Nonlinear nonhomogeneous Dirichlet equations with a superlinear reaction

Abstract

We consider a nonlinear elliptic Dirichlet equation driven by a nonlinear nonhomogeneous differential operator involving a Carath\'{e}odory reaction which is $(p-1)$-superlinear but does not satisfy the Ambrosetti-Rabinowitz condition. First we prove a three-solutions-theorem extending an earlier classical result of Wang (Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire 8 (1991), no. 1, 43--57). Subsequently, by imposing additional conditions on the reaction $f(x,\cdot)$, we produce two more nontrivial constant sign solutions and a nodal solution for a total of five nontrivial solutions. In the special case of $(p,2)$-equations we prove the existence of a second nodal solution for a total of six nontrivial solutions given with complete sign information. Finally, we study a nonlinear eigenvalue problem and we show that the problem has at least two nontrivial positive solutions for all parameters $\lambda>0$ sufficiently small where one solution vanishes in the Sobolev norm as $\lambda \to 0^+$ and the other one blows up (again in the Sobolev norm) as $\lambda \to 0^+$.