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Existence and multiplicity results for elliptic problems with critical growth and discontinuous nonlinearities. (English) Zbl 0877.35046

The paper provides existence and multiplicity results for the elliptic problem \[ -\Delta u=u^{p-1}+bh(u-a),\;u>0\quad\text{in }\Omega,\;u=0\quad\text{on} \partial\Omega\tag{1} \] where \(a,b>0\), \(\Omega\) is a bounded subset of \(\mathbb{R}^N\) and \(h\) is the Heaviside function \(h(s)=0\), if \(s<0\) and \(h(s)=1\), if \(s\geq 0\). The authors also study carefully the elliptic problem of the form (1) with nonmonotone nonlinearity, i.e. when the function \(h\) is replaced by a lower jump function \(g\) such that \(g(s)=1\), if \(s\leq 0\) and \(g(s)=0\), if \(s>0\). It is shown that for fixed positive \(b\) and \(N\geq 5\), problem (1) admits a weak solution \(u_a\) in \(W^{2,q}(\Omega)\cap H^1_0(\Omega)\) for every \(q\in[1,+\infty)\) and every \(a>0\), satisfying \(|u_a|_\infty>a\). If \(N=3,4\), the existence of such a solution to (1) is established only for \(a>0\) small. As far as the multiplicity of solutions is considered, it is proved that for \(b\in(0,b^*]\), with a suitable \(b^*>0\), there exists \(a^*=a^*(b)>0\) such that for every \(a\in(0,a^*)\) problem (1) admits, in the same class, three ordered weak solutions in \(\Omega\).
The approach used in this paper combines the sub- and super-solution method with the variational technique developed by Chang for a class of locally Lipschitz functionals.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
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