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The dual variational principle and elliptic problems with discontinuous nonlinearities. (English) Zbl 0687.35033
In this paper the problem $(*)\quad \Delta u+f(u)=p(x)\quad in\quad \Omega,\quad u=0\quad on\quad \partial \Omega,$ $$\Omega$$ a bounded domain in $${\mathbb{R}}^ N$$, is studied. The nonlinearity f is allowed to have discontinuities and $$p\in L^ 2(\Omega)$$. The discontinuities are such that
i) $$f\in C({\mathbb{R}}-A)$$ where $$A\subset {\mathbb{R}}$$ is a set with no finite accumulation points,
ii) $$h(s):=ms+f(s)$$ is strictly increasing for some $$m\geq 0.$$
Solutions of (*) are assumed to be in $$W^ 1_ 0(\Omega)\cap W^{2,2}(\Omega)$$ and $-\Delta u+p\in \hat f(u)\quad a.e.\quad in\quad \Omega,$ where $\hat f(s)=f(s),\quad s\not\in A;\quad \hat f(s)=[f(a- ),f(a+)],\quad u\in A.$ Using variational techniques the authors prove existence theorems in a number of interesting situations.
Reviewer: R.Sperb

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Variational methods for second-order elliptic equations 35D05 Existence of generalized solutions of PDE (MSC2000)
##### Keywords:
existence theorems
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##### References:
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