Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
bifurcation point; variational method; eigenvalues; exponential decay; standing waves
bifurcation point; variational method; eigenvalues; exponential decay; standing waves
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 $L^p$ $(2\leq p\leq \infty )$.
References:
[1] Anderson D.: Stability of time - dependent particle solutions in nonlinear field theories II. J. Math. Phys. 12 (1971), 945-952.
[2] Berestycki H., Lions P.L.: Nonlinear scalar field equations I: Existence of a ground state. Arch. Rat. Mech. Anal. 82 (1983), 313-345. MR 0695535 | Zbl 0533.35029
[3] Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order. SpringerVerlag, Berlin, Heidelberg, New York, 1983. MR 0737190 | Zbl 1042.35002
[4] Hewitt E., Stromberg K.: Real and Abstract Analysis. Springer-Verlag, Berlin, Heidelberg, New York, 1975. MR 0367121 | Zbl 0307.28001
[5] Rother W.: Bifurcation of nonlinear elliptic equations on $\Bbb R^N$. Bull. London Math. Soc. 21 (1989), 567-572. MR 1018205
[6] Rother W.: Bifurcation of nonlinear elliptic equations on $\Bbb R^N$ with radially symmetric coefficients. Manuscripta Math. 65 (1989), 413-426. MR 1019700
[7] Rother W.: The existence of infinitely many solutions all bifurcating from $\lambda =0$. Proc. Royal Soc. Edinburgh 118A (1991), 295-303. MR 1121669 | Zbl 0748.35029
[8] Rother W.: Nonlinear Scalar Field Equations. Differential and Integral Equations, to appear. MR 1167494 | Zbl 0755.35082
[9] Ruppen J.-H.: The existence of infinitely bifurcation branches. Proc. Royal Soc. Edinburgh 101A (1985), 307-320.
[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
[11] Stampacchia G.: Équations elliptiques du second ordre à coefficients discontinues. Séminaire de Mathématiques Supérieurs, No. 16, Montreal, 1965. MR 0251373
[12] Strauss W.A.: Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55 (1977), 149-162. MR 0454365 | Zbl 0356.35028
[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
[14] Stuart C.A.: Bifurcation for Dirichlet problems without eigenvalues. Proc. London Math. Soc. (3) 45 (1982), 169-192. MR 0662670 | Zbl 0505.35010
[15] Stuart C.A.: Bifurcation from the essential spectrum. Lecture Notes in Math. 1017 (1983), 575-596. MR 0726615 | Zbl 0527.35010
[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
[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
[18] Zhou H.-S., Zhu X.P.: Bifurcation from the essential spectrum of superlinear elliptic equations. Appl. Analysis 28 (1988), 51-61. MR 0960586 | Zbl 0621.35009
Search
Partner of
