A deformation lemma with an application to a mean field equation

Abstract

Given a Hilbert space $( {\mathcal H}, \langle \cdot,\cdot\rangle)$, $\Lambda$ an interval of $\mathbb R$ and $K \in C^{1,1} ({\mathcal H}, {\mathbb R})$ whose gradient is a compact mapping, we consider a family of functionals of the type: $$ I(\lambda, u) = \frac{1}{2} \langle u , u\rangle - \lambda K(u), \quad (\lambda,u) \in \Lambda \times {\mathcal H}. $$ Though the Palais-Smale condition may fail under just these assumptions, we present a deformation lemma to detect critical points. As a corollary, if $I(\overline \lambda,\cdot)$ has a ``mountain pass geometry'' for some $\overline \lambda \in \Lambda$, we deduce the existence of a sequence $\lambda_n \to \overline \lambda$ for which each $I(\lambda_n,\cdot)$ has a critical point. To illustrate such results, we consider the problem: $$ - \Delta u = \lambda \bigg( \frac{e^u}{\int_{\Omega} e^u } - \frac{T}{|\Omega|} \bigg), \quad u \in H_0^1 (\Omega), $$ where $\Omega \subset \subset {\mathbb R}^2$ and $T$ belongs to the dual $H^{-1}$ of $H^1_0 (\Omega)$. It is known that the associated energy functional does not satisfy the Palais-Smale condition. Nevertheless, we can prove existence of multiple solutions under some smallness condition on $\| T-1 \|_{H^{-1}}$, where $1$ denotes the constant function identically equal to $1$ in the domain.