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

Abstract

Abstract This paper studies Laplace’s equation −Δu=0 $-\Delta u=0$ in an exterior region U⊊RN $U\varsubsetneq {\mathbb{R}}^{N}$, when N≥3 $N\geq 3$, subject to the nonlinear boundary condition ∂u∂ν=λ|u|q−2u+μ|u|p−2u $\frac{\partial u}{\partial \nu }=\lambda \vert u \vert ^{q-2}u+\mu \vert u \vert ^{p-2}u$ on ∂U with 10 $\lambda >0$ and μ∈R $\mu \in \mathbb{R}$ arbitrary, then there exists a sequence {uk} $\{u_{k} \}$ of solutions with negative energy converging to 0 as k→∞ $k\to \infty $; on the other hand, when λ∈R $\lambda \in \mathbb{R}$ and μ>0 $\mu >0$ arbitrary, then there exists a sequence {u˜k} $\{\tilde{u}_{k} \}$ of solutions with positive and unbounded energy. Also, associated with the p-Laplacian equation −Δpu=0 $-\Delta _{p} u=0$, the exterior p-harmonic Steklov eigenvalue problems are described.

Keywords

• Exterior regions
• Laplace operator
• Concave and convex mixed nonlinear boundary conditions
• Fountain theorems
• Steklov eigenvalue problems