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Bifurcation for some semilinear elliptic equations when the linearization has no eigenvalues. (English) Zbl 0791.35094
The eigenvalue problem \(-\Delta u-q(x)| u|^{\sigma_ 1}\cdot u+r(x)| u|^{\sigma_ 2}\cdot u=\lambda u\) in \(\mathbb{R}^ N\) is studied. It is assumed that \(N\geq 2\) and \(\sigma_ 1\), \(\sigma_ 2\) are positive constants, with \(\sigma_ 1<4/N\). The problem arises, among other things, in connection with the Klein-Gordon equation and the Schrödinger equation. Existence and bifurcation results are proven.
Reviewer: R.Sperb (Zürich)
MSC:
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35Q40 PDEs in connection with quantum mechanics
35J60 Nonlinear elliptic equations
35A30 Geometric theory, characteristics, transformations in context of PDEs
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