Existence of solutions for perturbed fractional p-Laplacian equations

The purpose of this paper is to investigate the existence of weak solutions for a perturbed nonlinear elliptic equation driven by the fractional p-Laplacian operator as follows: (-Delta)(p)(s)u + V (x)vertical bar u vertical bar(p-2)u = lambda a(x)vertical bar u vertical bar(r-2)u - b(x)vertical bar u vertical bar(q-2)u in R-N, where A is a real parameter, (-Delta)(p)(s) is the fractional p-Laplacian operator with 0 < s < 1 < p < infinity, p < r < min{q, p(s)*} and V, a, b : R-N -> (0, infinity) are three positive weights. Using variational methods, we obtain nonexistence and multiplicity results for the above-mentioned equations depending on lambda and according to the integrability properties of the ratio a(q-p)/b(r-p). Our results extend the previous work of Autuori and Pucci (2013) [5] to the fractional p-Laplacian setting. Furthermore, we weaken one of the conditions used in their paper. Hence the results of this paper are new even in the fractional Laplacian case. (C) 2015 Elsevier Inc. All rights reserved.