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On the existence of a continuous branch of nodal solutions of elliptic equations with convex-concave nonlinearities. (English. Russian original) Zbl 1298.35069
Differ. Equ. 50, No. 6, 765-776 (2014); translation from Differ. Uravn. 50, No. 6, 768-779 (2014).
Summary: We study the existence of nodal solutions of a parametrized family of Dirichlet boundary value problems for elliptic equations with convex-concave nonlinearities. In the main result, we prove the existence of nodal solutions \(u_{\lambda}\) for \(\lambda \in (-\infty, \lambda ^*_{0})\). The critical value \(\lambda^*_{0} >0\) is found by a spectral analysis procedure according to Pokhozhaev’s fibering method. We show that the obtained solutions form a continuous branch (in the sense of level lines of the energy functional) with respect to the parameter \(\lambda\). Moreover, we prove the existence of an interval \((-\infty ,\tilde \lambda )\), where \(\tilde \lambda > 0\), on which this branch consists of solutions with exactly two nodal domains.
MSC:
35J57 Boundary value problems for second-order elliptic systems
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