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Existence of changing sign solutions for some semilinear problems with jumping nonlinearities at zero. (English) Zbl 0819.35054
Let \[ h(u)= \begin{cases} au- \alpha u^ 2, &u\geq 0\\ du+ \beta u^ 2, &u\leq 0 \end{cases}, \] where \(\alpha\geq 0\), \(\beta>0\) and \(a,d> \lambda_ 1\), here \(\lambda_ 1\) denotes the first eigenvalue of \(-\Delta\) under zero Dirichlet data on a bounded smooth domain \(D\) in \(\mathbb{R}^ n\). A domain in the \(ad\)-plane is found to characterize the existence of changing sign solutions of the equation \(-\Delta u= h(u)\), \(u|_{\partial D}=0\).

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data
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