Existence and non-existence phenomena for nonlinear elliptic equations with $L^1$ data and singular reactions
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2025
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| author | Oliva, Francescantonio Petitta, Francesco Stapenhorst, Matheus F. |
| author_facet | Oliva, Francescantonio Petitta, Francesco Stapenhorst, Matheus F. |
| contents | We study existence and non-existence of solutions for singular elliptic boundary value problems as \begin{equation}\label{eintro}\begin{cases}\tag{1}
\displaystyle -Δ_p u+ \frac{a(x)}{u^γ}=μf(x) \ &\text{ in }Ω, \newline
u>0&\text{ in }Ω, \newline
u = 0 \ &\text{ on }
\partialΩ,
\end{cases} \end{equation} where $Ω$ is a smooth bounded open subset of $\mathbb{R}^N$ ($N\ge 2$), $Δ_p u$ is the $p$-Laplacian with $p>1$, $0<γ\leq 1$, and $a\geq0$ is bounded and non-trivial. For any positive $ f\in L^{1}(Ω)$ we show that problem \eqref{eintro} is solvable for any $μ>μ_0>0$, for some $μ_0$ large enough. As a reciprocal outcome we also show that no finite energy solution exists if $0<μ<μ_{0*}$, for some small $μ_{0*}$.
This paper extends the celebrated one of J. I. Diaz, J. M. Morel and L. Oswald ([16]) to the case $p\neq2$. Our result is also new for $p=2$ provided the singular term has a critical growth near zero (i.e. $γ=1$). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_21131 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence and non-existence phenomena for nonlinear elliptic equations with $L^1$ data and singular reactions Oliva, Francescantonio Petitta, Francesco Stapenhorst, Matheus F. Analysis of PDEs We study existence and non-existence of solutions for singular elliptic boundary value problems as \begin{equation}\label{eintro}\begin{cases}\tag{1} \displaystyle -Δ_p u+ \frac{a(x)}{u^γ}=μf(x) \ &\text{ in }Ω, \newline u>0&\text{ in }Ω, \newline u = 0 \ &\text{ on } \partialΩ, \end{cases} \end{equation} where $Ω$ is a smooth bounded open subset of $\mathbb{R}^N$ ($N\ge 2$), $Δ_p u$ is the $p$-Laplacian with $p>1$, $0<γ\leq 1$, and $a\geq0$ is bounded and non-trivial. For any positive $ f\in L^{1}(Ω)$ we show that problem \eqref{eintro} is solvable for any $μ>μ_0>0$, for some $μ_0$ large enough. As a reciprocal outcome we also show that no finite energy solution exists if $0<μ<μ_{0*}$, for some small $μ_{0*}$. This paper extends the celebrated one of J. I. Diaz, J. M. Morel and L. Oswald ([16]) to the case $p\neq2$. Our result is also new for $p=2$ provided the singular term has a critical growth near zero (i.e. $γ=1$). |
| title | Existence and non-existence phenomena for nonlinear elliptic equations with $L^1$ data and singular reactions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.21131 |