On the Chern-Simons-Schrodinger Equation with Critical Exponential Growth

This paper is concerned with the following Chern-Simons-Schrodinger equation -Δu+V(|x|)u+(∫_(|x|)^(∞)h(s)/su^(2)(s)ds+h^(2)(|x|)/|x|^(2))u=a(|x|)f(u)in R^(2),where h(s)=∫_(0)^(s)l/2u^(2)(l)dl,V,a:R^(+)→R are radially symmetric potentials and the nonlinearity f:R→R is of subcritical or critical exponential growth in the sense of Trudinger-Moser.We give some new sufficient conditions on f to obtain the existence of nontrivial solutions or ground state solutions.In particular,some new estimates and techniques are used to overcome the difficulty arising from the critical growth of Trudinger-Moser type.

Existence of Semiclassical States for a Quasilinear Schr?dinger Equation on R^N with Exponential Critical Growth

We study a quasilinear Schrodinger equation {-εN△Nu＋V（x）｜u｜N-2= Q（x）f（u） in R^N,0〈u∈W1,N（RN）,u（x）^｜x｜→∞0,where V, Q are two continuous real functions on R^N and c 〉 0 is a real parameter. Assume that the nonlinearity f is of exponential critical growth in the sense of Trudinger-Moser inequality, we are able to establish the existence and concentration of the semiclassical solutions by variational methods. Keywords Exponential critical growth, semiclassical solutions, variational methods

The Existence of Ground State Solutions for Schrödinger-Kirchhoff Equations Involving the Potential without a Positive Lower Bound

In this paper, we study the following Schrödinger-Kirchhoff equation where V(x) ≥ 0 and vanishes on an open set of R2 and f has critical exponential growth. By using a version of Trudinger-Moser inequality and variational methods, we obtain the existence of ground state solutions for this problem.

The Ground State Solutions for Kirchhoff-Schrodinger Type Equations with Singular Exponential Nonlinearities in R^(N)

In this paper,the authors consider the following singular Kirchhoff-Schrodinger problem M(∫_(R^(N))|∇u|^(N)+V(x)|u|^(N)dx)(−Δ_(N)u+V(x)|u|^(N-2)u)=f(x,u)/|x|^(η)in R^(N),(P_(η))where 0<η<N,M is a Kirchhoff-type function and V(x)is a continuous function with positive lower bound,f(x,t)has a critical exponential growth behavior at infinity.Combining variational techniques with some estimates,they get the existence of ground state solution for(P_(η)).Moreover,they also get the same result without the A-R condition.

Existence of Solutions to Nonlinear Schr?dinger Equations Involving N-Laplacian and Potentials Vanishing at Infinity

We study the existence of solutions for the following class of nonlinear Schr?dinger equations-ΔN u+V(x)u=K(x)f(u)in R^N where V and K are bounded and decaying potentials and the nonlinearity f(s)has exponential critical growth.The approaches used here are based on a version of the Trudinger–Moser inequality and a minimax theorem.