arXiv subjects

Genival da Silva

Publications and source records attributed to Genival da Silva.

7 recordsLinked to original sources

arXiv2026

A $p$-Laplacian Schr\"odinger--Maxwell system with rapidly growing nonlinearities

Let $\Omega\subset\R^N$, $N\ge2$, be a bounded domain and let $1<p<\infty$. Inspired by the exponential case treated in \cite{BO2026}, we study the quasilinear Schr\"odinger--Maxwell system \[ \begin{cases} -\Delta_p u+\psi G'(u)=f &\text{in }\Omega,\\ -\Delta_p\psi=G(u) &\text{in }\Omega,\\ u=\psi=0 &\text{on }\partial\Omega, \end{cases} \] where $G\in C^1(\R)$ is even, convex, nonnegative and nontrivial, with $G(0)=0$, and satisfies \[ G(t)\lesssim 1+|G'(t)| \qquad\text{for large }|t|. \] Under the assumption \[ |f|\ln(1+|f|)\in L^1(\Omega), \] we prove the existence of a finite-energy weak solution and show that the solution is a saddle point of the associated functional.

math.AP↗

arXiv2026

An equation involving the 1-Laplacian and a singular nonlinearity

We prove existence of solutions to a nonlinear degenerate elliptic equation of the form \[ \begin{cases} -Δ_{1} u+ \frac{|D u|}{(1-u)^γ}=g & \mbox{in $Ω$,}\\ u=0 \hfill & \mbox{on $\partialΩ$,} \end{cases} \] in a suitable sense, where $Ω$ is a bounded open set of $\mathbb{R}^n$, $γ>0$ is a fixed parameter, $g\geq 0 $ is a function in some Lebesgue space.

math.AP↗

arXiv2024

An elliptic equation with power nonlinearity and degenerate coercivity

We discuss the existence and regularity of solutions to the following Dirichlet problem: $$\begin{equation} \begin{cases} -\textrm{div}\left(\frac{Du}{(1+|u|)^θ}\right)= -\textrm{div}\left(u^γE(x)\right)+f(x) \qquad & \mbox{in } Ω,\\ u (x) = 0 & \mbox{on } \partial Ω, \end{cases} \end{equation}$$ where $θ,γ>0$. An interesting feature of this problem is the interplay between the two nonlinearities, the degeneracy and the power nonlinearity.

math.AP↗

arXiv2024

An inhomogeneous p-laplacian equation with a Hardy potential

In this work we study the existence and regularity of solutions to the following equation: $$-Δ_p u + g(x) u = \fracλ{|x|^{p}} |u|^{p-2}u + f,$$ where $1< p < N$ and $f\in L^m$, where $m\ge 1$.

math.AP↗

arXiv2024

Quasi-linear elliptic equations with superlinear convection

We discuss the existence and regularity of solutions to a quasi-linear elliptic equation involving a Leray-Lions operator and a convection term with superlinear growth. In particular, equations involving the p-Laplacian are covered. This paper generalizes some of the results in Boccardo et al. (2024).

math.AP↗

arXiv2024

On a fully nonlinear k-Hessian system of Lane-Emden type

In this manuscript we prove the existence of solutions to a fully nonlinear system of (degenerate) elliptic equations of Lane-Emden type and discuss a inhomogeneous generalization.

math.AP↗

arXiv2024

Radially symmetric solutions to a Lane-Emden type system

The existence of radially symmetric solutions is discussed for a Lane-Emden type system. This answer a question posed by da Silva and do O (2024). We also comment on the inhomogeneous version of the same system and discuss some open questions.

math.AP↗