Multiplicity results for Schr\"odinger type fractional p-Laplacian boundary value problems

In this work, we study the existence and multiplicity of solutions for the following problem {equation}{probaa1} \{ {aligned} -(Δ)ₚˢ u + V(x)|u|ᵖ⁻²u &= λ f(u),&x∈Ω; u&=0,&x∈ NΩ, {aligned} . {equation} where Ω⊂N is an open bounded set with Lipschitz boundary ∂Ω, N 2, V∈ L∞(N), and (-Δ)ₚˢ denotes the fractional p-Laplacian with s∈(0,1), 1<p, $sp<N$, λ>0, and f:→ is a continuous function. We extend the results of Lopera { et al.} in {Lopera1} by proving the existence of a second weak solution for problem ({probaa1}). We apply a variant of the mountain-pass theorem due to Hofer {Hofer2} and infinite-dimensional Morse theory to obtain the existence of at least two solutions.