The convergence of nonnegative solutions for the family of problems − Δpu = λeu as p

• Published in: ESAIM. Control, optimisation and calculus of variations Vol. 24; no. 2; pp. 569 - 578
• Main Authors: Mihăilescu, Mihai, Stancu−Dumitru, Denisa, Varga, Csaba
• Format: Journal Article
• Language: English
• Published: Les Ulis EDP Sciences 01.04.2018
• Subjects: 35D30; 35D40; 35J60; 46E30; 47J30; asymptotic behavior; Asymptotic properties; Boundary conditions; Convergence; Dirichlet problem; distance function to the boundary; Mathematical analysis; nonlinear elliptic equations; Smooth boundaries; Weak solutionviscosity solution
• ISSN: 1292-8119; 1262-3377
• DOI: 10.1051/cocv/2017048

Let Ω ⊂ ℝN (N ≥ 2) be a bounded domain with smooth boundary. We show the existence of a positive real number λ* such that for each λ ∈ (0, λ*) and each real number p > N the equation −Δp u = λeu in Ω subject to the homogeneous Dirichlet boundary condition possesses a nonnegative solution up. Next, we analyze the asymptotic behavior of up as p → ∞ and we show that it converges uniformly to the distance function to the boundary of the domain.