Ground state solutions for some indefinite variational problems

We present a unified approach to the method of Nehari manifold for functionals which have a local minimum at 0 and we give several examples where this method is applied to the problem of finding ground states and multiple solutions for nonlinear elliptic boundary value problems. We also consider a recent generalization of this method to problems where 0 is a saddle point of the functional.

The method of Nehari manifold

We consider the semilinear Schrodinger equation $$\left\{ {\begin{array}{*{20}c}
 { - \Delta u + V(x)u = f(x,u), \mathbb{R}^N ,} \\
 {u \in H^1 (\mathbb{R}^N ),} \\
 \end{array} } \right.$$
 where f is a superlinear, subcritical nonlinearity. We mainly study the case where V(x) = V
 0(x) + V
 1(x), V
 0 ∈ C(ℝ
 N
 ), V
 0(x) is 1-periodic in each of x
 1, x
 2, …, x
 N
 and sup[σ(−Δ + V
 0) ∩ (−∞, 0)] < 0 < inf[σ(−Δ + V
 0) ∩ (0, ∞)], V
 1 ∈ C(ℝ
 N
 ) and lim|x|→∞
 V
 1(x) = 0. Inspired by previous work of Li et al. (2006), Pankov (2005) and Szulkin and Weth (2009), we develop a more direct approach to generalize the main result of Szulkin and Weth (2009) by removing the “strictly increasing” condition in the Nehari type assumption on f(x, t)/|t|. Unlike the Nahari manifold method, the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the Nehari-Pankov manifold N
 0 by using the diagonal method.

Non-Nehari manifold method for asymptotically periodic Schrödinger equations

This book provides a comprehensive treatment of the Gross–Pitaevskii equation with a periodic potential; in particular, the localized modes supported by the periodic potential. It takes the mean-field model of the Bose–Einstein condensation as the starting point of analysis and addresses the existence and stability of localized modes. The mean-field model is simplified further to the coupled nonlinear Schrodinger equations, the nonlinear Dirac equations, and the discrete nonlinear Schrodinger equations. One of the important features of such systems is the existence of band gaps in the wave transmission spectra, which support stationary localized modes known as the gap solitons. These localized modes realise a balance between periodicity, dispersion and nonlinearity of the physical system. Written for researchers in applied mathematics, this book mainly focuses on the mathematical properties of the Gross–Pitaevskii equation. It also serves as a reference for theoretical physicists interested in localization in periodic potentials.

Localization in Periodic Potentials: From Schrodinger Operators to the Gross-Pitaevskii Equation

It is shown that the periodic discrete nonlinear Schrodinger equation, with cubic nonlinearity, possesses gap solutions, i.e. standing waves, with the frequency in a spectral gap, that are exponentially localized in the spatial variable. The proof is based on the linking theorem in combination with periodic approximations.

https://iopscience.iop.org/article/10.1088/0951-7715/19/1/002/pdf

Gap solitons in periodic discrete nonlinear Schrödinger equations

We present some results on existence of nontrivial solutions of periodic stationary nonlinear Schrodinger equations. We also sketch an application to nonlinear optics and discuss some open problems.

A. Pankov

Papers

Periodic Nonlinear Schrödinger Equation with Application to Photonic Crystals