347 research outputs found
Solutions to a nonlinear Schr\"odinger equation with periodic potential and zero on the boundary of the spectrum
We study the following nonlinear Schr\"odinger equation where V and g are periodic in x. We assume that 0 is a right
boundary point of the essential spectrum of . The superlinear and
subcritical term g satisfies a Nehari type monotonicity condition. We employ a
Nehari manifold type technique in a strongly indefitnite setting and obtain the
existence of a ground state solution. Moreover we get infinitely many
geometrically distinct solutions provided that g is odd.Comment: To appear in Topol. Methods Nonlinear Ana
Normalized ground states of the nonlinear Schr\"{o}dinger equation with at least mass critical growth
We propose a simple minimization method to show the existence of least energy
solutions to the normalized problem \begin{cases}
-\Delta u + \lambda u = g(u) \quad \mathrm{in} \ \mathbb{R}^N, \ N \geq 3, \\
u \in H^1(\mathbb{R}^N), \\ \int_{\mathbb{R}^N} |u|^2 \, dx = \rho > 0,
\end{cases} where is prescribed and is to be determined. The new approach based on the direct
minimization of the energy functional on the linear combination of Nehari and
Pohozaev constraints is demonstrated, which allows to provide general growth
assumptions imposed on . We cover the most known physical examples and
nonlinearities with growth considered in the literature so far as well as we
admit the mass critical growth at
Bound states for the Schr\"{o}dinger equation with mixed-type nonlinearites
We prove the existence results for the Schr\"odinger equation of the form where is
superlinear and subcritical in some periodic set and linear in
for sufficiently large . The periodic potential
is such that lies in a spectral gap of . We find a solution
with the energy bounded by a certain min-max level, and infinitely many
geometrically distinct solutions provided that is odd in
- …