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.


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
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