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João Marcos do Ó

Publications and source records attributed to João Marcos do Ó.

44 records · Page 3Linked to original sources
arXiv2017

Concentration-compactness at the mountain pass level for nonlocal Schrödinger equations

The aim of this paper is to study a concentration-compactness principle for inhomogeneous fractional Sobolev space $H^s (\mathbb{R}^N)$ for $0<s\leq N/2.$ As an application we establish Palais-Smale compactness for the Lagrangian associated to the fractional Schrödinger equation $(-Δ)^{s} u + a(x)u= f(x,u)$ for $0<s<1.$ Moreover, we prove the existence of nontrivial nonnegative solutions to this class of elliptic equations for a wide class of possible singular potentials $a(x)$; not necessarily bounded away from zero. We consider possible oscillatory nonlinearities and that may not satisfy the Ambrosetti-Rabinowitz condition and for both cases; subcritical and critical growth range which are superlinear at origin.

math.AP↗
arXiv2016

Concentration-compactness principle for nonlocal scalar field equations with critical growth

The aim of this paper is to study a concentration-compactness principle for homogeneous fractional Sobolev space $\mathcal{D}^{s,2} (\mathbb{R}^N)$ for $0<s<\min\{1,N/2\}.$ As an application we establish Palais-Smale compactness for the Lagrangian associated to the fractional scalar field equation $(-Δ)^{s} u = f(x,u)$ for $0<s<1.$ Moreover, using an analytic framework based on $\mathcal{D}^{s,2}(\mathbb{R}^N),$ we obtain the existence of ground state solutions for a wide class of nonlinearities in the critical growth range.

math.AP↗
arXiv2015

Schrödinger-Poisson systems with a general critical nonlinearity

We consider a Schrödinger-Poisson system involving a general nonlinearity at critical growth and we prove the existence of positive solutions. The Ambrosetti-Rabinowitz condition is not required. We also study the asymptotics of solutions with respect to a parameter.

math.AP↗
arXiv2015

Concentration phenomena for fractional elliptic equations involving exponential critical growth

In this paper, we deal with the following singular perturbed fractional elliptic problem $ ε^{} (-Δ)^{1/2}{u}+V(z)u=f(u)\,\,\, \mbox{in} \,\,\, \mathbb{R}, $ where $ (-Δ)^{1/2}u$ is the square root of the Laplacian and $f(s)$ has exponential critical growth. Under suitable conditions on $f(s)$, we construct a localized bound state solution concentrating at an isolated component of the positive local minimum points of the potential of $V$ as $ε$ goes to $0.$

math.AP↗
arXiv2015

Ground states of nonlocal scalar field equations with Trudinger-Moser critical nonlinearity

We investigate the existence of ground state solutions for a class of nonlinear scalar field equations defined on whole real line, involving a fractional Laplacian and nonlinearities with Trudinger-Moser critical growth. We handle the lack of compactness of the associated energy functional due to the unboundedness of the domain and the presence of a limiting case embedding.

math.AP↗
arXiv2014

Critical and subcritical fractional problems with vanishing potentials

We investigate a class of nonlinear nonautonomous scalar field equations with fractional diffusion, critical power nonlinearity and a subcritical term. The involved potentials are allowed for vanishing behavior at infinity. The problem is set on the whole space and compactness issues have to be tackled.

math.AP↗