Browsing by Subject "logarithmic nonlinearity"

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Access status: Open Access ,
Ground states for fractional nonlocal equations with logarithmic nonlinearity
(Wydawnictwa AGH, 2022) Guo, Lifeng; Sun, Yan; Shi, Guannan
In this paper, we study on the fractional nonlocal equation with the logarithmic nonlinearity formed by $\begin{cases}\mathcal{L}_{K}u(x)+u\log|u|+|u|^{q-2}u=0, & x\in\Omega,\\ u=0, & x\in\mathbb{R}^{n}\setminus\Omega,\end{cases}$ where $2\lt q\lt 2^{*}_s$, $L_{K}$ is a non-local operator, $\Omega$ is an open bounded set of $\mathbb{R}^{n}$ with Lipschitz boundary. By using the fractional logarithmic Sobolev inequality and the linking theorem, we present the existence theorem of the ground state solutions for this nonlocal problem.
Access status: Open Access ,
Local existence for a viscoelastic Kirchhoff type equation with the dispersive term, internal damping, and logarithmic nonlinearity
(Wydawnictwa AGH, 2024) Cordeiro, Sebastião; Raposo, Carlos; Ferreira, Jorge; Rocha, Daniel; Shahrouzi, Mohammad
This paper concerns a viscoelastic Kirchhoff-type equation with the dispersive term, internal damping, and logarithmic nonlinearity. We prove the local existence of a weak solution via a modified lemma of contraction of the Banach fixed-point theorem. Although the uniqueness of a weak solution is still an open problem, we proved uniqueness locally for specifically suitable exponents. Furthermore, we established a result for local existence without guaranteeing uniqueness, stating a contraction lemma.