On a Schrödinger equation with periodic potential and critical growth in \mathbb{R}^{2} (original) (raw)

On a Schrodinger equation with periodic potential and spectrum point zero

Indiana University Mathematics Journal, 2003

The main purpose of this paper is to establish the existence of a solution of the semilinear Schrödinger equation −∆u + V (x)u = f (u), in R 2 where V is a 1-periodic functions with respect to x, 0 lies in a gap of the spectrum of −∆ + V , and f (s) behaves like ± exp(αs 2) when s → ±∞.

Existence of solutions to asymptotically periodic Schrodinger equations

Electronic Journal of Differential Equations, 2017

We show the existence of a nonzero solution for the semilinear Schrodinger equation −Deltau+V(x)u=f(x,u)-\Delta u+V(x)u=f(x,u)−Deltau+V(x)u=f(x,u). The potential V is periodic and 0 belongs to a gap of sigma(−Delta+V)\sigma(-\Delta +V)sigma(−Delta+V). The function f is superlinear and asymptotically periodic with respect to x variable. In the proof we apply a new critical point theorem for strongly indefinite functionals proved in [3].

On a periodic Schrödinger equation involving periodic and nonperiodic nonlinearities in R2

Journal of Mathematical Analysis and Applications, 2018

We study the existence of solutions for the nonlinear Schrödinger equation −Δu + V (x)u = f (x, u) in R 2 , where the potential V is 1-periodic, 0 lies in a spectral gap from the spectrum of the Schrödinger operator S = −Δ + V and the nonlinearity f (x, t) has exponential growth in the sense of Trudinger-Moser. The main feature here is that f (x, t) is allowed to be both periodic and nonperiodic in the x variable. Our proofs rely on a linking theorem and the Lions concentration compactness principle.

On a Class of Nonlinear Schrödinger Equations in R2 Involving Critical Growth

Journal of Differential Equations, 2001

In this paper we deal with semilinear elliptic problem of the form &= 2 2u+V(z) u= f (u), u # C 2 (R 2) & H 1 (R 2) , u>0, in R 2 in R 2 , where = is a small positive parameter, V: R 2 Ä R is a positive potential bounded away from zero, and f (u) behaves like exp(:s 2) when s Ä +. We prove the existence of solutions concentrating around a local minima not necessarily nondegenerate of V(x), when = tends to 0.

Schrödinger equations with asymptotically periodic terms

Proceedings of the Royal Society of Edinburgh: Section A Mathematics

We study the existence of non-trivial solutions for a class of asymptotically periodic semilinear Schrödinger equations in ℝN. By combining variational methods and the concentration-compactness principle, we obtain a non-trivial solution for the asymptotically periodic case and a ground state solution for the periodic one. In the proofs we apply the mountain pass theorem and its local version.

Ground State Solutions for Quasilinear Schrödinger Equations with Periodic Potential

2020

This article concerns the quasilinear Schrödinger equation −∆u− u∆(u) + V (x)u = K(x)|u|2·2 −2u + g(x, u), x ∈ R , u ∈ H(R ), u > 0, where V and K are positive, continuous and periodic functions, g(x, u) is periodic in x and has subcritical growth. We use the generalized Nehari manifold approach developed by Szulkin and Weth to study the ground state solution, i.e. the nontrivial solution with least possible energy.

Kirchhoff–Schrödinger equations in ℝ2 with critical exponential growth and indefinite potential

Communications in Contemporary Mathematics, 2020

We prove the existence of ground state solution for the nonlocal problem [Formula: see text] where [Formula: see text] is a Kirchhoff type function, [Formula: see text] may be negative and noncoercive, [Formula: see text] is locally bounded and the function [Formula: see text] has critical exponential growth. We also obtain new results for the classical Schrödinger equation, namely the local case [Formula: see text]. In the proofs, we apply Variational Methods besides a new Trudinger–Moser type inequality.

Positive Solutions for a Quasilinear Schrödinger Equation with Critical Growth

Journal of Dynamics and Differential Equations, 2011

We consider the quasilinear problem −ε p div(|∇u| p−2 ∇u) + V (z)u p−1 = f (u) + u p * −1 , u ∈ W 1,p (R N), where ε > 0 is a small parameter, 1 < p < N , p * = N p/(N − p), V is a positive potential and f is a superlinear function. Under a local condition for V we relate the number of positive solutions with the topology of the set where V attains its minimum. In the proof we apply Ljusternik-Schnirelmann theory.

One-Dimensional Schrödinger Equation with an Almost Periodic Potential

Physical Review Letters, 1983

Recent theories of scaling in quasiperiodic dynamical systems are applied to the behavior of a particle in an almost periodic potential. A special tight-binding model is solved exactly by a renormalization group whose fixed points determine the scaling properties of both the energy spectrum and certain features of the eigenstates. Similar results are found empirically for Harper's equation. In addition to ordinary extended and localized states, "critical" states are found which are neither extended nor localized according to conventional criteria.