Perturbations of hemivariational inequalities with constraints
Rǎdulescu, Vicenţiu, 1958-
The author considers a nonlinear eigenvalue problem of the form (P) ▫$$(u,\lambda) \in V \times \bold R,$$▫ ▫$$a(u,v) + C(u,v) + \int\sb{\Omega} (j\sp{0}(x,u(x); v(x)) + g\sp{0}(x,u(x);v(x)))dx ...+\langle\varphi, v\rangle\sb{V} \geq \lambda(u,v) \quad \forall v \in V,$$▫ ▫$$\|u\| = r.$$▫ In the case ▫$g = 0$▫ and ▫$\varphi = 0$▫, it has been proved in [D. Motreanu and P. D. Panagiotopoulos, J. Global Optim. 6 (1995), no. 2, 163--177] that, under suitable assumptions involving symmetry, problem (P) admits infinitely many solutions. Here the author first announces a result concerning the perturbed problem, in which ▫$g$▫ satisfies no symmetry assumption. Namely, the number of solutions of (P) becomes greater and greater as $g$ and ▫$\varphi$▫ become smaller and smaller in a suitable sense. In dimension one and two, he also states the existence of infinitely many solutions for problem (P). Finally, a result of the former type is also announced when ▫$g=0$▫, ▫$\varphi = 0$▫, but the space ▫$V$▫ is substituted by a sequence ▫$(V_n)$▫ of convex closed subsets of ▫$V$▫ converging to ▫$V$▫ in the sense of [U. Mosco, Advances in Math. 3 (1969), 510--585].
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