Pairs of positive solutions for [formula omitted]-Laplacian equations with sublinear and superlinear nonlinearities which do not satisfy the AR-condition

Abstract

We consider a nonlinear Dirichlet problem driven by the p -Laplacian differential. The right-hand-side nonlinearity, exhibits a ( p − 1 ) -sublinear term of the form m ( z ) | x | r − 2 x , r < p (concave term), and a Carathéodory term f ( z , x ) which is ( p − 1 ) -superlinear near + ∞ . However, it does not satisfy the usual Ambrosetti–Rabinowitz condition (AR-condition). Instead we employ a more general condition. Using a variational approach based on the critical point theory and the Ekeland variational principle, we show the existence of two nontrivial positive smooth solutions and then the existence of two nontrivial negative smooth solutions.