Multiple critical points for nondifferentiable functionals involving Hardy potentials
The author studies general functionals of the calculus of variations with the presence of the Hardy potentials, improving many results obtained in the semilinear framework. The functional \(J\) is defined on the Sobolev space \(W^{1,p}_0 (\Omega)\), where \(\Omega\) is a bounded subset of \({\mathbb R}^N\), \(N \geq 2\), and it is defined in the following way: \[ J(v) = \int_\Omega j(x,v,\nabla v) - {\lambda \over p} \int_\Omega {{|v|^p}\over {|x|^p}} - \int_\Omega G(x,v), \] where \(1 < p < N\), \(j(x,s,\xi) \colon \Omega \times {\mathbb R} \times {\mathbb R}^N \rightarrow {\mathbb R}\) is a measurable function with respect to \(x\) for any \((s,\xi) \in {\mathbb R} \times {\mathbb R}^N\), and of class \(C^1\) with respect to \((s,\xi)\) for almost everywhere \(x \in \Omega\), and \(G \colon \Omega \times {\mathbb R} \rightarrow {\mathbb R}\) is a continuous function. The author first proves a general weak lower semicontinuity result for the functional \(J\), which will imply the existence of a critical point whenever the functional \(J\), under suitable assumptions on \(J\) and \(G\), is coercive. Moreover existence and multiplicity results for critical points of \(J\) are obtained, even if the functional is nonsmooth. The author apply a nonsmooth critical point theory developed in \textit{J.-N. Corvellec} et al. [Topol. Methods Nonlinear Anal. 1, No. 1, 151--171 (1993; Zbl 0789.58021)] and \textit{M. Degiovanni} and \textit{M. Marzocchi} [ Ann. Mat. Pura Appl., IV. Ser. 167, 73--100 (1994; Zbl 0828.58006)].
scientific article; zbMATH DE number 2167830
scientific article