Superharmonic functions in the upper half space with a nonlocal boundary condition
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915135151407104 |
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| author | Ghergu, Marius |
| author_facet | Ghergu, Marius |
| contents | We discuss the existence of positive superharmonic functions $u$ in $\mathbb{R}^N_+=\mathbb{R}^{N-1}\times (0, \infty)$, $N\geq 3$, in the sense $-Δu=μ$ for some Radon measure $μ$, so that $u$ satisfies the nonlocal boundary condition $$ \frac{\partial u}{\partial n}(x',0)=λ\int\limits_{\mathbb{R}^{N-1}}\frac{u(y',0)^p}{|x'-y'|^k}dy' \quad\mbox{ on }\partial \mathbb{R}^N_+, $$ where $p,λ>0$ and $k\in (0, N-1)$. First, we show that no solutions exist if $0<k\leq 1$. Next, if $1<k<N-1$, we obtain a new critical exponent given by $p^*=\frac{N-1}{k-1}$ for the existence of such solutions. If $μ\equiv 0$ we construct an exact solution for $p>p^*$ and discuss the existence of regular solutions, case in which we identify a second critical exponent given by $p^{**}=2\cdot \frac{N-1}{k-1}-1$. Our approach combines various integral estimates with the properties of the newly introduced $α$-lifting operator and fixed point theorems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_01566 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Superharmonic functions in the upper half space with a nonlocal boundary condition Ghergu, Marius Analysis of PDEs 45M20, 45G05, 31B10, 45E10, 45M05 We discuss the existence of positive superharmonic functions $u$ in $\mathbb{R}^N_+=\mathbb{R}^{N-1}\times (0, \infty)$, $N\geq 3$, in the sense $-Δu=μ$ for some Radon measure $μ$, so that $u$ satisfies the nonlocal boundary condition $$ \frac{\partial u}{\partial n}(x',0)=λ\int\limits_{\mathbb{R}^{N-1}}\frac{u(y',0)^p}{|x'-y'|^k}dy' \quad\mbox{ on }\partial \mathbb{R}^N_+, $$ where $p,λ>0$ and $k\in (0, N-1)$. First, we show that no solutions exist if $0<k\leq 1$. Next, if $1<k<N-1$, we obtain a new critical exponent given by $p^*=\frac{N-1}{k-1}$ for the existence of such solutions. If $μ\equiv 0$ we construct an exact solution for $p>p^*$ and discuss the existence of regular solutions, case in which we identify a second critical exponent given by $p^{**}=2\cdot \frac{N-1}{k-1}-1$. Our approach combines various integral estimates with the properties of the newly introduced $α$-lifting operator and fixed point theorems. |
| title | Superharmonic functions in the upper half space with a nonlocal boundary condition |
| topic | Analysis of PDEs 45M20, 45G05, 31B10, 45E10, 45M05 |
| url | https://arxiv.org/abs/2502.01566 |