# Article

 Title: Bifurcation for some semilinear elliptic equations when the linearization has no eigenvalues (English) Author: Rother, Wolfgang Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 34 Issue: 1 Year: 1993 Pages: 125-138 . Category: math . Summary: We prove existence and bifurcation results for a semilinear eigenvalue problem in $\Bbb R^N$ $(N\geq 2)$, where the linearization --- $\vartriangle$ has no eigenvalues. In particular, we show that under rather weak assumptions on the coefficients $\lambda =0$ is a bifurcation point for this problem in $H^1, H^2$ and $L^p$ $(2\leq p\leq \infty )$. (English) Keyword: bifurcation point Keyword: variational method Keyword: eigenvalues Keyword: exponential decay Keyword: standing waves MSC: 35A30 MSC: 35B32 MSC: 35J60 MSC: 35P30 MSC: 35Q40 idZBL: Zbl 0791.35094 idMR: MR1240210 . Date available: 2009-01-08T18:01:43Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/118562 . Reference: [1] Anderson D.: Stability of time - dependent particle solutions in nonlinear field theories II.J. Math. Phys. 12 (1971), 945-952. Reference: [2] Berestycki H., Lions P.L.: Nonlinear scalar field equations I: Existence of a ground state.Arch. Rat. Mech. Anal. 82 (1983), 313-345. Zbl 0533.35029, MR 0695535 Reference: [3] Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order.SpringerVerlag, Berlin, Heidelberg, New York, 1983. Zbl 1042.35002, MR 0737190 Reference: [4] Hewitt E., Stromberg K.: Real and Abstract Analysis.Springer-Verlag, Berlin, Heidelberg, New York, 1975. Zbl 0307.28001, MR 0367121 Reference: [5] Rother W.: Bifurcation of nonlinear elliptic equations on $\Bbb R^N$.Bull. London Math. Soc. 21 (1989), 567-572. MR 1018205 Reference: [6] Rother W.: Bifurcation of nonlinear elliptic equations on $\Bbb R^N$ with radially symmetric coefficients.Manuscripta Math. 65 (1989), 413-426. MR 1019700 Reference: [7] Rother W.: The existence of infinitely many solutions all bifurcating from $\lambda =0$.Proc. Royal Soc. Edinburgh 118A (1991), 295-303. Zbl 0748.35029, MR 1121669 Reference: [8] Rother W.: Nonlinear Scalar Field Equations.Differential and Integral Equations, to appear. Zbl 0755.35082, MR 1167494 Reference: [9] Ruppen J.-H.: The existence of infinitely bifurcation branches.Proc. Royal Soc. Edinburgh 101A (1985), 307-320. Reference: [10] Stampacchia G.: Le probleème de Dirichlet pour les équations elliptique du second ordre à coefficients discontinues.Annls Inst. Fourier Univ. Grenoble 15 (1965), 189-257. MR 0192177 Reference: [11] Stampacchia G.: Équations elliptiques du second ordre à coefficients discontinues.Séminaire de Mathématiques Supérieurs, No. 16, Montreal, 1965. MR 0251373 Reference: [12] Strauss W.A.: Existence of solitary waves in higher dimensions.Commun. Math. Phys. 55 (1977), 149-162. Zbl 0356.35028, MR 0454365 Reference: [13] Stuart C.A.: Bifurcation from the continuous spectrum in the $L^2$ - theory of elliptic equations on $\Bbb R^N$.Recent Methods in Nonlinear Analysis and Applications, Proc. SAFA IV, Liguori, Napoli, 1981, pp. 231-300. MR 0819032 Reference: [14] Stuart C.A.: Bifurcation for Dirichlet problems without eigenvalues.Proc. London Math. Soc. (3) 45 (1982), 169-192. Zbl 0505.35010, MR 0662670 Reference: [15] Stuart C.A.: Bifurcation from the essential spectrum.Lecture Notes in Math. 1017 (1983), 575-596. Zbl 0527.35010, MR 0726615 Reference: [16] Stuart C.A.: Bifurcation in $L^p(\Bbb R^N)$ for a semilinear elliptic equation.Proc. London Math. Soc. (3) 57 (1988), 511-541. MR 0960098 Reference: [17] Stuart C.A.: Bifurcation from the essential spectrum for some non-compact non-linearities.Math. Methods Appl. Sci. 11 (1989), 525-542. MR 1001101 Reference: [18] Zhou H.-S., Zhu X.P.: Bifurcation from the essential spectrum of superlinear elliptic equations.Appl. Analysis 28 (1988), 51-61. Zbl 0621.35009, MR 0960586 .

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