The authors study the following perturbed hemivariational eigenvalue inequality with constraints: find ▫$(u,\lambda) \in V \times {\bf R}$▫ such that ▫$a(u,v) + \int_{\Omega}(j^0(x,u(x);v(x)) + ...g(x,u(x);v(x)))dx + \langle\phi,v\rangle \ge \lambda (u,v)$▫ and ▫$\|u\| = r$▫, where ▫$V$▫ is a Hilbert space, densely and compactly imbedded in ▫$L^p(\Omega;{\bf R}^N)$▫ and ▫$g$▫ and ▫$\phi$▫ play the role of small perturbations. It is proved that the number of solutions of the above problem tends to infinity when the perturbations approach zero. In the limit, the problem coincides with the inequality studied by D. Motreanu and P. D. Panagiotopoulos [J. Math. Anal. Appl. 197 (1996), no. 1, 75--89] (where the existence of infinitely many solutions is shown).
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