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Liliane A. Maia

Publications and source records attributed to Liliane A. Maia.

12 recordsLinked to original sources
arXiv2023

The Nehari manifold for a degenerate logistic parabolic equation

The present paper analyses the behavior of solutions to a degenerate logistic equation with a nonlinear term of the form b(x)f(u), where the weight function b is assumed to be nonpositive. We exploit variational techniques and comparison principle in order to study the evolutionary dynamics. A crucial role is then played by the Nehari manifold, as we note how it changes as the parameter λ in the equation or the function b vary, affecting the existence and non-existence of stationary solutions. We describe a detailed picture of the positive dynamics and also address the local behavior of solutions near a nodal equilibrium, which sheds some further light on the study of the evolution of sign-changing solutions.

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arXiv2022

An upper bound for the least energy of a sign-changing solution to a zero mass problem

We give an upper bound for the least energy of a sign-changing solution to the the nonlinear scalar field equation $$-Δu = f(u), \qquad u\in D^{1,2}(\mathbb{R}^{N}),$$ where $N\geq5$ and the nonlinearity $f$ is subcritical at infinity and supercritical near the origin. More precisely, we establish the existence of a nonradial sign-changing solution whose energy is smaller that $12c_0$ if $N=5,6$ and smaller than $10c_0$ if $N\geq 7$, where $c_0$ is the ground state energy.

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arXiv2019

An Indefinite Elliptic Problem on RN Autonomous at Infinity: The Crossing Effect of the Spectrum and the Nonlinearity

We present a new approach to solve a Schrödinger Equation autonomous at infinity, by identifying the relation between the arrangement of the spectrum of the concerned operator and the behavior of the nonlinearity at zero and at infinity. In order to apply variational methods, we set up a suitable linking structure depending on the growth of the nonlinear term and making use of information about the autonomous problem at infinity. Our method allows us to circumvent the lack of compactness. The main novelty is that none monotonicity assumption is required on the nonlinearity, which may be sign-changing as well as the potential. Furthermore, depending on the nonlinearity, the limit of the potential at infinity may be non-positive, so that zero may be an interior point in the essential spectrum of the Schrödinger operator.

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arXiv2019

A Note on Nonlinear Schrödinger Equations: Unveiling the Relation Between Spectral Gaps and the Nonlinearity Behavior

We establish the existence of a nontrivial weak solution to strongly indefinite asymptotically linear and superlinear Schrödinger equations. The novelty is to identify the essential relation between the spectrum of the operator and the behavior of the nonlinear term, in order to weaken the necessary assumptions to obtain a linking structure to the problem, for instance to allow zero being in the spectrum or the nonlinearity being sign-changing. Our main difficulty is to overcome the lack of monotonicity on the nonlinear term, as well, as the lack of compactness since the domain is unbounded. With this purpose, we require periodicity on $V$.

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arXiv2019

An abstract linking theorem applied to indefinite problems via spectral properties

An abstract linking result for Cerami sequences is proved without the Cerami condition. It is applied directly in order to prove the existence of critical points for a class of indefinite problems in infinite dimensional Hilbert Spaces. The main applications are given to Hamiltonian systems and Schrodinger equations. Here spectral properties of the operators are exploited and hypotheses of monotonicity on the nonlinearities are discarded.

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arXiv2018

Non-Cooperative Systems Modeling Repulsive Interaction of Bose-Einstein Condesates in RN

Inspired by so many possible applications of this class of problems, we seek solution for non-cooperative elliptic systems of two Schrodinger equations. General conditions are assumed under the potentials, which produces convenient spectral properties on the elliptic operator concerned, hence the non-cooperation characterizes it as a strongly indefinite problem. An Abstract Linking Theorem developed previously by the first two authors is the main tool, since variational approach is applied. Furthermore, super and asymptotically quadratic nonlinearities are considered.

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arXiv2018

Positive multipeak solutions to a zero mass problem in exterior domains

We establish the existence of positive multipeak solutions to the nonlinear scalar field equation with zero mass $$-Δu = f(u), \qquad u\in D_0^{1,2}(Ω_R),$$ where $Ω_R:=\{x \in \mathbb{R}^N:|u|>R\}$ with $R>0$, $N\geq4$, and the nonlinearity $f$ is subcritical at infinity and supercritical near the origin. We show that the number of positive multipeak solutions becomes arbitrarily large as $R \to \infty$.

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arXiv2017

Existence of a positive solution to a nonlinear scalar field equation with zero mass at infinity

We establish the existence of a positive solution to the problem $$-Δu+V(x)u=f(u),\qquad u\in D^{1,2}(\mathbb{R}^{N}),$$ for $N\geq3$, 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.

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arXiv2014

On fractional p-Laplacian problems with weight

We investigate the existence of nonnegative solutions for a nonlinear problem involving the fractional p-Laplacian operator. The problem is set on a unbounded domain, and compactness issues have to be handled.

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arXiv2014

Asymptotically linear fractional Schrodinger equations

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

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