Isolated singularities for elliptic equations with convolution terms in a punctured ball
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914166430760960 |
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| author | Ghergu, Marius Yu, Zhe |
| author_facet | Ghergu, Marius Yu, Zhe |
| contents | The purpose of this article is two-fold. First, we investigate the inequality $$ -Δu+V(x) u\geq f\quad\mbox{ in } B_1\setminus\{0\}\subset \mathbb{R}^N , N \geq 2, $$ where $f\in L^1_{loc}(B_1)$. If $V\geq 0$ is radially symmetric, we provide optimal conditions for which any solution $0\leq u\in \mathcal{C}^2(B_1\setminus\{0\})$ of the above inequality satisfies $u, Δu, V(x)u\in L^1_{loc}(B_1)$. This extends a result of H. Brezis and P.-L. Lions (1982), originally established for constant potentials $V$. Second, we investigate the equation $$\displaystyle -Δu + λV(x) u = (K_{α, β} * u^p) u^q \quad\text{in } B_1 \setminus \{0\},$$ where $0\leq V\in \mathcal{C}^{0, ν}( \overline B_1\setminus\{0\})$, $0<ν<1$, $λ, p, q>0$ and $$K_{α, β}(x) = |x|^{-α}\log^β\frac{2e}{|x|}, \quad\text{where } 0 \leq α< N, β\in \mathbb{R}.$$ For $N \geq 3$, we establish sharp conditions on the exponents $α, β, p, q$ under which singular solutions exist and exhibit the asymptotic behavior $u(x) \simeq |x|^{2-N}$ near the origin. For $N = 2$, we provide a classification of the existence and boundedness of solutions based on the local behavior of the potential $V(x)$ near the origin. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_17149 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Isolated singularities for elliptic equations with convolution terms in a punctured ball Ghergu, Marius Yu, Zhe Analysis of PDEs 35J61, 35A23, 35A21, 35B33, 35A08, 35J75 The purpose of this article is two-fold. First, we investigate the inequality $$ -Δu+V(x) u\geq f\quad\mbox{ in } B_1\setminus\{0\}\subset \mathbb{R}^N , N \geq 2, $$ where $f\in L^1_{loc}(B_1)$. If $V\geq 0$ is radially symmetric, we provide optimal conditions for which any solution $0\leq u\in \mathcal{C}^2(B_1\setminus\{0\})$ of the above inequality satisfies $u, Δu, V(x)u\in L^1_{loc}(B_1)$. This extends a result of H. Brezis and P.-L. Lions (1982), originally established for constant potentials $V$. Second, we investigate the equation $$\displaystyle -Δu + λV(x) u = (K_{α, β} * u^p) u^q \quad\text{in } B_1 \setminus \{0\},$$ where $0\leq V\in \mathcal{C}^{0, ν}( \overline B_1\setminus\{0\})$, $0<ν<1$, $λ, p, q>0$ and $$K_{α, β}(x) = |x|^{-α}\log^β\frac{2e}{|x|}, \quad\text{where } 0 \leq α< N, β\in \mathbb{R}.$$ For $N \geq 3$, we establish sharp conditions on the exponents $α, β, p, q$ under which singular solutions exist and exhibit the asymptotic behavior $u(x) \simeq |x|^{2-N}$ near the origin. For $N = 2$, we provide a classification of the existence and boundedness of solutions based on the local behavior of the potential $V(x)$ near the origin. |
| title | Isolated singularities for elliptic equations with convolution terms in a punctured ball |
| topic | Analysis of PDEs 35J61, 35A23, 35A21, 35B33, 35A08, 35J75 |
| url | https://arxiv.org/abs/2511.17149 |