Existence of positive solutions for a semipositone $p(\cdot)$-Laplacian problem
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| Format: | Preprint |
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2024
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| _version_ | 1866910641095180288 |
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| author | Vallejos, Lucas A. Vidal, Raúl E. |
| author_facet | Vallejos, Lucas A. Vidal, Raúl E. |
| 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_06081 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence of positive solutions for a semipositone $p(\cdot)$-Laplacian problem Vallejos, Lucas A. Vidal, Raúl E. Analysis of PDEs 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. |
| title | Existence of positive solutions for a semipositone $p(\cdot)$-Laplacian problem |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2410.06081 |