Existence and nonexistence of solutions for singular quadratic quasilinear equations
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2025
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| author | Arcoya, David Carmona, José Leonori, Tommaso Martínez-Aparicio, Pedro J. Orsina, Luigi Petitta, Francesco |
| author_facet | Arcoya, David Carmona, José Leonori, Tommaso Martínez-Aparicio, Pedro J. Orsina, Luigi Petitta, Francesco |
| contents | We study both existence and nonexistence of nonnegative solutions for nonlinear elliptic problems with singular lower order terms that have natural growth with respect to the gradient, whose model is $$ \begin{cases}
-Δu + \frac{|\nabla u|^2}{u^γ} = f & \mbox{in } Ω,\newline
\hfill u=0 \hfill & \mbox{on } \partial Ω, \end{cases} $$ where $Ω$ is an open bounded subset of $\mathbb{R}^N $, $γ> 0$ and $f$ is a function which is strictly positive on every compactly contained subset of $Ω$. As a consequence of our main results, we prove that the condition $γ<2$ is necessary and sufficient for the existence of solutions in $H^{1}_{0}(Ω)$ for every sufficiently regular $f$ as above. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_06375 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence and nonexistence of solutions for singular quadratic quasilinear equations Arcoya, David Carmona, José Leonori, Tommaso Martínez-Aparicio, Pedro J. Orsina, Luigi Petitta, Francesco Analysis of PDEs We study both existence and nonexistence of nonnegative solutions for nonlinear elliptic problems with singular lower order terms that have natural growth with respect to the gradient, whose model is $$ \begin{cases} -Δu + \frac{|\nabla u|^2}{u^γ} = f & \mbox{in } Ω,\newline \hfill u=0 \hfill & \mbox{on } \partial Ω, \end{cases} $$ where $Ω$ is an open bounded subset of $\mathbb{R}^N $, $γ> 0$ and $f$ is a function which is strictly positive on every compactly contained subset of $Ω$. As a consequence of our main results, we prove that the condition $γ<2$ is necessary and sufficient for the existence of solutions in $H^{1}_{0}(Ω)$ for every sufficiently regular $f$ as above. |
| title | Existence and nonexistence of solutions for singular quadratic quasilinear equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.06375 |