Quasilinear asymptotically periodic Schrödinger equations with critical growth

It is established the existence of solutions for a class of asymptotically periodic quasilinear elliptic equations in ${\mathbb{R}^N}$ with critical growth. Applying a change of variable, the quasilinear equations are reduced to semilinear equations, whose respective associated functionals are well defined in ${H^1(\mathbb{R}^N)}$ and satisfy the geometric hypotheses of the Mountain Pass Theorem. The Concentration–Compactness Principle and a comparison argument allow to verify that the problem possesses a nontrivial solution.     

Positive solutions of a Schrödinger equation with critical nonlinearity

We study the nonlinear Schröodinger equation with critical exponent 2^*= 2 N/( N-2), N 4, where a 0, has a potential well. Using variational methods we establish existence and multiplicity of positive solutions which localize near the potential well for small and large.     

Quasi-linear Schrödinger–Poisson system under an exponential critical nonlinearity: existence and asymptotic behaviour of solutions

We prove the generalized $$L^p$$ -Poincaré inequalities and Sobolev type inequalities on graphs with polynomial volume growth. They are optimal on Vicsek graphs.     

Positive solutions of a Schr?dinger equation with critical nonlinearity

We study the nonlinear Schröodinger equation -Du+la(x)u=mu+u^2*-1,  u ? \mathbbR^N,-\Delta u+\lambda a(x)u=\mu u+u^{2^{\ast }-1},{ \ }u\in \mathbb{R}^{N}, with critical exponent 2^*= 2 N/( N-2), N 4, where a 0, has a potential well. Using variational methods we establish existence and multiplicity of positive solutions which localize near the potential well for small and large.     

Ground state solutions for asymptotically periodic Schrödinger equations with indefinite linear part

Using the method of the Nehari manifold, we study the existence of ground state solutions for asymptotically periodic Schrödinger equations with indefinite linear part and superlinear nonlinearity. Copyright © 2014 John Wiley & Sons, Ltd.     

Infinitely many positive solutions for a Schrödinger–Poisson system

We are interested in the existence of infinitely many positive non-radial solutions of a Schrödinger–Poisson system with a positive radial bounded external potential decaying at infinity.     

Fractional Schrödinger–Poisson system with critical growth and potentials vanishing at infinity

In this paper, we are concerned with the existence of positive solutions for a class of fractional Schrödinger–Poisson system with critical nonlinearity and multiple competing potentials, which may decay and vanish at infinity. Under some local conditions, we show the existence and concentration of positive solutions by using the modified penalization method and concentration–compactness principle.     

Existence and multiplicity of positive solutions to Schrödinger–Poisson type systems with critical nonlocal term

The existence, nonexistence and multiplicity of positive radially symmetric solutions to a class of Schrödinger–Poisson type systems with critical nonlocal term are studied with variational methods. The existence of both the ground state solution and mountain pass type solutions are proved. It is shown that the parameter ranges of existence and nonexistence of positive solutions for the critical nonlocal case are completely different from the ones for the subcritical nonlocal system.     

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.