative Omega, 1/2 < s < 1, lambda. > 0 is a real parameter, 0 < q< 1, N> 2s, 2(s)* = 2N/(N - 2s) and A(u) = u chi(Sigma D) + partial derivative(v)u chi(Sigma N) partial derivative(v) = partial derivative/partial derivative v. Here Sigma(D), Sigma(N) are smooth (N - 1) dimensional submanifolds of partial derivative Omega such that Sigma(D) boolean OR Sigma(N) = partial derivative Omega, Sigma(D) boolean AND Sigma(N) = empty set and Sigma(D) boolean AND (Sigma(N)) over bar = tau' is a smooth (N - 2) dimensional submanifold of partial derivative Omega. Within a suitable range of lambda, we establish existence of at least two opposite energy solutions for (P-lambda) using the standard Nehari manifold technique. (c) 2023 Elsevier Inc. All rights reserved.