Infinite multiplicity of positive solutions of an inhomogeneous supercritical elliptic equation on $\mathbb{R}^N$
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| Format: | Preprint |
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2025
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| _version_ | 1866918021668274176 |
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| author | Katayama, Sho Miyamoto, Yasuhito |
| author_facet | Katayama, Sho Miyamoto, Yasuhito |
| contents | We are concerned with positive radial solutions of the inhomogeneous elliptic equation $Δu+K(|x|)u^p+μf(|x|)=0$ on $\mathbb{R}^N$, where $N\ge 3$, $μ>0$ and $K$ and $f$ are nonnegative nontrivial functions. If $K(r)\sim r^α$, $α>-2$, near $r=0$, $K(r)\sim r^β$, $β>-2$, near $r=\infty$ and certain assumptions on $f$ are imposed, then the problem has a unique positive radial singular solution for a certain range of $μ$. We show that existence of a positive radial singular solution is equivalent to existence of infinitely many positive bounded solutions which are not uniformly bounded, if $p$ is between the critical Sobolev exponent $p_S(α)$ and Joseph-Lundgren exponent $p_{JL}(α)$. Using these theorems, we establish existence of infinitely many positive bounded solutions which are not uniformly bounded, for $p_S(α)<p<p_{JL}(α)$ if $K(r)=r^{-α}$, $α>-2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_10503 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Infinite multiplicity of positive solutions of an inhomogeneous supercritical elliptic equation on $\mathbb{R}^N$ Katayama, Sho Miyamoto, Yasuhito Analysis of PDEs Primary: 35J60, 35B33, secondary 34D05, 35B05 We are concerned with positive radial solutions of the inhomogeneous elliptic equation $Δu+K(|x|)u^p+μf(|x|)=0$ on $\mathbb{R}^N$, where $N\ge 3$, $μ>0$ and $K$ and $f$ are nonnegative nontrivial functions. If $K(r)\sim r^α$, $α>-2$, near $r=0$, $K(r)\sim r^β$, $β>-2$, near $r=\infty$ and certain assumptions on $f$ are imposed, then the problem has a unique positive radial singular solution for a certain range of $μ$. We show that existence of a positive radial singular solution is equivalent to existence of infinitely many positive bounded solutions which are not uniformly bounded, if $p$ is between the critical Sobolev exponent $p_S(α)$ and Joseph-Lundgren exponent $p_{JL}(α)$. Using these theorems, we establish existence of infinitely many positive bounded solutions which are not uniformly bounded, for $p_S(α)<p<p_{JL}(α)$ if $K(r)=r^{-α}$, $α>-2$. |
| title | Infinite multiplicity of positive solutions of an inhomogeneous supercritical elliptic equation on $\mathbb{R}^N$ |
| topic | Analysis of PDEs Primary: 35J60, 35B33, secondary 34D05, 35B05 |
| url | https://arxiv.org/abs/2505.10503 |