Critical points for some functionals of the calculus of variations
Keywords
Nonsmooth critical point theory, integral functionalsAbstract
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 $L^1(\mathbb R)$.Downloads
Published
2001-06-01
How to Cite
1.
PELLACCI, Benedetta. Critical points for some functionals of the calculus of variations. Topological Methods in Nonlinear Analysis. Online. 1 June 2001. Vol. 17, no. 2, pp. 285 - 305. [Accessed 29 March 2024].
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