Singular limits in Liouville-type equations

Abstract

Abstract. We consider the boundary value problem ∆u + ε 2 k(x) e u = 0 in a bounded, smooth domain Ω in R 2 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 2 Ω ke u ε → 8mπ. In particular, we find that if k ∈ C 2 (Ω), infΩ k > 0 and Ω is not simply connected then such a solution exists for any given m ≥ 1

Multiple blowing-up and concentrating solutions for Liouville-type equations with singular sources under mixed boundary conditions

“…and for m = 0 and 1 ≤ n ≤ N, we find the solution j of the nonlinear Equation (26). The following result can be proved using standard arguments as in [26,28].…”

“…In this section, we will carry out the finite dimensional reduction to solve the Equation (26). First of all, we need to get the desired invertibility of linearized operation L. Set…”

“…□ Proof of Proposition 3.1. Case (i): From Lemma 3.5 (i), and the Fredholm's alternative theory with Robin boundary condition instead of Dirichlet boundary condition if necessary (see [34,38] and the references therein), the proof can be similarly given through those in [ [26], pp. 61-63].…”

“…It is necessary to point out that the key step of the finite dimensional reduction is the analysis of the bounded invertibility of the corresponding linearized operator L of the Equation (3) at the suitable approximate solution. In [26,27], the authors construct the approximate solution, carry out the finite dimensional reduction and use some stability assumptions of critical points of 0,m to get the existence of multiple blowing-up and concentrating solutions for the Equation (3)| t = 1 with = ∅, namely a i = 0 for any i = 1,2, ..., N. When {α 1 , . .…”

Deformation retracts to the fat diagonal and applications to the existence of peak solutions of nonlinear elliptic equations

“…The function Z plays the same role in constructing single and multiple peak solutions for problems defined on two-dimensional domains; see [Esposito et al 2005;del Pino et al 2005;Esposito et al 2006].…”