A class of functionals possessing multiple global minima

Abstract

We get a new multiplicity result for gradient systems. Here is a very particular corollary: Let $\Omega\subset
{\bf R}^n$ ($n\geq 2$) be a smooth bounded domain and let $\Phi:{\bf R}^2\to {\bf R}$ be a $C^1$ function, with $\Phi(0,0)=0$,
such that
$$\sup_{(u,v)\in {\bf R}^2}\frac{|\Phi_u(u,v)|+|\Phi_v(u,v)|}{1+|u|^p+|v|^p}<+\infty$$
where $p>0$, with $p=\frac{2}{n-2}$ when $n>2$.\\
Then, for every convex set $S\subseteq L^{\infty}(\Omega)\times L^{\infty}(\Omega)$ dense in $L^2(\Omega)\times
L^2(\Omega)$, there exists $(\alpha,\beta)\in S$ such that the problem
$$-\Delta u=(\alpha(x)\cos(\Phi(u,v))-\beta(x)\sin(\Phi(u,v)))\Phi_u(u,v)\hskip 5pt \hbox {\rm in}\hskip 5pt \Omega$$
$$-\Delta v= (\alpha(x)\cos(\Phi(u,v))-\beta(x)\sin(\Phi(u,v)))\Phi_v(u,v)\hskip 5pt \hbox {\rm in}\hskip 5pt \Omega$$
$$u=v=0\hskip 5pt \hbox {\rm on}\hskip 5pt \partial\Omega$$
has at least three weak solutions, two of which are global minima in $H^1_0(\Omega)\times H^1_0(\Omega)$ of the functional
$$(u,v)\to \frac{1}{2}\left ( \int_{\Omega}|\nabla u(x)|^2dx+\int_{\Omega}|\nabla v(x)|^2dx\right )$$
$$-\int_{\Omega}(\alpha(x)\sin(\Phi(u(x),v(x)))+\beta(x)\cos(\Phi(u(x),v(x))))dx\ .$$