Nonexistence of positive radial solutions for semipositone $ϕ$-Laplacian problems with superlinear reaction term
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| Format: | Preprint |
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2025
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| _version_ | 1866911098053066752 |
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| author | Herrón, Sigifredo Lopera, Emer Sánchez, Diana |
| author_facet | Herrón, Sigifredo Lopera, Emer Sánchez, Diana |
| contents | The aim of this paper is to prove the nonexistence of positive radial solutions to the problem $-Δ_ϕu=λf(u)$, $x\in B_1(0)$, $u(x)=0$ on $|x|=1$, for $λ>0$ sufficiently large. Here, $ϕ$ is a continuous function, $Δ_ϕ$ denotes the $ϕ$-Laplacian operator which is defined by $Δ_ϕ(u):=div (ϕ(|\nabla u|) \nabla u)$, and $B_1(0)$ is the unit ball in $\mathbb{R}^N$, with $N>1$. Furthermore, $f$ is a continuous, nondecreasing function such that $f(0)<0$, and its behavior at infinity is intimately related to $ϕ$. Our findings are presented in a combined format, employing both an indirect argument and an energy analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_05930 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonexistence of positive radial solutions for semipositone $ϕ$-Laplacian problems with superlinear reaction term Herrón, Sigifredo Lopera, Emer Sánchez, Diana Analysis of PDEs 35A01, 35A24, 35B09, 35B33, 35G31, 35J62 The aim of this paper is to prove the nonexistence of positive radial solutions to the problem $-Δ_ϕu=λf(u)$, $x\in B_1(0)$, $u(x)=0$ on $|x|=1$, for $λ>0$ sufficiently large. Here, $ϕ$ is a continuous function, $Δ_ϕ$ denotes the $ϕ$-Laplacian operator which is defined by $Δ_ϕ(u):=div (ϕ(|\nabla u|) \nabla u)$, and $B_1(0)$ is the unit ball in $\mathbb{R}^N$, with $N>1$. Furthermore, $f$ is a continuous, nondecreasing function such that $f(0)<0$, and its behavior at infinity is intimately related to $ϕ$. Our findings are presented in a combined format, employing both an indirect argument and an energy analysis. |
| title | Nonexistence of positive radial solutions for semipositone $ϕ$-Laplacian problems with superlinear reaction term |
| topic | Analysis of PDEs 35A01, 35A24, 35B09, 35B33, 35G31, 35J62 |
| url | https://arxiv.org/abs/2508.05930 |