PARAMETRIC p-LAPLACIAN EQUATIONS WITH SUPERLINEAR REACTIONS

We consider a parametric nonlinear Dirichlet problem driven by the p-Laplacian and with a Carath´eodory reaction which is (p − 1)-superlinear near ±∞ (but without satisfying the Ambrosetti-Rabinowitz condition) and (p − 1)-sublinear near zero. We show that for all values of the parameter λ > 0, the problem has at least three nontrivial solutions (two of constant sign). If we alter the geometry near the origin by introducing a “concave” nonlinearity (problem with combined nonlinearities), we show the existence of at least five nontrivial solutions (four of constant sign and the fifth nodal), when the parameter λ > 0 is small. Also, we produce extremal constant sign solutions u^∗_λ ∈ -int C₊ and v^∗_λ ∈ −int C+. We investigate the monotonicity and continuity properties of the map λ→ u^∗_λ .