Abstract
We study a nonlinear, nonhomogeneous elliptic equation with an asymmetric reaction term depending on a positive parameter, coupled with Robin boundary conditions. Under appropriate hypotheses on both the leading differential operator and the reaction, we prove that, if the parameter is small enough, the problem admits at least four nontrivial solutions: two of such solutions are positive, one is negative, and one is sign-changing. Our approach is variational, based on critical point theory, Morse theory, and truncation techniques.
Highlights
We study the following nonlinear, nonhomogeneous Robin problem:−div a(∇u) + ξ(x)|u|p−2u = λg(x, u) + f (x, u) in Ω ∂u ∂na + β(x)|u|p−2u = (1.1) on ∂Ω.Here Ω ⊂ RN (N > 1) is a bounded domain with a C2-boundary ∂Ω, p > 1, and a : RN → RN is a continuous, monotone mapping which satisfies certain growth and regularity conditions
Nonhomogeneous elliptic equation with an asymmetric reaction term depending on a positive parameter, coupled with Robin boundary conditions
Under appropriate hypotheses on both the leading differential operator and the reaction, we prove that, if the parameter is small enough, the problem admits at least four nontrivial solutions: two of such solutions are positive, one is negative, and one is sign-changing
Summary
We study the following nonlinear, nonhomogeneous Robin problem:. (1.1) on ∂Ω. Ω ⊂ RN (N > 1) is a bounded domain with a C2-boundary ∂Ω, p > 1, and a : RN → RN is a continuous, monotone mapping ( maximal monotone too) which satisfies certain growth and regularity conditions (see hypotheses Ha below) These conditions are mild enough to include in our framework many non-linear operators of interest, such as the pLaplacian, the (p, q)-Laplacian, and the generalized mean curvature operator. Compared with the existing literature, our result is novel in a twofold sense: unlike most of the aforementioned works, our result proves existence of four nontrivial solutions with precise sign information; and it holds for a very general problem, incorporating Robin and Neumann boundary conditions and several nonlinear leading differential operators as special cases (the only exception is represented by [16], which provides four solutions but only for Dirichlet conditions and the p-Laplace operator). The paper has the following structure: in Section 2 we introduce our hypotheses and main result, and we establish some preliminary results and notations; in Section 3 we deal with constant sign solutions; and in Section 4 we investigate extremal constant sign solutions and nodal solutions
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