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 $$\sup_{(u,v)\in {\bf R}^2}{{|\Phi_u(u,v)|+|\Phi_v(u,v)|}\over {1+|u|^p+|v|^p}}<+\infty$$ where $p>0$, with $p={{2}\over {n-2}}$ 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 $$\cases {-\Delta u=(\alpha(x)\cos(\Phi(u,v))-\beta(x)\sin(\Phi(u,v)))\Phi_u(u,v) & in $\Omega$ \cr & \cr -\Delta v= (\alpha(x)\cos(\Phi(u,v))-\beta(x)\sin(\Phi(u,v)))\Phi_v(u,v) & in $\Omega$ \cr & \cr u=v=0 & on $\partial\Omega$\cr}$$ 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 $$(u,v)\to {{1}\over {2}}\left ( \int_{\Omega}|\nabla u(x)|^2dx+\int_{\Omega}|\nabla v(x)|^2dx\right )$$ $$-\int_{\Omega}(\alpha(x)\sin(\Phi(u(x),v(x)))+\beta(x)\cos(\Phi(u(x),v(x))))dx\ .$$