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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
Keywords:
existence; bifurcation
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