Exact multiplicity result for the perturbed scalar curvature problem in $\mathbb {R}^N$ $(N \geq 3)$
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- by S. Prashanth PDF
- Proc. Amer. Math. Soc. 135 (2007), 201-209 Request permission
Abstract:
Let $D^{1,2} (\mathbb {R}^N)$ denote the closure of $C_0^\infty (\mathbb {R}^N)$ in the norm $\|u\|_{D^{1,2} (\mathbb {R}^N)}^2 = \int \limits _{\mathbb {R}^N} |\nabla u|^2.$ Let $N \geq 3$ and define the constants $\alpha _N = N (N-2)$ and $C_N = (N (N-2))^{\frac {N-2}{4}}.$ Let $K \in C^2 (\mathbb {R}^N).$ We consider the following problem for $\varepsilon \geq 0:$ \begin{equation} \tag {$P_\varepsilon $} \left \{\begin {array}{llll} \mbox { Find } u \in D^{1, 2} (\mathbb {R}^N) \mbox { solving}: \\ \left . \begin {array}{lll}-\Delta u &=& \alpha _N (1+ \varepsilon K (x)) u^{\frac {N+2}{N-2}},\\ u &>& 0 \end{array}\right \} \mbox { in } \mathbb {R}^N. \end{array} \right . \end{equation} We show an exact multiplicity result for $(P_\varepsilon )$ for all small $\varepsilon >0$.References
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Additional Information
- S. Prashanth
- Affiliation: TIFR Centre, Indian Institute of Science Campus, P.B. No. 1234, Bangalore - 560 012, India
- Received by editor(s): April 14, 2005
- Received by editor(s) in revised form: July 30, 2005
- Published electronically: June 28, 2006
- Communicated by: David S. Tartakoff
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 201-209
- MSC (2000): Primary 35B32, 35B33
- DOI: https://doi.org/10.1090/S0002-9939-06-08644-8
- MathSciNet review: 2280188