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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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References:
 [1] DOI: 10.1112/blms/21.6.567 · Zbl 0662.35081 [2] DOI: 10.1016/0362-546X(79)90073-7 · Zbl 0388.47039 [3] DOI: 10.1016/0022-1236(73)90051-7 · Zbl 0273.49063 [4] DOI: 10.1080/00036818808839748 · Zbl 0621.35009 [5] Toland, Trans. Amer. Math. Soc. 282 pp 335– (1984) [6] DOI: 10.1016/0022-0396(82)90026-2 · Zbl 0455.34015 [7] DOI: 10.1007/BF01172789 · Zbl 0699.35016 [8] DOI: 10.1007/BF01457083 · Zbl 0513.35068 [9] DOI: 10.1112/plms/s3-45.1.169 · Zbl 0505.35010 [10] DOI: 10.1007/BFb0103282 [11] DOI: 10.1016/0022-1236(80)90063-4 · Zbl 0458.47048 [12] Stuart, C.R. Acad. Sci. Paris. 288 pp 761– (1979) [13] Ruppen, Proc. Roy. Soc. Edinburgh Sect. A 101 pp 307– (1985) · Zbl 0603.35006 [14] DOI: 10.1112/plms/s3-57.3.511 · Zbl 0673.35005
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