Liliane A. Maia

SOLUTIONS FOR A RESONANT ELLIPTIC SYSTEM WITH COUPLING IN

ABSTRACT Existence and multiplicity of solutions are established, via the Variational Method, for a class of resonant semilinear elliptic system in under a local nonquadraticity condition at infinity. The main goal is to consider systems with coupling where one of the potentials does not satisfy any coercivity condition. The existence of solution is proved under a critical growth condition on the nonlinearity.

Orbital Stability Property for Coupled Nonlinear Schrodinger Equations

Abstract Orbital stability property for weakly coupled nonlinear Schrödinger equations is investigated. Different families of orbitally stable standing waves solutions will be found, generated by different classes of solutions of the associated elliptic problem. In particular, orbitally stable standing waves can be generated by least action solutions, but also by solutions with one trivial component whether or not they are ground states. Moreover, standing waves with components propagating with the same frequencies are orbitally stable if generated by vector solutions of a suitable Schrödinger weakly coupled system, even if they are not ground states.

Infinitely many nodal solutions for a weakly coupled nonlinear Schrödinger system

Existence of radial solutions with a prescribed number of nodes is established, via variational methods, for a system of weakly coupled nonlinear Schrodinger equations. The main goal is to obtain a nodal solution with all vector components not identically zero and an estimate on their energies.

A sign-changing solution for an asymptotically linear Schrödinger equation

The aim of this paper is to present a sign-changing solution for a class of radially symmetric asymptotically linear Schrodinger equations. The proof is variational and the Ekeland variational principle is employed as well as a deformation lemma combined with Miranda’s theorem.

Asymptotically linear fractional Schrödinger equations

By exploiting a variational technique based upon projecting over the Pohožaev manifold, we prove existence of positive solutions for a class of nonlinear fractional Schrödinger equations having a nonhomogenous nonautonomous asymptotically linear nonlinearity.

Positive solutions for asymptotically linear problems in exterior domains

The existence of a positive solution for a class of asymptotically linear problems in exterior domains is established via a linking argument on the Nehari manifold and by means of a barycenter function.

A quasi-linear Schrödinger equation with indefinite potential

In this paper, we investigate the existence of solution for the quasi-linear Schrödinger equation: in , where g has a subcritical growth and a kind of monotonicity and V is a potential, which changes sign. We employ the mountain pass theorem to obtain the existence of a non-trivial solution.

Positive and Nodal Solutions of Nonlinear Schrödinger Equations in a Saturable Medium

Abstract We consider the nonlinear Schrödinger equation −Δu + V(x)u = f (x, u), x ∈ ℝN, where V is invariant under an orthogonal involution and converges to a positive constant as |x| → +∞ and the nonlinearity f is asymptotically linear at infinity. Existence of a positive solution, as well as some results on the existence of a class of sign-changing solutions is presented. The new idea is to use the projection on the Pohozaev manifold associated with the problem at infinity in order to show that a min-max energy level is in the appropriate range for compactness.

A Note on the Existence of a Positive Solution for a Non-autonomous Schrödinger–Poisson System

We consider the system \( \left\{\begin{array}{cl}{{-\Delta u + V(x)u + K(x)\phi}(x)u = a(x){|u|^{p-1}}u, \quad x \in \mathbb{R}^{3}} \\{-\Delta \phi = K(x)u^{2},}\qquad\qquad\qquad\qquad\qquad\qquad\quad{x\in \mathbb{R}^{3}}\end{array} \right.\) where 3 < p < 5 and the potentials \( K(x), a(x) ]\rm {and} V(x)\) has finite limits as \( |x|\rightarrow + \infty .\) By imposing some conditions on the decay rate of the potentials we obtain the existence of a ground state solution. In the proof we apply variational methods.

Existence of a Positive Solution to a Nonlinear Scalar Field Equation with Zero Mass at Infinity

Abstract We establish the existence of a positive solution to the problem - Δ ⁢ u + V ⁢ ( x ) ⁢ u = f ⁢ ( u ) , u ∈ D 1 , 2 ⁢ ( ℝ N ) , -\Delta u+V(x)u=f(u),\quad u\in D^{1,2}(\mathbb{R}^{N}), for N ≥ 3 {N\geq 3} , when the nonlinearity f is subcritical at infinity and supercritical near the origin, and the potential V vanishes at infinity. Our result includes situations in which the problem does not have a ground state. Then, under a suitable decay assumption on the potential, we show that the problem has a positive bound state.