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Title: | Singular semilinear elliptic equations in the half-space | Authors: | Tintarev, Kyril | Keywords: | singular elliptic operators; semilinear elliptic equations; critical exponent; convergence; concentration compactness | Issue Date: | 2001 | Publisher: | Università degli Studi di Trieste. Dipartimento di Scienze Matematiche | Source: | Kyril Tintarev, "Singular semilinear elliptic equations in the half-space", in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 33 (2001), pp. 327-337. | Series/Report no.: | Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics 33 (2001) |
Abstract: | We show that equation $x_{N}^{q}\Delta u+u^{p-1}=0$ on the half-space $Y=\mathbf{R}^{N-1}\times\left(0,\infty\right)$ and on some of its subsets has a ground state solution for $q=N-\frac{p\left(N-2\right)}{2},\; p\;\epsilon\left(2,2*\right)$. For N $\geq$ 3 the end point cases p=2 and p=2{*} correspond to eh Hardy inequality and the limit exponent Sobolev inequality respectively. For N=2 the problem can be interpreted in terms of Laplace-Beltrami operator on the hyperbolic half-plane. |
Type: | Article | URI: | http://hdl.handle.net/10077/4230 | ISSN: | 0049-4704 |
Appears in Collections: | Rendiconti dell'Istituto di Matematica dell'Università di Trieste: an International Journal of Mathematics vol.33 (2001) |
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