### Critical points for some functionals of the calculus of variations

DOI: http://dx.doi.org/10.12775/TMNA.2001.017

#### Abstract

In this paper we prove the existence of critical

points of non differentiable functionals of the kind

$$

J(v)=\frac_\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 $L^1(\mathbb R)$.

points of non differentiable functionals of the kind

$$

J(v)=\frac_\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 $L^1(\mathbb R)$.

#### Keywords

Nonsmooth critical point theory; integral functionals

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