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Hemivariational inequalities associated to multivalued problems with strong resonance
Rǎdulescu, Vicenţiu, 1958-
Existence results for a non-smooth problem with strong resonance at infinity are obtained. This is achieved by applying tools developed for hemivariational inequalities [see e.g., Z. Naniewicz and P. ...D. Panagiotopoulos, Mathematical theory of hemivariational inequalities and applications, Dekker, New York, 1995]. The hemivariational problem considered here is the following: Given ▫$\Omega$▫ an open and bounded set in ▫${\bf R}^N$▫, ▫$N \ge 2$▫, ▫$V \in L^1_{\rm{loc}}(\Omega)$▫, ▫$f \in L^{\infty}(R)$▫ and ▫$F(t) = \int_0^t f(s)ds$▫, find ▫$u \in H^1_0(\Omega) \setminus {0}$▫ such that ▫$\int_{\Omega} (Du Dv - \lambda_1^V V(x) uv)\,dx - \int_{\Omega} F^0(u(x),v(x))\,dx \ge 0$▫, for every ▫$v \in H^1_0(\Omega)$▫. Here we denote ▫$$F^0(u,v) = \limsup_{y \to u, \lambda\searrow 0} {{F(y+\lambda v)-F(y)}\over{\lambda}}$$▫ and ▫$\lambda_1^V > 0$▫ the first eigenvalue of the problem ▫$-\Delta u = \lambda V(x) u$▫ in ▫$\Omega$▫, ▫$u=0$▫ on ▫$\partial\Omega$▫. Additional assumptions on the data are made, the basic one on ▫$f$▫ being ▫$\text{ess\,lim}_{t \to +\infty} f(t) = \text{ess\,lim}_{t \to +\infty} F(t)=0$▫ (i.e., strong resonance at ▫$+\infty$)▫. This hemivariational inequality is equivalent to the following multivalued elliptic problem: ▫$-\Delta u - \lambda_1^V V(x) u\ in [\underline{f}(u(x)), \overline{f}(u(x))]$▫ in ▫$\Omega$▫, ▫$u=0$▫ on ▫$\partial\Omega▫$, ▫$u \not\equiv 0$▫ in ▫$\Omega$▫, where ▫$\underline{f}(t) = \lim_{\varepsilon \searrow 0} {\rm ess\,inf} {f(s); |t-s| < \varepsilon}$▫ and ▫$\overline{f}(t) = \lim_{\varepsilon\searrow 0} {\rm ess\,sup} {f(s); |t-s| < \varepsilon}$▫. Under the assumptions in the paper, the energy associated to the hemivariational problem is a locally Lipschitz functional on ▫$H^1_0(\Omega)$▫. Existence results are then obtained using non-smooth critical point theory.
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