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 equationprobaa1 \ aligned - () ₏^{s u + V (x) |u|^p-2u u&=0, &x ^N, aligned. equation where ^N is an open bounded set with Lipschitz boundary, N 2, V L^ (^N), and (-) ₚˢ denotes the fractional p-Laplacian with s (0, 1), 10, 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.