# The convergence of nonnegative solutions for the family of problems - Δ p u = λ eu as p → ∞

Mihai Mihǎilescu, Denisa Stancu-Dumitru, C. Varga

Research output: Contribution to journalArticle

### Abstract

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 ⋯ 2 (0, λ∗) and each real number p > N the equation -Δpu = 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 → 1 and we show that it converges uniformly to the distance function to the boundary of the domain.

Original language English 569-578 10 ESAIM - Control, Optimisation and Calculus of Variations 24 2 https://doi.org/10.1051/cocv/2017048 Published - Apr 1 2018

### Fingerprint

Nonnegative Solution
Boundary conditions
Distance Function
Dirichlet Boundary Conditions
Bounded Domain
Asymptotic Behavior
Converge
Family

### Keywords

• Asymptotic behavior
• Distance function to the boundary
• Nonlinear elliptic equations
• Weak solutionviscosity solution

### ASJC Scopus subject areas

• Control and Systems Engineering
• Control and Optimization
• Computational Mathematics

### Cite this

The convergence of nonnegative solutions for the family of problems - Δ p u = λ eu as p → ∞. / Mihǎilescu, Mihai; Stancu-Dumitru, Denisa; Varga, C.

In: ESAIM - Control, Optimisation and Calculus of Variations, Vol. 24, No. 2, 01.04.2018, p. 569-578.

Research output: Contribution to journalArticle

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