Laplace’s equation with concave and convex boundary nonlinearities on an exterior region

Abstract

This paper studies Laplace's equation −∆ u = 0 in an exterior region U R N , when N ≥ 3, subject to the nonlinear boundary condition ∂u ∂ν = λ |u| q−2 u + µ |u| p−2 u on ∂U with 1 < q < 2 < p < 2 * . In the function space H (U ), one observes when λ > 0 and µ ∈ R arbitrary, then there exists a sequence {u k } of solutions with negative energy converging to 0 as k → ∞; on the other hand, when λ ∈ R and µ > 0 arbitrary, then there exists a sequence {ũ k } of solutions with positive and unbounded energy. Also, associated with the p-Laplacian equation −∆ p u = 0, the exterior p-harmonic Steklov eigenvalue problems are described.N −2 is the critical Sobolev index and ∇u := (D 1 u, D 2 u, . . . , D N u) is the weak gradient of u.A region is a nonempty, open, connected subset U of R N , and is said to be an exterior region provided that its complement R N \ U is a nonempty, compact subset. Without loss of generality, we simply assume that 0 / ∈ U. The boundary of a set A is denoted by ∂A.Our general assumption on U is the following condition.Condition B1. U R N is an exterior region, with 0 / ∈ U, whose boundary ∂U is the union of finitely many disjoint, closed, Lipschitz surfaces, each of finite surface area.One may want to notice here that the following prototypical problemin Ω has originally been investigated by Ambrosetti, Brézis and Cerami [1] in 1994 andthen in 1995 by Bartsch and Willem [7], in the function space H 1 0 (Ω) on a bounded region Ω with a smooth boundary ∂Ω. Since then, there has been a large number of papers appearing on some related Key words and phrases. Exterior regions, Laplace operator, Concave and convex mixed nonlinear boundary conditions, Fountain theorems, Steklov eigenvalue problems.

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