Existence of solutions for perturbed fractional p-Laplacian equations

Abstract

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:(−Δ)psu+V(x)|u|p−2u=λa(x)|u|r−2u−b(x)|u|q−2uin RN, where λ is a real parameter, (−Δ)ps is the fractional p-Laplacian operator with 0<s<1<p<∞, p<r<min⁡{q,ps⁎} and V,a,b:RN→(0,∞) are three positive weights. Using variational methods, we obtain nonexistence and multiplicity results for the above-mentioned equations depending on λ and according to the integrability properties of the ratio aq−p/br−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.