On the instability of positive solution of an elliptic equation

Electronic Journal of Differential Equations

In this article, we study a class of nonlinear elliptic systems in regular domains of R n (n ≥ 3) with compact boundary. More precisely, we prove the existence of bounded positive continuous solutions to the system ∆u = λf (., u, v), ∆v = µg(., u, v), subject to some Dirichlet conditions. Our approach is essentially based on properties of functions in a Kato class K ∞ (D) and the Schauder fixed point theorem.

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.

Proceedings of the American Mathematical Society, 2021

We consider non-negative distributional solutions u ∈ C 1 ( B R ¯ ) u\in C^1 (\overline {B_R } ) to the equation − div ⁡ [ g ( | ∇ u | ) | ∇ u | − 1 ∇ u ] = f ( | x | , u ) -\operatorname {div} [g(|\nabla u|)|\nabla u|^{-1} \nabla u ] = f(|x|,u) in a ball B R B_R , with u = 0 u=0 on ∂ B R \partial B_R , where f f is continuous and non-increasing in the first variable and g ∈ C 1 ( 0 , + ∞ ) ∩ C [ 0 , + ∞ ) g\in C^1 (0,+\infty )\cap C[0, +\infty ) , with g ( 0 ) = 0 g(0)=0 and g ′ ( t ) > 0 g’(t)>0 for t > 0 t>0 . According to a result of the first author, the solutions satisfy a certain ‘local’ type of symmetry. Using this, we first prove that the solutions are radially symmetric provided that f f satisfies appropriate growth conditions near its zeros. In a second part we study the autonomous case, f = f ( u ) f=f(u) . The solutions of the equation are critical points for an associated variation problem. We show under rather mild conditions that global and local minimize...

Nonlinear Analysis: Theory, Methods & Applications, 1999

Abstract and Applied Analysis, 2006

LetDbe a bounded domain inℝn(n≥2). We consider the following nonlinear elliptic problem:Δu=f(⋅,u)inD(in the sense of distributions),u|∂D=ϕ, whereϕis a nonnegative continuous function on∂Dandfis a nonnegative function satisfying some appropriate conditions related to some Kato class of functionsK(D). Our aim is to prove that the above problem has a continuous positive solution bounded below by a fixed harmonic function, which is continuous onD¯. Next, we will be interested in the Dirichlet problemΔu=−ρ(⋅,u)inD(in the sense of distributions),u|∂D=0, whereρis a nonnegative function satisfying some assumptions detailed below. Our approach is based on the Schauder fixed-point theorem.

2003

In this paper, we study the existence and regularity of positive solution for an elliptic system on a bounded and regular domain. The non linearities in this equation are functions of Caratheodory type satisfying some exponential growth conditions. * Mathematics Subject Classifications: 35J20, 35J45, 35J50, 35J70.

Comptes Rendus Mathematique, 2007

We study the behavior near x0 of any positive solution of (E) −∆u = u q in Ω which vanishes on ∂Ω \ {x0}, where Ω ⊂ R N is a smooth domain, q ≥ (N + 1)/(N − 1) and x0 ∈ ∂Ω. Our results are based upon a priori estimates of solutions of (E) and existence, non-existence and uniqueness results for solutions of some nonlinear elliptic equations on the upper-half unit sphere. To cite this article:

Proceedings of the American Mathematical Society, 1998

In the present paper, we study the existence of solutions to the problem    ∆u + f (x, u) = 0 in D u > 0 i nD u= 0 on ∂D where D is an unbounded domain in R 2 with a compact nonempty boundary ∂D consisting of finitely many Jordan curves. The goal is to prove an existence theorem for the above problem in a general setting by using Brownian path integration and potential theory.

Differential and Integral Equations

This paper is devoted to the study of the structure of positive radial solutions for the following semi-linear equation: $$\Delta u + f(u,|x|)=0 .$$ We require $f$ to be nonnegative and to exhibit both subcritical and supercritical behavior with respect to the Sobolev critical exponent. More precisely we assume that $f$ is subcritical for $u$ small and $|x|$ large and supercritical for $u$ large and $|x|$ small, and we give existence and non-existence results for ground states regular and singular, with either fast or slow decay. We find a surprisingly rich structure, which is characterized by two different patterns of bifurcations. We perform a Fowler transformation and we use a dynamical approach, exploiting some ideas borrowed from Bamon, Del Pino, and Flores, combining them with the use of the translation of the Pohozaev function for this dynamical context.

Applicable Analysis, 1989