Asymmetric (p,2)-equations, superlinear at +∞, resonant at −∞

Abstract

We consider a Dirichlet problem driven by the sum of a p-Laplacian and a Laplacian (known as a (p,2)-equation) and with a nonlinearity which exhibits asymmetric behavior as s→±∞. More precisely, it is (p−1)-superlinear near +∞ (but without satisfying the Ambrosetti–Rabinowitz condition) and it is (p−1)-sublinear near −∞ and possibly resonant with respect to the principal eigenvalue of the p-Laplacian. Using variational tools along with Morse theory we prove a multiplicity theorem generating five nontrivial solutions (one is negative, two are positive, one is nodal and for the fifth we do not have any information about its sign).