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 ) p s u + V ( x ) | u | p − 2 u = λ a ( x ) | u | r − 2 u − b ( x ) | u | q − 2 u in  R N , where λ is a real parameter, ( − Δ ) p s is the fractional p -Laplacian operator with 0 s 1 p ∞ , p r min ⁡ { q , p s ⁎ } and V , a , b : R N → ( 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 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.