The Sattinger iteration method for 1-Laplace type problems and its application to concave-convex nonlinearities
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| Format: | Preprint |
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2024
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| _version_ | 1866915075078488064 |
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| author | Aparicio, Antonio J. Martínez Oliva, Francescantonio Petitta, Francesco |
| author_facet | Aparicio, Antonio J. Martínez Oliva, Francescantonio Petitta, Francesco |
| contents | In this paper we extend the classical sub-supersolution Sattinger iteration method to $1$-Laplace type boundary value problems of the form \begin{equation*}
\begin{cases}
\displaystyle -Δ_1 u = F(x,u) & \text{in}\;Ω,\\ \newline
u=0 & \text{on}\;\partialΩ,
\end{cases} \end{equation*} where $Ω$ is an open bounded domain of $\mathbb{R}^N$ ($N\geq 2$) with Lipschitz boundary and $F(x,s)$ is a Caratheódory function. This goal is achieved through a perturbation method that overcomes structural obstructions arising from the presence of the $1$-Laplacian and by proving a weak comparison principle for these problems. As a significant application of our main result we establish existence and non-existence theorems for the so-called ``concave-convex'' problem involving the $1$-Laplacian as leading term. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_16608 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Sattinger iteration method for 1-Laplace type problems and its application to concave-convex nonlinearities Aparicio, Antonio J. Martínez Oliva, Francescantonio Petitta, Francesco Analysis of PDEs In this paper we extend the classical sub-supersolution Sattinger iteration method to $1$-Laplace type boundary value problems of the form \begin{equation*} \begin{cases} \displaystyle -Δ_1 u = F(x,u) & \text{in}\;Ω,\\ \newline u=0 & \text{on}\;\partialΩ, \end{cases} \end{equation*} where $Ω$ is an open bounded domain of $\mathbb{R}^N$ ($N\geq 2$) with Lipschitz boundary and $F(x,s)$ is a Caratheódory function. This goal is achieved through a perturbation method that overcomes structural obstructions arising from the presence of the $1$-Laplacian and by proving a weak comparison principle for these problems. As a significant application of our main result we establish existence and non-existence theorems for the so-called ``concave-convex'' problem involving the $1$-Laplacian as leading term. |
| title | The Sattinger iteration method for 1-Laplace type problems and its application to concave-convex nonlinearities |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.16608 |