Journal of innovative applied mathematics and computational sciences
Volume 3, Numéro 2, Pages 142-155
2024-01-21
Existence Of Solutions For A Class Of Kirchhoff Type Problem With Triple Regime Logarithmic Nonlinearity
Authors : Boudjeriou Tahir .
Abstract
In this paper, we use variational methods to study the existence of nontrivial solutions for a class of Kirchhoff-type elliptic problems driven by the $p(x)$-Laplacian with triple regime and sign-changing nonlinearity. Specifically, we consider the following equation \begin{equation*} \left\{\begin{array}{llc} -m\left(\int_{B_{R}}\frac{1}{p(x)} |\nabla u|^{p(x)}\,dx\right)\Delta_{p(x)} u=|u|^{q(x)-2}u\log(|u|)+|u|^{q(x)-2}u & \text{in}\ & B_{R}, \\ u=0, & \text{in}\ & \partial B_{R}.\\ \end{array}\right. \end{equation*} Here, $B_{R}$ represents the open ball in $\mathbb{R}^{N}$ $(N\geq 1)$ centered at zero with a radius of $R>0$. The functions $m :\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$, and $p,q :\overline{B}_{R}\rightarrow \mathbb{R}^{+}$ are continuous and satisfy certain conditions. The main novelty of this paper is our ability to establish an existence result for a class of Kirchhoff-type problems in which the reaction term is sign-changing and exhibits a triple regime (subcritical, critical, and supercritical).
Keywords
Variational methods ; p(x)-Kirchhoff-type equation ; Nonlocal problems
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