Elliptic equation with singular potential
We consider the following problem of finding a nonnegative function $u(x)$ in a ball $B = B(O, R) ⊂ R^n,\; n ≥ 3:$ $$−Δu=V(x)u,u|∂B=ϕ(x),$$ where $Δ$ is the Laplace operator, $x = (x 1, x 2,…, x n )$, and $∂B$ is the boundary of the ball $B$. It is assumed that $0 ≤ V(x) ∈ L 1(B), 0 ≤ φ(x) ∈ L 1(∂B...
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| Дата: | 2010 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2010
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/2994 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| Резюме: | We consider the following problem of finding a nonnegative function $u(x)$ in a ball $B = B(O, R) ⊂ R^n,\; n ≥ 3:$ $$−Δu=V(x)u,u|∂B=ϕ(x),$$
where $Δ$ is the Laplace operator, $x = (x 1, x 2,…, x n )$, and $∂B$ is the boundary of the ball $B$. It is assumed that $0 ≤ V(x) ∈ L 1(B), 0 ≤ φ(x) ∈ L 1(∂B)$, and $φ(x)$ is continuous on $∂B$.
We study the behavior of nonnegative solutions of this problem and prove that there exists a constant $C_{*} (n) = (n − 2)^2/4$ such that if $V_0 (x) = \frac{c}{|x|^2}, then, for $0 ≤ c ≤ $C_{*} (n)$ and $V(x) ≤ V_0 (x)$ in the ball $B$, this problem has a nonnegative solution for any nonnegative continuous boundary function $φ(x) ∈ L_1(∂B)$, whereas, for $c > C_{*} (n)$ and $V(x) ≥ V_0(x)$, the ball $B$ does not contain nonnegative solutions if $φ(x) > 0$. |
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