Singular limits in Liouville-type equations

Singular Limits in Liouville-type Equations

We consider the boundary value problem ∆u+ε k(x) e = 0 in a bounded, smooth domain Ω in R with homogeneous Dirichlet boundary conditions. Here ε > 0, k(x) is a non-negative, not identically zero function. We find conditions under which there exists a solution uε which blows up at exactly m points as ε→ 0 and satisfies ε ∫ Ω keε → 8mπ. In particular, we find that if k ∈ C(Ω̄), infΩ k > 0 and Ω is...

Liouville type results for semilinear elliptic equations in unbounded domains

This paper is devoted to the study of some class of semilinear elliptic equations in the whole space: −aij(x)∂iju(x)− qi(x)∂iu(x) = f(x, u(x)), x ∈ R . The aim is to prove uniqueness of positive bounded solutions Liouville type theorems. Along the way, we establish also various existence results. We first derive a sufficient condition, directly expressed in terms of the coefficients of the line...

On Singular Limits of Mean - Field Equations July

Singular solutions of perturbed logistic-type equations

We are concerned with the qualitative analysis of positive singular solutions with blow-up boundary for a class of logistic-type equations with slow diffusion and variable potential. We establish the exact blow-up rate of solutions near the boundary in terms of Karamata regular variation theory. This enables us to deduce the uniqueness of the singular solution. 2011 Elsevier Inc. All rights res...

The Uniqueness Theorem for the Solutions of Dual Equations of Sturm-Liouville Problems with Singular Points and Turning Points

In this paper, linear second-order differential equations of Sturm-Liouville type having a finite number of singularities and turning points in a finite interval are investigated. First, we obtain the dual equations associated with the Sturm-Liouville equation. Then, we prove the uniqueness theorem for the solutions of dual initial value problems.