Existence of solutions for singular quasilinear elliptic problems with dependence of the gradient
In this paper we establish existence of solutions to the following boundary value problem involving a $p$-gradient term $$\displaystyle -\Delta_{p} u + g(u)|\nabla u|^p = \lambda u^\sigma+ \Psi(x), \quad u>0 \quad\mbox{in} ~\Omega, \quad u = 0 \quad\mbox{on} ~ \partial\Omega,$$ where $\Delt...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2025-05-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=11410 |
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| Summary: | In this paper we establish existence of solutions to the following boundary value problem involving a $p$-gradient term
$$\displaystyle -\Delta_{p} u + g(u)|\nabla u|^p = \lambda u^\sigma+ \Psi(x), \quad u>0 \quad\mbox{in} ~\Omega, \quad u = 0 \quad\mbox{on} ~ \partial\Omega,$$
where $\Delta_{p}:= \operatorname{div}(|\nabla u|^{p-2}\nabla u)$ is $p$-Laplacian operator, $\Omega\subset \mathbb{R}^N$ $\left (N\geq 3\right )$ is a bounded domain with smooth boundary, $1<p<N$, $0<\sigma<p^*-1$ with $p^*:= pN/\left ( N-p\right )$, $\Psi$ is a measurable function and $g(s)\geq 0$ is a continuous function on the interval $(0,+\infty)$ which may have a singularity at the origin, i.e.~$g(s)\to +\infty$ as $s\to
0$. Using the topological degree theory, under certain assumptions on $\Psi$, we prove the existence of a continuum of positive solutions. |
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| ISSN: | 1417-3875 |