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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.06081 |
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Table of Contents:
- In this paper we find a positive weak solution for a semipositone $p(\cdot )$- Laplacian problem. More precisely, we find a solution for the problem \[ \left\{ \begin{array}{cc} -Δ_{p(\cdot )}u=f(u)-λ& \text{in }Ω\\ u>0 & \text{in }Ω\\ u=0 & \text{on }\partial Ω\end{array}% \right. , \] where $Ω\subset \mathbb{R}^{N}$, $N\geq 2$ is a smooth bounded domain, $f$ is a contiuous function with subcritical growth, $λ>0$ and $Δ_{p(\cdot )}u=\text{div}(\left\vert \nabla u\right\vert ^{p(\cdot )-2}\nabla u)$. Also, we assume an Ambrosetti-Rabinowitz type of condition and using the Mountain Pass arguments, comparision principles and regularity principles we prove the existence of positive weak solution for $λ$ small enough.