Critical points for some functionals of the calculus of variations - Pellacci, Benedetta (Topological Methods in Nonlinear Analysis, 2001)
In this paper we prove the existence of criticalpoints of non differentiable functionals of the kind $$ J(v)=\frac_Ω A(x,v) abla v· abla v-\frac1{p+1}\int_Ω (v^+)^{p+1}, $$ where $1< p< (N+2)/(N-2)$ if $N> 2$, $p> 1$ if $N≤ 2$ and $v^+$ stands for the positive part of the function $v$. Thecoefficient $A(x,s)=(a_{ij}(x,s))$ 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 $H^1_0(Ω)$. Indeed, $J$ is only derivable along directions of $H^1_0(Ω)\cap L^{∞}(Ω)$so that the classical critical point theory cannot be applied.We will prove the existence of a critical point of $J$ byassuming that there exist positive continuous functions $α(s)$, $β(s)$ and a positive constants $α_0$ and $M$ satisfying $α_0|ξ|^2≤ α(s)|ξ|^2≤ A(x,s)ξ· ξ$, $A(x,0)≤ M$, $|A'_s(x,s)|≤ β(s)$, with $β(s)$ in $L^1(\mathbb R)$.