Finite energy solutions for nonlinear elliptic equations with competing gradient, singular and $L^1$ terms
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866909383799078912 |
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| author | Balducci, Francesco Oliva, Francescantonio Petitta, Francesco |
| author_facet | Balducci, Francesco Oliva, Francescantonio Petitta, Francesco |
| contents | In this paper we deal with the following boundary value problem
\begin{equation*}
\begin{cases}
-Δ_{p}u + g(u) | \nabla u|^{p} = h(u)f & \text{in $Ω$,} \newline
u\geq 0 & \text{in $Ω$,} \newline
u=0 & \text{on $\partial Ω$,} \
\end{cases}
\end{equation*}
in a domain $Ω\subset \mathbb{R}^{N}$ $(N \geq 2)$, where $1\leq p<N $, $g$ is a positive and continuous function on $[0,\infty)$, and $h$ is a continuous function on $[0,\infty)$ (possibly blowing up at the origin). We show how the presence of regularizing terms $h$ and $g$ allows to prove existence of finite energy solutions for nonnegative data $f$ only belonging to $L^1(Ω)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_16129 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Finite energy solutions for nonlinear elliptic equations with competing gradient, singular and $L^1$ terms Balducci, Francesco Oliva, Francescantonio Petitta, Francesco Analysis of PDEs In this paper we deal with the following boundary value problem \begin{equation*} \begin{cases} -Δ_{p}u + g(u) | \nabla u|^{p} = h(u)f & \text{in $Ω$,} \newline u\geq 0 & \text{in $Ω$,} \newline u=0 & \text{on $\partial Ω$,} \ \end{cases} \end{equation*} in a domain $Ω\subset \mathbb{R}^{N}$ $(N \geq 2)$, where $1\leq p<N $, $g$ is a positive and continuous function on $[0,\infty)$, and $h$ is a continuous function on $[0,\infty)$ (possibly blowing up at the origin). We show how the presence of regularizing terms $h$ and $g$ allows to prove existence of finite energy solutions for nonnegative data $f$ only belonging to $L^1(Ω)$. |
| title | Finite energy solutions for nonlinear elliptic equations with competing gradient, singular and $L^1$ terms |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2308.16129 |