Nontrivial Solutions for Asymptotically Linear Variational Inequalities

The aim of this paper is to extend to variational inequalities a result of Amann and Zehnder’s, refined by Chang (see [1, 6, 14]), concerning the existence of nontrivial solutions for a semilinear elliptic boundary value problem, when the nonlinearity is asymptotically linear, is zero at zero and its derivative has a suitable jump between zero and infinity. In the constrained problem studied here (see Theorem 4.1) the presence of the “obstacle” enters into the discussion and what determines the required “jump of behaviour” is a combination of the nonlinearity and the obstacle. Following the ideas of [1] we use the Conley index and show that the index of zero as invariant set in the associated parabolic flow is different from the index of the maximal invariant set in a suitable large ball. Then there exists an invariant set larger than {0}, so, by the variational nature of the flow, there exists a second invariant point. For computing these indices it is natural to use a continuation argument, passing to some “limit flows”. This requires proving an index continuation result for flows with moving domains, which is done in Section 2, generalizing the work of Rybakowski [13, 15]. In Section 3 the result described above is proved in a general abstract setting. We use a nonsmooth variational approach which consists in viewing solutions as