Abstract
We consider a Robin problem driven by the (p, q)-Laplacian plus an indefinite potential term. The reaction is either resonant with respect to the principal eigenvalue or (p-1)-superlinear but without satisfying the Ambrosetti-Rabinowitz condition. For both cases we show that the problem has at least five nontrivial smooth solutions ordered and with sign information. When q=2 (a (p, 2)-equation), we show that we can slightly improve the conclusions of the two multiplicity theorems.
Highlights
Let ⊆ RN be a bounded domain with a C2-boundary ∂
We study the following nonlinear, nonhomogeneous Robin problem
For every r ∈ (1, ∞), by r we denote the r -Laplace differential operator defined by r u = div |Du|r−2 Du for all u ∈ W 1,r ( )
Summary
Let ⊆ RN be a bounded domain with a C2-boundary ∂. In problem (1.1), in addition to the ( p, q)-differential operator there is a potential term ξ(z)|u|p−2u, with the potential function ξ ∈ L∞( ) being in general indefinite (that is, sign-changing). First we assume that f (z, ·) exhibits ( p − 1)-linear growth as x → ±∞ (that is, f (z, ·) is asymptotically ( p − 1)-homogeneous) In this case we permit resonance with respect to the principal eigenvalue of u → − pu+ξ(z)|u|p−2u with Robin boundary condition. We mention the very recent work of Vetro [32], where the author examines perturbations (both sublinear and superlinear) of the eigenvalue problem for the operator u → − pu + ξ(z)|u|p−2u with Robin boundary condition. We should mention the works of Amster [2], Papageorgiou-Radulescu-Repovš [22,23] and Papageorgiou-Zhang [28], where the authors deal with problems involving concave terms
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