Critical points for some functionals of the calculus of variations

Abstract

In this paper we prove the existence of critical points of non differentiable functionals of the kind $$ J(v)=\frac12\int_\Omega A(x,v)\nabla v\cdot\nabla v-\frac1{p+1}\int_\Omega (v^+)^{p+1}, $$ where $1< p< (N+2)/(N-2)$ if $N> 2$, $p> 1$ if $N\leq 2$ and $v^+$ stands for the positive part of the function $v$. The coefficient $A(x,s)=(a_{ij}(x,s))$ is a Carathéodory matrix derivable with respect to the variable $s$. Even if both $A(x,s)$ and $A'_s(x,s)$ are uniformly bounded by positive constants, the functional $J$ fails to be differentiable on $H^1_0(\Omega)$. Indeed, $J$ is only derivable along directions of $H^1_0(\Omega)\cap L^{\infty}(\Omega)$ so that the classical critical point theory cannot be applied.

We will prove the existence of a critical point of $J$ by assuming that there exist positive continuous functions $\alpha(s),\beta(s)$ and a positive constants $\alpha_0$ and $M$ satisfying $\alpha_0|\xi|^2\leq \alpha(s)|\xi|^2 \leq A(x,s)\xi\cdot \xi$, $A(x,0)\leq M$, $|A'_s(x,s)|\leq \beta(s)$, with $\beta(s)$ in $L^1(\mathbb R)$.