Asymmetric, Noncoercive, Superlinear (p,2)-Equations

We examine a nonlinear nonhomogeneous Dirichlet problem driven by the sum of a

-Laplacian

(p\geq 2) ( p ≥ 2 )

and a Laplacian (a

(p,2) ( p , 2 )

-equation). The reaction term is asymmetric and it is superlinear in the positive direction and sublinear in the negative direction. The superlinearity is not expressed using the Ambrosetti-Rabinowitz condition, while the asymptotic behavior as

x\rightarrow-\infty x → − ∞

permits resonance with respect to any nonprincipal eigenvalue of

(-\Delta_p,W^{1,p}_{0}(\Omega)) ( − Δ p , W 0 1 , p ( Ω ) )

. Using variational methods based on the critical point theory and Morse theory (critical groups), we prove a multiplicity theorem producing three nontrivial solutions.