arXiv subjects

Diego Ferraz

Publications and source records attributed to Diego Ferraz.

8 recordsLinked to original sources

arXiv2026

Supercritical Schr\"odinger equations involving integro-differential operators and vanishing potentials

This paper is devoted to the study of the existence of positive and bounded solutions for a Schr\"odinger type equation defined on the entire Euclidean space, involving a general integro-differential operator. We consider the case where the potential is nonnegative and vanishes at infinity with a nonlinearity exhibiting critical or supercritical growth in the Sobolev sense. To overcome the lack of compactness and the difficulties imposed by the general structure of the nonlinearity, we employ variational methods combined with a penalization technique. Unlike the classical fractional Laplacian framework, where specific regularity results, decay estimates, and the $s$-harmonic extension are available, our approach relies on a weak Maximum Principle combined with the construction of a supersolution based on the truncated fundamental solution of the fractional Laplacian to control the asymptotic behavior of the solutions. We prove that, for sufficiently small perturbation parameters and under suitable decay conditions on the potential, the equation admits a nontrivial solution.

math.AP↗

arXiv2026

Application of a profile decomposition theorem to elliptic equations with critical growth

This paper introduces new variational methods centered on the direct application of a profile decomposition theorem for bounded sequences in Sobolev spaces. We employ these methods to prove the existence of ground state solutions for a class of semilinear elliptic equations in $\mathbb{R}^N$ with critical Sobolev growth, set in an asymptotically periodic framework where the coefficients converge to periodic functions at infinity. Our approach successfully addresses highly general nonlinearities, including a subcritical term that does not need to satisfy the classical Ambrosetti-Rabinowitz condition and a critical term that extends far beyond the standard pure power assumption to include functions with oscillatory behavior. We prove the existence of ground states under two alternative conditions: either a strict energy gap between the minimax levels of the original and asymptotic problems or a direct energy comparison between the associated functionals. Some restrictive assumptions, such as specific decay rates for the coefficients or monotonicity properties of the nonlinearities, are not required in our results.

math.AP↗

arXiv2024

Ground states of nonlocal elliptic equations with general nonlinearities via Rayleigh quotient

It is established ground states and multiplicity of solutions for a nonlocal Schr\"{o}dinger equation $(-\Delta )^s u + V(x) u = \lambda a(x) |u|^{q-2}u + b(x)f(u)$ in $\mathbb{R}^N,$ $u \in H^s(\mathbb{R}^N),$ where $0 0,$ under general conditions over the measurable functions $a,$ $b$, $V$ and $f.$ The nonlinearity $f$ is superlinear at infinity and at the origin, and does not satisfy any Ambrosetti-Rabinowitz type condition. It is considered that the weights $a$ and $b$ are not necessarily bounded and the potential $V$ can change sign. We obtained a sharp $\lambda^*> 0$ which guarantees the existence of at least two nontrivial solutions for each $\lambda \in (0, \lambda^*)$. Our approach is variational in its nature and is based on the nonlinear Rayleigh quotient method together with some fine estimates. Compactness of the problem is also considered.

math.AP↗

arXiv2024

Concentration-compactness via profile decomposition for systems of coupled Schrödinger equations of Hamiltonian type

We analyse Hamiltonian-type systems of second-order elliptic PDE invariant under a non-compact group and, consequently, involve a lack of compactness of the Sobolev embedding. We show that the loss of compactness can be compensated by using a concentration-compactness principle via weak profile decomposition for bounded Palais-Smale sequences in Banach spaces. Our analysis to prove the existence of ground states involves a reduction by the inversion method of the system to a fourth-order equation combined with a variational principle of a minimax nature. Among other results, including regularity and a Pohozaev-type identity, we also prove the non-existence of weak solutions for a class of Lane-Emden systems.

math.AP↗

arXiv2023

Existence of bound states for quasilinear elliptic problems involving critical growth and frequency

In this paper we study the existence of bound states of the following class of quasilinear problems, \begin{equation*} \left\{ \begin{aligned} &-\varepsilon ^pΔ_pu+V(x)u^{p-1}=f(u)+u^{p^\ast -1},\ u>0,\ \text{in}\ \mathbb{R}^{N}, &\lim _{|x|\rightarrow \infty }u(x) = 0 , \end{aligned} \right. \end{equation*} where $\varepsilon>0$ is small, $1<p<N,$ $f$ is a nonlinearity with general subcritical growth in the Sobolev sense, $p^{\ast } = pN/(N-p)$ and $V$ is a continuous nonnegative potential. By introducing a new set of hypotheses, our analysis includes the critical frequency case which allows the potential $V$ to not be necessarily bounded below away from zero. We also study the regularity and behavior of positive solutions as $|x|\rightarrow \infty$ or $\varepsilon \rightarrow 0,$ proving that they are uniformly bounded and concentrate around suitable points of $\mathbb{R}^N,$ that may include local minima of $V$.

math.AP↗

arXiv2022

Simpler and efficient characterizations of tree t-spanners for graphs with few P4's and (k, l)-graphs

A tree $t$-spanner of a graph $G$ is a spanning tree $T$ in which the distance between any two adjacent vertices of $G$ is at most $t$. The smallest $t$ for which $G$ has a tree $t$-spanner is called tree stretch index. The $t$-admissibility problem aims to decide whether the tree stretch index is at most $t$. Regarding its optimization version, the smallest $t$ for which $G$ is $t$-admissible is the stretch index of $G$, denoted by $σ_T(G)$. Given a graph with $n$ vertices and $m$ edges, the recognition of $2$-admissible graphs can be done $O(n+m)$ time, whereas $t$-admissibility is NP-complete for $σ_T(G) \leq t$, $t \geq 4$ and deciding if $t = 3$ is an open problem, for more than 20 years. Since the structural knowledge of classes can be determinant to classify $3$-admissibility's complexity, in this paper we present simpler and faster algorithms to check $2$ and $3$-admissibility for families of graphs with few $P_4$'s and $(k,\ell)$-graphs. Regarding $(0,\ell)$-graphs, we present lower and upper bounds for the stretch index of these graphs and characterize graphs whose stretch indexes are equal to the proposed upper bound. Moreover, we prove that $t$-admissibility is NP-complete even for line graphs of subdivided graphs.

cs.DM↗

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↗