Jorge García-Melián

Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up

In this paper we prove uniqueness of positive solutions to logistic singular problems −∆u = λ(x)u − a(x)up, u|∂Ω = +∞, p > 1, a > 0 in Ω, where the main feature is the fact that a|∂Ω = 0. More importantly, we provide exact asymptotic estimates describing, in the form of a two-term expansion, the blow-up rate for the solutions near ∂Ω. This expansion involves both the distance function d(x) = dist(x, ∂Ω) and the mean curvature H of ∂Ω.

Existence and uniqueness of positive large solutions to some cooperative elliptic systems

Abstract In this work we consider positive solutions to cooperative elliptic systems of the form -Δu = λu - u2 + buυ, -Δυ = μυ - υ2 + cuυ a bounded smooth domain Ω ⊂ ℝN (λ, μ ∈ ℝ, b, c > 0) which blow up on the boundary ∂Ω, that is u(x), v(x) → ∞ as dist(x, ∂Ω) → 0. We show existence and nonexistence of solutions, and give sufficient conditions for uniqueness. We also provide an exact estimate of the behaviour of the solutions near the boundary in terms of dist(x, ∂Ω).

Existence, Asymptotic Behavior and Uniqueness for Large Solutions to Δu = eq(x)u

Abstract In this paper we consider existence, asymptotic behavior near the boundary and uniqueness for solutions to Δu = eq(x)u in a bounded smooth domain Ω with the boundary condition u(x) → + ∞ as dist(x, ∂Ω) → 0. The exponent q(x) is assumed to be a Hölder continuous function which is either positive on ∂Ω or is positive in a neighborhood of ∂Ω maybe vanishing on ∂Ω. When dealing with nonnegative exponents q we are allowing nonempty interior regions Ω0 ⊂ Ω where q vanishes. Changing sign exponents q will be also considered.

EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO SOME INHOMOGENEOUS NONLOCAL DIFFUSION PROBLEMS

We consider the nonlocal evolution Dirichlet problem

Existence and non-existence of solutions to elliptic equations with a general convection term

u_t(x,t)=\int_{\Omega}J(\frac{x-y}{g(y)})\frac{u(y,t)}{g(y)^N}dy-u(x,t)

Limit cases in an elliptic problem with a parameter-dependent boundary condition

,

Local bifurcation from the second eigenvalue of the Laplacian in a square

x\in\Omega

Layer Profiles of Solutions to Elliptic Problems under Parameter-Dependent Boundary Conditions

,

Quasilinear Equations With Boundary Blow-up and Exponential Reaction

t>0

Analysis of an elliptic system with infinitely many solutions

;