Positive solutions for a class of quasilinear problems with indefinite potential

Abstract We study classes of quasilinear problems involving the p -Laplacian operator and an indefinite potential with homogeneous Dirichlet boundary condition. The nonlinear reaction term satisfies superlinear subcritical growth at infinity as well as Ambrosetti and Rabinowitz type condition. Independent of the behavior of the nonlinearity near the origin, we establish the existence of positive solutions for a sufficiently small multiplicative parameter of the reaction term. If the nonlinearity is positive, we prove the existence of at least two distinct positive solutions when this parameter is small. We also provide a sufficient condition on the nonlinearity to guarantee the nonexistence of positive solutions when this parameter is large. We use bootstrap arguments, mountain pass theorem, sub- and supersolution method, Sturm theorem, and Picone’s identity to establish our results.