Marius Ghergu

Sublinear singular elliptic problems with two parameters

Abstract We establish several existence and nonexistence results for the boundary value problem −Δu+K(x)g(u)=λf(x,u)+μh(x) in Ω , u=0 on ∂Ω , where Ω is a smooth bounded domain in R N , λ and μ are positive parameters, h is a positive function, while f has a sublinear growth. The main feature of this paper is that the nonlinearity g is assumed to be unbounded around the origin. Our analysis shows the importance of the role played by the decay rate of g combined with the signs of the extremal values of the potential K(x) on Ω . The proofs are based on various techniques related to the maximum principle for elliptic equations.

Multi-parameter bifurcation and asymptotics for the singular Lane-Emden-Fowler equation with a convection term

to the second variable and g is unbounded around the origin. The asymptotic behaviour of the solution around the bifurcation point is also established, provided g(u) behaves like u −α around the origin, for some 0 <α< 1. Our approach relies on finding explicit sub- and supersolutions combined with various techniques related to the maximum principle for elliptic equations. The analysis we develop in this paper shows the key role played by the convection term |∇u|p.

On a class of sublinear singular elliptic problems with convection term

Abstract We establish several results related to existence, nonexistence or bifurcation of positive solutions for the boundary value problem − Δ u + K ( x ) g ( u ) + | ∇ u | a = λ f ( x , u ) in Ω, u = 0 on ∂Ω, where Ω ⊂ R N ( N ⩾ 2 ) is a smooth bounded domain, 0 a ⩽ 2 , λ is a positive parameter, and f is smooth and has a sublinear growth. The main feature of this paper consists in the presence of the singular nonlinearity g combined with the convection term | ∇ u | a . Our approach takes into account both the sign of the potential K and the decay rate around the origin of the singular nonlinearity g. The proofs are based on various techniques related to the maximum principle for elliptic equations.

Bifurcation and asymptotics for the Lane–Emden–Fowler equation

We are concerned with the Lane–Emden–Fowler equation −Δu=λf(u)+a(x)g(u) in Ω, subject to the Dirichlet boundary condition u=0 on ∂Ω, where Ω⊂RN is a smooth bounded domain, λ is a positive parameter, a:Ω→[0,∞) is a Holder function, and f is a positive nondecreasing continuous function such that f(s)/s is nonincreasing in (0,∞). The singular character of the problem is given by the nonlinearity g which is assumed to be unbounded around the origin. In this Note we discuss the existence and the uniqueness of a positive solution of this problem and we also describe the precise decay rate of this solution near the boundary. The proofs rely essentially on the maximum principle and on elliptic estimates. To cite this article: M. Ghergu, V.D. Rădulescu, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Explosive solutions of elliptic equations with absorption and nonlinear gradient term

AbstractLetf be a non-decreasing C1-function such that

A singular Gierer-Meinhardt system with different source terms

f > 0 on (0,\infty ), f(0) = 0, \int_1^\infty {1/\sqrt {F(t)} dt< \infty }

TURING PATTERNS IN GENERAL REACTION-DIFFUSION SYSTEMS OF BRUSSELATOR TYPE

andF(t)/f2a(t)→ 0 ast → ∞, whereF(t)=∫0tf(s) ds anda ∈ (0, 2]. We prove the existence of positive large solutions to the equationΔu +q(x)|Δu|a=p(x)f(u) in a smooth bounded domain Ω ⊂RN, provided thatp, q are non-negative continuous functions so that any zero ofp is surrounded by a surface strictly included in Ω on whichp is positive. Under additional hypotheses onp we deduce the existence of solutions if Ω is unbounded.

Steady-state solutions for Gierer-Meinhardt type systems with Dirichlet boundary condition

We are concerned with the following stationary system: subject to homogeneous Neumann boundary conditions. Here (N ≥ 1) is a smooth and bounded domain and a, b, m, λ, θ are positive parameters. The particular case m = 2 corresponds to the steady-state Brusselator system. We establish existence and non-existence results for non-constant positive classical solutions. In particular, we provide upper and lower bounds for solutions which allows us to extend the previous works in the literature without any restriction on the dimension N ≥ 1. Our analysis also emphasizes the role played by the nonlinearity um. The proofs rely essentially on various types of a priori estimates.