Ground states for quasilinear Schrödinger equations with critical growth

For a class of quasilinear Schrödinger equations with critical exponent we establish the existence of both one-sign and nodal ground states of soliton type solutions by the Nehari method. The method is to analyze the behavior of solutions for subcritical problems from our earlier work (Liu et al. Commun Partial Differ Equ 29:879–901, 2004) and to pass limit as the exponent approaches to the critical exponent.     

Bound states to critical quasilinear Schr?dinger equations

In this paper, we consider the critical quasilinear Schr?dinger equations of the form

Positive bound states of systems of nonlinear Schrödinger equations

We study the existence of positive bound states of non-autonomous systems of nonlinear Schrödinger equations. Both the singular case and the regular case are discussed. The proof is based on a nonlinear alternative principle of Leray–Schauder.     

Multi-peak positive solutions for nonlinear Schrödinger equations with critical frequency

We study the nonlinear Schrödinger equations: \(-\epsilon^{2}\Delta u + V(x)u=u^p,\quad u > 0\quad \mbox{in } {\bf R}^{N},\quad u\in H^{1} ({\bf R}^{N}).\) where p > 1 is a subcritical exponent and V(x) is nonnegative potential function which has “critical frequency” \(\inf_{x\in{\bf R}^{N}} V(x)=0\). We also assume that V(x) satisfies \(0 < \liminf_{|x|\to\infty}V(x)\le \sup_{x\in{\bf R}^{N}}V(x) < \infty\) and V(x) has k local or global minima. In critical frequency cases, Byeon-Wang [5,6] showed the existence of single-peak solutions which concentrating around global minimum of V(x). Their limiting profiles—which depend on the local behavior of the potential V(x)—are quite different features from non-critical frequency case. We show the existence of multi-peak positive solutions joining single-peak solutions which concentrate around prescribed local or global minima of V(x). Moreover, under additional conditions on the behavior of V(x), we state the limiting profiles of peaks of solutions u _ε(x) as follows: rescaled function \(w_\epsilon(y)=\left(\frac{g(\epsilon)}{\epsilon}\right)^{\frac{2}{p-1}} u_\epsilon(g(\epsilon)y+x_\epsilon)\) converges to a least energy solution of ?Δw + V ₀(y) w = w ^p , w > 0 in Ω₀, \(w\in H^{1}_0(\Omega_0)\). Here g(ε), V ₀(x) and Ω₀ depend on the local behaviors of V(x).     

Ground states for asymptotically periodic Schrödinger-Poisson systems with critical growth

For a class of asymptotically periodic Schrödinger-Poisson systems with critical growth, the existence of ground states is established. The proof is based on the method of Nehari manifold and concentration compactness principle.     

Ground states for Schrödinger–Poisson type systems

In this paper we consider the following elliptic system in \mathbbR³{\mathbb{R}^3}

Expansion with respect to ħ for bound states of the Schrödinger equation

A WKB-complementing expansion for bound states of the radial Schrödinger equation is discussed. A recursive method for calculating the quantum corrections of any order to the energy of the classical motion is presented. The use of quantization conditions makes it possible to write down recursion relations in an equally simple form for the ground and radially excited states. The connection between the approach and the 1/N expansion is considered. It is shown that the method can also be used for analysis in thel, E) plane in the form of a expansion for Regge trajectories.Dnepropetrovsk State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 2, pp. 208–217, February, 1992.     

Standing waves with a critical frequency for nonlinear Schrödinger equations,II

For elliptic equations of the form , where the potential V satisfies , we develop a new variational approach to construct localized bound state solutions concentrating at an isolated component of the local minimum of V where the minimum value of V can be positive or zero. These solutions give rise to standing wave solutions having a critical frequency for the corresponding nonlinear Schrödinger equations. Our method allows a fairly general class of nonlinearity f(u) including ones without any growth restrictions at large.Received: 5 July 2002, Accepted: 24 October 2002, Published online: 14 February 2003The research of the first author was supported by Grant No. 1999-2-102-003-5 from the Interdisciplinary Research Program of the KOSEF.     

Standing waves with large frequency for 4-superlinear Schrödinger–Poisson systems

Annali di Matematica Pura ed Applicata (1923 -) - We consider standing waves with frequency $$omega $$ for 4-superlinear Schrödinger–Poisson system. For large $$omega $$ , the problem...