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

Abstract

It is established existence and multiplicity of solutions to the fractional p-Laplacian problem in the whole space RN. More precisely, we consider the nonlocal elliptic problem with sign changing nonlinearities in the following form:{(−Δ)psu+V(x)|u|p−2u=λf(x)|u|q−2u+g(x)|u|r−2uinRN,u∈Ws,p(RN), where λ∈(γ⁎,0)∪(0,λ⁎),γ⁎<0, λ⁎>0 and N>ps with s∈(0,1) fixed. Furthermore, we assume that 1<q<p<r<ps⁎=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 λ∈(γ⁎,0)∪(0,λ⁎). More specifically, the numbers λ⁎>0 and γ⁎<0 are sharp in order to consider the Nehari method, that is, the number λ⁎ is the largest positive number such that the Nehari method can be applied for each λ∈(0,λ⁎). The same assertion is verified for γ⁎, that is, the number γ⁎<0 is the smallest negative number such that the Nehari method can be employed for each λ∈(γ⁎,0).