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


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)
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