Local uniqueness of $m$-bubbling sequences for the Gel'fand equation

Abstract

We consider the Gel'fand problem, $$ \begin{cases} Δw_{\varepsilon}+\varepsilon^2 h e^{w_{\varepsilon}}=0\quad&\mbox{in}\quadΩ, w_{\varepsilon}=0\quad&\mbox{on}\quad\partialΩ, \end{cases} $$ where $h$ is a nonnegative function in $\Omega \subset {\mathbb{R}}^2$. Under suitable assumptions on $h$ and $\Omega$, we prove the local uniqueness of $m-$bubbling solutions for any $\epsilon >0$ small enough.

References in corpus (2)

• Supercritical Mean Field Equations on convex domains and the Onsager's statistical description of two-dimensional turbulence
• Non degeneracy, Mean Field Equations and the Onsager theory of 2D turbulence

Cited by in corpus (4)

• On the global bifurcation diagram of the Gel'fand problem
• Uniqueness of bubbling solutions of mean field equations with non-quantized singularities
• Local uniqueness and non-degeneracy of blow up solutions of mean field equations with singular data
• On the global bifurcation diagram of the equation $-\Delta u=\mu |x{|}^{2\alpha }{e}^u$ in dimension two