Multiplicity of singular solutions for semilinear elliptic equations with superlinear source terms
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908469324414976 |
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| author | Fujishima, Yohei Ioku, Norisuke |
| author_facet | Fujishima, Yohei Ioku, Norisuke |
| contents | This paper investigates the multiplicity of singular solutions for the nonlinear elliptic equation $-Δu =f(u)$ near the origin. Applying the classification of nonlinear functions and the transformation, which were developed by the authors, we generalize the multiplicity results known for the concrete model nonlinearity $f(u)=u^p$ with $\frac{N}{N-2}<p<\frac{N+2}{N-2}$. Our result applies to various nonlinearities, such as $f(s)=s^p+s^r$ with $0<r<p$, $f(s)=s^p(\log s)^r$ with $r\in \mathbb{R}$, $f(s)=s^p\exp((\log s)^r)$ with $0<r<1$ and $f(s)=s^p+s^r(\log s)^β$ with $0<r<p$ and $β\in \mathbb{R}$, for $\frac{N}{N-2}<p<\frac{N+2}{N-2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20450 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiplicity of singular solutions for semilinear elliptic equations with superlinear source terms Fujishima, Yohei Ioku, Norisuke Analysis of PDEs This paper investigates the multiplicity of singular solutions for the nonlinear elliptic equation $-Δu =f(u)$ near the origin. Applying the classification of nonlinear functions and the transformation, which were developed by the authors, we generalize the multiplicity results known for the concrete model nonlinearity $f(u)=u^p$ with $\frac{N}{N-2}<p<\frac{N+2}{N-2}$. Our result applies to various nonlinearities, such as $f(s)=s^p+s^r$ with $0<r<p$, $f(s)=s^p(\log s)^r$ with $r\in \mathbb{R}$, $f(s)=s^p\exp((\log s)^r)$ with $0<r<1$ and $f(s)=s^p+s^r(\log s)^β$ with $0<r<p$ and $β\in \mathbb{R}$, for $\frac{N}{N-2}<p<\frac{N+2}{N-2}$. |
| title | Multiplicity of singular solutions for semilinear elliptic equations with superlinear source terms |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.20450 |