arXiv subjects

Ismael S. da Silva

Publications and source records attributed to Ismael S. da Silva.

3 recordsLinked to original sources

arXiv2023

Existence of a positive solution for a class of Schrödinger logarithmic equations on exterior domains

In this paper we will prove the existence of a positive solution for a class of Schrödinger logarithmic equation of the form \begin{equation} \left\{\begin{aligned} -Δu &+ u =Q(x)u\log u^2,\;\;\mbox{in}\;\;Ω,\nonumber &\mathcal{B}u=0 \,\,\, \mbox{on} \,\,\, \partial Ω, \end{aligned} \right. \end{equation} where $Ω\subset \mathbb{R}^N$, $N \geq 3$, is an \textit{exterior domain}, i.e., $Ω^c=\mathbb{R}^N \setminus Ω$ is a bounded smooth domain where $\mathcal{B}u=u$ or $\mathcal{B}u=\frac{\partial u}{\partial ν}$. We have used new approach that allows us to apply the usual $C^1$-variational methods to get a nontrivial solutions for these classes of problems.

math.AP↗

arXiv2023

[Existence of multiple solutions for a Schröndiger logarithmic equation

This paper concerns the existence of multiple solutions for a Schrödinger logarithmic equation of the form \begin{equation} \left\{\begin{aligned} -\varepsilon^2Δu + V(x)u & =u\log u^2,\;\;\mbox{in}\;\;\mathbb{R}^{N},\nonumber u \in H^{1}(\mathbb{R}^{N}), \end{aligned} \right.\leqno{(P_\varepsilon)} \end{equation} where $V:\mathbb{R}^N\longrightarrow \mathbb{R}$ is a continuous function that satisfies some technical conditions and $\varepsilon$ is a positive parameter. We will establish the multiplicity of solution for $(P_\varepsilon)$ by using the notion of Lusternik-Schnirelmann category, by introducing a new function space where the energy functional is $C^1$.

math.AP↗

arXiv2023

New minimax theorems for lower semicontinuous functions and applications

In this paper we prove a version of the Fountain Theorem for a class of nonsmooth functionals that are sum of a $C^1$ functional and a convex lower semicontinuous functional, and also a version of a theorem due to Heinz for this class of functionals. The new abstract theorems will be used to prove the existence of infinitely many solutions for some elliptic problems whose the associated energy functional is of the above mentioned type. We study problems with logarithmic nonlinearity and a problem involving the 1-Laplacian operator.

math.AP↗