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 critical points of non differentiable functionals of the kind J(v)=_Ω A(x,v) abla v· abla v-1p+1_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. 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(Ω). Indeed, J is only derivable along directions of H^1_0(Ω)L^∞(Ω) 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 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(R).