Lower and upper solutions for elliptic problems in nonsmooth domains (original) (raw)

Existence of Strictly Positive Solutions for Sublinear Elliptic Problems in Bounded Domains

Advanced Nonlinear Studies, 2014

Let Ω be a smooth bounded domain in RN and let m be a possibly discontinuous and unbounded function that changes sign in Ω. Let f : [0,∞) → [0,∞) be a nondecreasing continuous function such that k1ξp ≤ f (ξ) ≤ k2ξp for all ξ ≥ 0 and some k1, k2 > 0 and p ∈ (0, 1). We study existence and nonexistence of strictly positive solutions for nonlinear elliptic problems of the form −Δu = m(x) f (u) in Ω, u = 0 on ∂Ω.

Existence of two nontrivial solutions for semilinear elliptic problems

Electronic Journal of Differential Equations, 2006

This paper concerns the existence of multiple nontrivial solutions for some nonlinear problems. The first nontrivial solution is found using a minimax method, and the second by computing the Leray-Schauder index and the critical group near 0.

Existence of Positive Bounded Solutions of Semilinear Elliptic Problems

International Journal of Differential Equations, 2010

This paper is concerned with the existence of bounded positive solution for the semilinear elliptic problemΔu=λp(x)f(u)inΩsubject to some Dirichlet conditions, whereΩis a regular domain inℝn (n≥3)with compact boundary. The nonlinearityfis nonnegative continuous and the potentialpbelongs to some Kato classK(Ω). So we prove the existence of a positive continuous solution depending onλby the use of a potential theory approach.

Multiplicity of positive weak solutions to subcritical singular elliptic Dirichlet problems

Electronic Journal of Qualitative Theory of Differential Equations, 2017

We study a superlinear subcritical problem at infinity of the form −∆u = a (x) u −α + f (λ, x, u) in Ω, u = 0 on ∂Ω, u > 0 in Ω, where Ω is a bounded domain in R n , 0 ≤ a ∈ L ∞ (Ω) , and 0 < α < 3. Under suitable assumptions on f , we prove that there exists Λ > 0 such that this problem has at least one weak solution in H 1 0 (Ω) if and only if λ ∈ [0, Λ] ; and also that there exists Λ * such that for any λ ∈ (0, Λ *), at least two solutions exist.

On the extremal solutions of semilinear elliptic problems

Abstract and Applied Analysis, 2005

We investigate here the properties of extremal solutions for semilinear elliptic equation −∆u = λ f (u) posed on a bounded smooth domain of R n with Dirichlet boundary condition and with f exploding at a finite positive value a. 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. (C 1) For any λ ∈ (0,λ *), there exists a positive, minimal, classical solution u λ ∈ C 2 (Ω). The function u λ is minimal in the following sense: for every solution u of (P λ), we have u λ ≤ u on Ω. In addition, the function λ → u λ is increasing and λ 1 (−∆ − λ f (u λ)) > 0, for example, for any ϕ ∈ H 1 0 (Ω)\{0}, λ Ω f u λ ϕ 2 dx < Ω |∇ϕ| 2 dx. (1.3) (C 2) For any λ > λ * , there exists no classical solution for (P λ).

Positive solutions to sublinear elliptic problems

Colloquium Mathematicum, 2018

Let L be a second order elliptic operator L with smooth coefficients defined on a domain Ω in R d , d ≥ 3, such that L1 ≤ 0. We study existence and properties of continuous solutions to the following problem (0.1) Lu = ϕ(•, u), in Ω, where Ω is a Greenian domain for L (possibly unbounded) in R d and ϕ is a nonnegative function on Ω × [0, +∞[ increasing with respect to the second variable. By means of thinness, we obtain a characterization of ϕ for which (0.1) has a nonnegative nontrivial bounded solution.