A potential well for an integral functional with singular Lagrangian

Abstract

This paper concerns a functional of the form $$\begin{aligned} \Phi (u)=\int _\Omega L(x,u(x),\nabla u(x))\, dx \end{aligned}$$ on the Sobolev space $$H_0^1(\Omega )$$ where $$\Omega $$ is a bounded open subset of $${\mathbb {R}}^N$$ with $$N\ge 3$$ and $$0\in \Omega $$ . The hypotheses on L ensure that $$u\equiv 0$$ is a critical point of $$\Phi $$ , but allow the Lagrangian to be singular at $$x=0$$ . It is shown that, under these assumptions, the usual conditions associated with Jacobi (positive definiteness of the second variation of $$\Phi $$ at $$u\equiv 0$$ ), Legendre (ellipticity at $$u\equiv 0$$ ) and Weierstrass [strict convexity of $$L(x,s,\xi )$$ with respect to $$\xi $$ ] from the calculus of variations are not sufficient ensure that $$u\equiv 0$$ is a local minimum of $$\Phi $$ . Using recent criteria for the existence of a potential well of a $$C^1$$ -functional on a real Hilbert space, conditions implying that $$u\equiv 0$$ lies in a potential well of $$\Phi $$ are established. They are shown to be sharp in some cases.