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

• Published in: Boundary value problems Vol. 2019; no. 1; pp. 1 - 12
• Main Authors: Mao, Jinxiu, Zhao, Zengqin, Qian, Aixia
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 13.03.2019; Hindawi Limited; SpringerOpen
• Subjects: Analysis; Approximations and Expansions; Boundary conditions; Concave and convex mixed nonlinear boundary conditions; Difference and Functional Equations; Eigenvalues; Exterior regions; Fountain theorems; Function space; Laplace equation; Laplace operator; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Steklov eigenvalue problems; Laplace operator; 35J65; 35J20; Exterior regions; 46E22; Fountain theorems; Steklov eigenvalue problems; 49R99; Concave and convex mixed nonlinear boundary conditions
• ISSN: 1687-2770; 1687-2762; 1687-2770
• DOI: 10.1186/s13661-019-1163-7

This paper studies Laplace’s equation − Δ 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 that, 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 { u ˜ k } of solutions with positive and unbounded energy. Also, associated with the p -Laplacian equation − Δ p u = 0 , the exterior p -harmonic Steklov eigenvalue problems are described.