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The existence of infinitely many solutions all bifurcating from \(\lambda{} = 0\). (English) Zbl 0748.35029
The author considers the differential equation \[ -\Delta u-q(x)| u(x)|^ \sigma u=\lambda u \quad \text{ in } \mathbb{R}^ N(N\geq 2), \] and states conditions for \(q\) which guarantee the existence of infinitely many distinct pairs of weak solutions \((\pm u^ \lambda_ k)_{k\in \mathbb{N}}\) satisfying \(\lim_{\lambda\to 0-}\| u^ \lambda_ k\|_{H^ 1}=0\) for all \(k\in\mathbb{N}\).
The main tools are results from A. Ambrosetti and P. H. Rabinowitz [J. Funct. Anal. 14, 349-381 (1973; Zbl 0273.49063)]. The results here are generalisations of the results obtained by H.-J. Ruppen [Proc. R. Soc. Edinb., Sect. A 101, 307-320 (1985; Zbl 0603.35006)].
MSC:
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J60 Nonlinear elliptic equations
35B32 Bifurcations in context of PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
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