×

Normalized solutions of nonlinear Schrödinger equations. (English) Zbl 1260.35098

Summary: We consider the problem \[ \left\{\begin{aligned} & -\Delta u - g(u) = \lambda u, \\ & u \in H^1(\mathbb{R}^N), \int_{\mathbb{R}^N} u^2 = 1, \lambda \in \mathbb{R},\end{aligned}\right. \] in dimension \(N \geq 2\). Here \(g\) is a superlinear, subcritical, possibly nonhomogeneous, odd nonlinearity. We deal with the case where the associated functional is not bounded below on the \(L ^{2}\)-unit sphere, and we show the existence of infinitely many solutions.

MSC:

35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
35J60 Nonlinear elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] T. Bartsch, Topological methods for variational problems with symmetries, Lecture Notes in Mathematics 1560, Springer-Verlag, Berlin 1993. · Zbl 0789.58001
[2] Bartsch T.: Infinitely many solutions of a symmetric Dirichlet Problem. Nonlin. Anal. 20, 1205–1216 (1993) · Zbl 0799.35071
[3] Berestycki H, Lions P.-L: Nonlinear scalar field equations I. Arch. Rat. Mech. Anal. 82, 313–345 (1983) · Zbl 0533.35029
[4] Berestycki H, Lions P.-L: Nonlinear scalar field equations, II. Arch. Rat. Mech. Anal. 82, 347–375 (1983) · Zbl 0556.35046
[5] P. E. Conner and E.E. Floyd, Fixed point free involutions and equivariant maps. II., Trans. Amer. Math. Soc. 105, 222–228. · Zbl 0114.14402
[6] Fadell E.R, Rabinowitz P.H: Bifurcation for odd potential operators and an alternative topological index. J. Funct. Anal. 26, 48–67 (1977) · Zbl 0363.47029
[7] Hajaiej H, Stuart C: Existence and non-existence of Schwarz symmetric ground states for elliptic eigenvalue problems, Ann. Mat. Pura Appl. 184, 297–314 (2005) · Zbl 1099.49002
[8] Hirata J, Ikoma N, Tanaka K: Nonlinear scalar field equations in $${\(\backslash\)mathbb{R}\^N}$$ : mountain pass and symmetric mountain pass approaches, Top. Meth. Nonlin. Anal. 35, 253–276 (2010) · Zbl 1203.35106
[9] Jeanjean L.: Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlin. Anal. 28, 1633–1659 (1997) · Zbl 0877.35091
[10] P.G. Kevrekidis, d.J. Frantzeskakis, and R. Carretero-Gonzalez (eds.), Emergent Nonlinear Phenomena in Bose–Einstein Condensation, Springer-Verlag, Berlin 2008. · Zbl 1137.82003
[11] P. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf. Ser. Math. 65, Amer. Math. Soc., Providence 1986. · Zbl 0609.58002
[12] M. Willem, Minimax Methods, Birkhäuser, Boston 1996.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.