An indefinite concave-convex equation under a Neumann boundary condition II
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arXiv
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| Format: | Preprint |
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2017
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| _version_ | 1866914645373091840 |
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| author | Quoirin, Humberto Ramos Umezu, Kenichiro |
| author_facet | Quoirin, Humberto Ramos Umezu, Kenichiro |
| contents | We proceed with the investigation of the problem $(P_λ): $ $-Δu = λb(x)|u|^{q-2}u +a(x)|u|^{p-2}u \ \mbox{ in } Ω, \ \ \frac{\partial u}{\partial \mathbf{n}} = 0 \ \mbox{ on } \partial Ω$, where $Ω$ is a bounded smooth domain in $\mathbb{R}^N$ ($N \geq2$), $1<q<2<p$, $λ\in \mathbb{R}$, and $a,b \in C^α(\overlineΩ)$ with $0<α<1$. Dealing now with the case $b \geq 0$, $b \not \equiv 0$, we show the existence (and several properties) of a unbounded subcontinuum of nontrivial non-negative solutions of $(P_λ)$. Our approach is based on a priori bounds, a regularization procedure, and Whyburn's topological method. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1703_04229 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | An indefinite concave-convex equation under a Neumann boundary condition II Quoirin, Humberto Ramos Umezu, Kenichiro Analysis of PDEs We proceed with the investigation of the problem $(P_λ): $ $-Δu = λb(x)|u|^{q-2}u +a(x)|u|^{p-2}u \ \mbox{ in } Ω, \ \ \frac{\partial u}{\partial \mathbf{n}} = 0 \ \mbox{ on } \partial Ω$, where $Ω$ is a bounded smooth domain in $\mathbb{R}^N$ ($N \geq2$), $1<q<2<p$, $λ\in \mathbb{R}$, and $a,b \in C^α(\overlineΩ)$ with $0<α<1$. Dealing now with the case $b \geq 0$, $b \not \equiv 0$, we show the existence (and several properties) of a unbounded subcontinuum of nontrivial non-negative solutions of $(P_λ)$. Our approach is based on a priori bounds, a regularization procedure, and Whyburn's topological method. |
| title | An indefinite concave-convex equation under a Neumann boundary condition II |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/1703.04229 |