### A deformation lemma with an application to a mean field equation

DOI: http://dx.doi.org/10.12775/TMNA.2007.021

#### 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.

$\Lambda$ 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.

#### Keywords

Deformation lemma; Palais-Smale condition; nonlinear PDE; mean field equation

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