On the extremal solutions of semilinear elliptic problems

Lamia Ben Chaabane

doi:10.1155/aaa.2005.1

Abstract

We investigate here the properties of extremal solutions for semilinear elliptic equation −Δu = λf(u) posed on a bounded smooth domain of ℝn with Dirichlet boundary condition and with f exploding at a finite positive value a.

Highlights

We consider the following semilinear elliptic problem: (Pλ)−∆u = λ f (u) in Ω, u > 0 in Ω, (1.1)u = 0 on ∂Ω, where λ > 0, Ω ⊂ Rn is a bounded smooth domain and f satisfies the following condition: (H) f is a C2 positive nondecreasing convex function on [0, ∞) such that lim t→+∞ f (t) t = +∞. (1.2)It is well known that under this condition (H), there exists a critical positive value λ∗ ∈(0,∞) for the parameter λ such that the following holds. (C1) For any λ ∈ (0, λ∗), there exists a positive, minimal, classical solution uλ ∈ C2(Ω ).The function uλ is minimal in the following sense: for every solution u of (Pλ), we have uλ ≤ u on Ω
We investigate here the properties of extremal solutions for semilinear elliptic equation −∆u = λ f (u) posed on a bounded smooth domain of Rn with Dirichlet boundary condition and with f exploding at a finite positive value a
In [3], Brezis et al have introduced a notion of weak solution as follows: we say u is a weak solution for (Pλ) if u ∈ L1(Ω), u ≥ 0, f (u)δ ∈ L1(Ω) with δ(x) = dist(x, ∂Ω), and u(−∆ξ)dx = λ f (u)ξ dx, Ω

Summary

Introduction

U = 0 on ∂Ω, where λ > 0, Ω ⊂ Rn is a bounded smooth domain and f satisfies the following condition: (H) f is a C2 positive nondecreasing convex function on [0, ∞) such that lim t→+∞. It is well known that under this condition (H), there exists a critical positive value λ∗ ∈. (0,∞) for the parameter λ such that the following holds. (C1) For any λ ∈ (0, λ∗), there exists a positive, minimal, classical solution uλ ∈ C2(Ω ). The function uλ is minimal in the following sense: for every solution u of (Pλ), we have uλ ≤ u on Ω. Ω (C2) For any λ > λ∗, there exists no classical solution for (Pλ)

On the extremal solutions of semilinear elliptic problems

Application