Topological Methods in Nonlinear Analysis

Critical points for some functionals of the calculus of variations

Benedetta Pellacci

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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 $L^1(\mathbb R)$.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 17, Number 2 (2001), 285-305.

Dates
First available in Project Euclid: 22 August 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1471875821

Mathematical Reviews number (MathSciNet)
MR1868902

Zentralblatt MATH identifier
0987.35041

Citation

Pellacci, Benedetta. Critical points for some functionals of the calculus of variations. Topol. Methods Nonlinear Anal. 17 (2001), no. 2, 285--305. https://projecteuclid.org/euclid.tmna/1471875821


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