A class of functionals possessing multiple global minima

We get a new multiplicity result for gradient systems. Here is a very particular corollary: Let $\mathrm{\Omega }\subset {\mathbf{R}}^n$ ($n\ge 2$) be a smooth bounded domain and let $\mathrm{\Phi }:{\mathbf{R}}^2\to \mathbf{R}$ be a $C^1$ function, with $\mathrm{\Phi }(0,0)=0$, such that $$\underset{(u,v)\in {\mathbf{R}}^2}{\sup }\frac{\mathrm{\mid }{\mathrm{\Phi }}_u(u,v)\mathrm{\mid }+\mathrm{\mid }{\mathrm{\Phi }}_v(u,v)\mathrm{\mid }}{1+\mathrm{\mid }u{\mathrm{\mid }}^p+\mathrm{\mid }v{\mathrm{\mid }}^p}<+\mathrm{\infty }$$ where $p>0$, with $p=\frac{2}{n-2}$ when $n>2$. Then, for every convex set $S\subseteq L^{\mathrm{\infty }}(\mathrm{\Omega })\times L^{\mathrm{\infty }}(\mathrm{\Omega })$ dense in $L^2(\mathrm{\Omega })\times L^2(\mathrm{\Omega })$, 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 $H_0^1(\mathrm{\Omega })\times H_0^1(\mathrm{\Omega })$ of the functional $$(u,v)\to \frac{1}{2}({\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }u(x){\mathrm{\mid }}^{2}dx+{\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }v(x){\mathrm{\mid }}^{2}dx)$$ $$-{\int }_{\mathrm{\Omega }}(\alpha (x)\sin (\mathrm{\Phi }(u(x),v(x)))+\beta (x)\cos (\mathrm{\Phi }(u(x),v(x))))dx\mathrm{.}$$