Laplace\u2019s equation with concave and convex boundary nonlinearities on an exterior region

Jinxiu Mao,Zengqin Zhao,Aixia Qian

doi:10.1186/s13661-019-1163-7

Abstract

This paper studies Laplace’s equation -Delta u=0 in an exterior region Uvarsubsetneq {mathbb{R}}^{N}, when Ngeq 3, subject to the nonlinear boundary condition frac{partial u}{partial nu }=lambda vert u vert ^{q-2}u+mu vert u vert ^{p-2}u on ∂U with 1< q<2<p<2_{*}. In the function space mathscr{H} (U ), one observes that, when lambda >0 and mu in mathbb{R} arbitrary, then there exists a sequence {u_{k} } of solutions with negative energy converging to 0 as kto infty ; on the other hand, when lambda in mathbb{R} and mu >0 arbitrary, then there exists a sequence {tilde{u}_{k} } of solutions with positive and unbounded energy. Also, associated with the p-Laplacian equation -Delta _{p} u=0, the exterior p-harmonic Steklov eigenvalue problems are described.

Highlights

This paper discusses the existence of infinitely many harmonic functions in an exterior region U RN when N ≥ 3, subject to a nonlinear boundary condition on ∂U that combines concave and convex terms with 1 < q < 2 < p < 2∗, described as below ⎧ ⎨– u(x) = 0 ⎩ ∂ ∂u ν (z) λ|u(z)|q–2u(z) μ|u(z)|p–2u(z) in U, on ∂U, (1.1) in the space E1 (U )of functions where u ∈ L2∗ (U)
This paper studies Laplace’s equation – u = 0 in an exterior region U RN, when
The main result of this paper is described as below

Summary

Introduction

This paper discusses the existence of infinitely many harmonic functions in an exterior region U RN when N ≥ 3, subject to a nonlinear boundary condition on ∂U that combines concave and convex terms with 1 < q < 2 < p < 2∗, described as below. There is no result for the sequence of p-harmonic Steklov eigenvalue problems on an exterior region U, so we will study this in Sect. Auchmuty and Han [3, 4, 12] recently introduced a new function space E1,p(U) suitable for the study of harmonic boundary value problems on an exterior region U which satisfies (2019) 2019:51. Theorem 2.2 ([1]) Let G be a compact group, X = j∈N X (j) a Banach space with norm · , and ψ ∈ C1(X , R) an invariant functional; for each k ∈ N, let Yk, Zk be defined as in (2.4), and ρk > k > 0 some constants. As a result, these ukl are bounded, and converge weakly without loss of generality to a function u ∈ H (U); via Lemma 2.1 again, we may assume that ukl → uin.

Proof Since t

Findings

Keep in mind