Fractional p-Laplacian elliptic problems with sign changing nonlinearities via the nonlinear Rayleigh quotient

It is established existence and multiplicity of solutions to the fractional p-Laplacian problem in the whole space R-N. More precisely, we consider the nonlocal elliptic problem with sign changing nonlinearities in the following form: {(-Delta)(p)(s) u + V (x) vertical bar u vertical bar(p-2) u = lambda f(x) |u|(q-2) u + g(x) |u|(r-2) u in R-N, u is an element of W-s,W-p(R-N), where lambda is an element of (gamma*, 0) boolean OR (0, lambda*), gamma* < 0, lambda* > 0 and N > ps with s is an element of (0,1) fixed. Furthermore, we assume that 1 < q < p < r < p(s)* = Np/(N - ps). The potential V is a continuous function which is bounded from below by a positive constant. The main objective here is to consider nonlinearities f and g that can be sign changing functions. In this case, by using the nonlinear Rayleigh quotient, we prove that our main problem has at least two nontrivial solutions for each lambda is an element of (gamma*, 0) boolean OR (0, lambda*). More specifically, the numbers lambda* > 0 and gamma* < 0 are sharp in order to consider the Nehari method, that is, the number lambda* is the largest positive number such that the Nehari method can be applied for each lambda is an element of (0, lambda*). The same nassertion is verified for gamma*, that is, the number gamma* < 0 is the smallest negative number such that the Nehari method can be employed for each lambda is an element of (gamma*, 0). (c) 2023 Elsevier Inc. All rights reserved.