Multiplicity of dead core solutions in indefinite elliptic problems
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916849981063168 |
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| author | Bobkov, Vladimir Quoirin, Humberto Ramos |
| author_facet | Bobkov, Vladimir Quoirin, Humberto Ramos |
| contents | We investigate nonnegative solutions of indefinite elliptic problems which enjoy the dead core phenomenon. Our model is the subhomogeneous problem $$ -Δ_p u = (a^+(x) - μa^-(x))|u|^{q-2}u, \quad u \in W_0^{1,p}(Ω), $$ where $Ω$ is a bounded domain in $\mathbb{R}^N$, $1<q<p$, and $μ> 0$. Thanks to a dead core formation property, we obtain multiple nonnegative dead core solutions of this problem for $μ$ large enough. This assertion, in combination with a uniqueness feature, yields an exact multiplicity result, which gives a precise description of the nonnegative solutions set of this problem for large values of $μ$. We also extend the multiplicity result to other classes of problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_14016 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiplicity of dead core solutions in indefinite elliptic problems Bobkov, Vladimir Quoirin, Humberto Ramos Analysis of PDEs We investigate nonnegative solutions of indefinite elliptic problems which enjoy the dead core phenomenon. Our model is the subhomogeneous problem $$ -Δ_p u = (a^+(x) - μa^-(x))|u|^{q-2}u, \quad u \in W_0^{1,p}(Ω), $$ where $Ω$ is a bounded domain in $\mathbb{R}^N$, $1<q<p$, and $μ> 0$. Thanks to a dead core formation property, we obtain multiple nonnegative dead core solutions of this problem for $μ$ large enough. This assertion, in combination with a uniqueness feature, yields an exact multiplicity result, which gives a precise description of the nonnegative solutions set of this problem for large values of $μ$. We also extend the multiplicity result to other classes of problems. |
| title | Multiplicity of dead core solutions in indefinite elliptic problems |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.14016 |