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Existence and bifurcation of solutions for nonlinear perturbations of the periodic Schrödinger equation. (English) Zbl 0767.35006

The authors combine earlier work of theirs on abstract bifurcation theorems to establish the existence of bifurcating solutions in \(H^ 1\) of the equations \[ -\Delta u+p(x)u\pm r(x)| u|^ \sigma u=\lambda u \] on \(\mathbb{R}^ N\). Here \(p\) is periodic and the problem is subcritical. The main difficulty is to construct test functions so that the condition \(T(\delta)\) of the abstract theorem is satisfied. The bifurcation occurs at the ends of gaps in the spectrum of the linear part.

MSC:

35B32 Bifurcations in context of PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35Q40 PDEs in connection with quantum mechanics
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