arXiv subjects

Artur Jorge Marinho

Publications and source records attributed to Artur Jorge Marinho.

6 recordsLinked to original sources

arXiv2026

Solutions to critical equations with a superposition of nonlocal Hartree-type nonlinearities

We study a class of nonlinear nonlocal elliptic equations in $\mathbb{R}^N$ involving superpositions of Hartree-type nonlinearities. Motivated by the Schrödinger-Poisson-Slater system, these equations arise as natural generalizations of problems with a single nonlocal interaction term. More precisely, we consider equations driven by a family of Riesz potentials weighted by a positive Borel measure, which gives rise to a superposed nonlocal operator. To treat this problem variationally, we introduce suitable functional settings, namely the superposed Coulomb space and the associated superposed Coulomb-Sobolev space, and study their main properties. Combining variational methods with a recently developed scaling-based critical point theory, we prove existence and multiplicity results for radial solutions. We also investigate a Brezis-Nirenberg-type problem and obtain multiplicity results near eigenvalues of an associated nonlinear eigenvalue problem. Our results extend previous works on single Hartree-type equations and provide a unified framework for treating superpositions of nonlocal interactions of Hartree type.

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arXiv2026

Anisotropic quasilinear elliptic systems involving nonlinearities with negative critical terms

In this paper we investigate existence and multiplicity of solutions for a critical growth anisotropic quasilinear elliptic system, depending on two real parameters $λ,\,μ$, that is coupled through a subcritical perturbation term and involves a negative critical part. We identify a specific scaling for the system and introduce a parameter $γ$ associated with this scaling, which governs the geometry of the corresponding variational functional. We establish three distinct types of multiplicity results, depending on the regimes $γ= 1$, $γ> 1$, and $γ< 1$. The multiple results of multiplicity of solutions are based on a recent abstract critical point theorem for symmetric functionals on product spaces when $γ\geq1$ and on a truncation argument for $γ<1$.

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arXiv2026

Critical quasilinear elliptic systems with convex subcritical perturbations

We study a class of Dirichlet problems for coupled quasilinear elliptic systems driven by the $p$--Laplacian in a bounded domain, where the nonlinear terms split into a critical homogeneous part and a convex subcritical perturbation. Using variational methods, a nonlinear eigenvalue theory based on the Fadell--Rabinowitz $\mathbb Z_2$--cohomological index, and a refined linking construction, we prove the existence of a nontrivial solution for every parameter value under suitable dimensional assumptions. We also treat a nonhomogeneous version with small forcing terms by applying a recent abstract perturbation theorem, yielding the existence of two distinct nontrivial solutions.

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arXiv2026

Doubly critical mixed-order quasilinear equations: existence, multiplicity and regularity

We investigate a class of mixed-order quasilinear elliptic equations in $\mathbb R^N$ driven by the $q$-biharmonic and $p$-Laplacian operators, \[ Δ_q^2u-Δ_pu = λ\frac{|u|^{r_σ-2}u}{|x|^σ} +|u|^{p^*-2}u +|u|^{q^{**}-2}u, \] where $1<q<N/2$, $1<p<q^*$, $p\ne q$, $q\ne p^*$, and the weighted exponent is determined by the natural scaling of the equation. We establish compact weighted embeddings compatible with this mixed scaling and develop the corresponding variational framework. As an application, we prove existence and multiplicity of nontrivial solutions under suitable assumptions on the parameters. We also establish a local regularity result for the more general equation $Δ_q^2u-Δ_pu=f(x,u),$ where $f$ is a Carathéodory function with local critical growth. By combining nonlinear potential estimates, a regularity-lifting argument, and a finite bootstrap for an associated second-order system, we obtain higher local regularity of both $u$ and the nonlinear flux $|Δu|^{q-2}Δu$. This regularity result, which is also of independent interest, enables us to derive a Pohozaev-type identity for the equations under consideration.

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arXiv2026

New solutions to Schrödinger-Poisson-Slater equations in Coulomb-Sobolev spaces

We prove existence and multiplicity results for the nonlinear and nonlocal PDE $$ - Δu + (I_α\star |u|^p)\, |u|^{p-2}\, u = f(|x|,u) \quad \textrm{in} \,\,\mathbb {R}^N, $$ where $N \geq 2$, $I_α: \mathbb{R}^N \setminus \{0\} \rightarrow \mathbb{R}$ is the Riesz potential of order $α\in (1,N),$ $p>1,$ and the local nonlinearity $f: [0,\infty) \times \mathbb{R} \rightarrow \mathbb R$ is subject to a new class of assumptions. We find solutions to this zero-mass problem in a Coulomb-Sobolev space using a new scaling based approach in critical point theory, by which we classify the possibly different behaviour of the nonlinearity $f$ at zero and at infinity in terms of the scaling properties of the left hand side of the equation. This is accomplished identifying a scaling invariant PDE which can be interpreted as a nonlinear eigenvalue problem, for which a sequence of eigenvalues $\{λ_k\}$ is conveniently defined via the ${\mathbb{Z}}_2$-cohomological index of Fadell and Rabinowitz. This index allows us to use new critical group estimates (and scaling-based linking sets) which might not be possible via the classical genus. Within a fairly broad set of parameters $N,α, p$ and class of assumptions on the local nonlinearity $f,$ we establish compactness results for an associated action functional and find multiple solutions as critical points, whose existence and number is sensitive to the ''resonance'' of $f$ with the sequence of eigenvalues for the scaling invariant problem, a construction which is at places reminiscent, in the present nonlinear setting, of the classical Fredholm alternative. As a byproduct of our analysis, letting $p\neq 2$ allows us to capture general nonlinearities $f$ of Sobolev-subcritical, critical, or supercritical growth.

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arXiv2024

Local and nonlocal critical growth anisotropic quasilinear elliptic systems

In this paper we prove new multiplicity results for a critical growth anisotropic quasilinear elliptic system that is coupled through a subcritical perturbation term. We identify a certain scaling for the system and a parameter γ related to this scaling that determines the geometry of the associated variational functional. This leads to a natural classification of different nonlinear regimes for the system in terms of scaling properties of the perturbation term. We give three different types of multiplicity results in the three regimes γ = 1, γ > 1, and γ < 1. Proofs of our multiplicity results are based on a new abstract critical point theorem for symmetric functionals on product spaces, which we prove using the piercing property of the Z2-cohomological index of Fadell and Rabinowitz. This abstract result only requires a local (PS) condition and is therefore applicable to systems with critical growth. It is of independent interest as it has wide applicability to many different types of critical elliptic systems. We also indicate how it can be applied to obtain similar multiplicity results for nonlocal systems.

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