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March  2005, 4(1): 143-164. doi: 10.3934/cpaa.2005.4.143

Existence of sign changing solutions for some critical problems on $\mathbb R^N$

Norimichi Hirano 1, , A. M. Micheletti 2, and  A. Pistoia 3,

1. 

Department of Mathematics, Yokohama National University, 156 Tokiwadai Hodogaya-ku - Yokohama, Japan
2. 

Dipartimento di Matematica Applicata "U.Dini", Università di Pisa, Via Bonanno 25B - 56126 Pisa, Italy
3. 

Dipartimento di Metodi e Modelli Matematici, Università di Roma "La Sapienza", Via Scarpa, 16 - 00166 Roma

Received  February 2004 Revised  December 2004 Published  December 2004

The main purpose of this paper is to construct families of positive and changing-sign solutions for both the slightly subcritical and slightly supercritical equations

$-\Delta u+V(x)u=N(N-2)|u|^{\frac{4}{N-2}\pm\varepsilon}u$ in $\mathbb R^N,$

which blow-up and concentrate at different points of $\mathbb R^N$ as $\varepsilon$ goes to 0, under certain conditions on the potential $V.$

Keywords: blowing-up solution, sign changing solutions., critical Sobolev exponent.
Mathematics Subject Classification: 35J6.
Citation: Norimichi Hirano, A. M. Micheletti, A. Pistoia. Existence of sign changing solutions for some critical problems on $\mathbb R^N$. Communications on Pure and Applied Analysis, 2005, 4 (1) : 143-164. doi: 10.3934/cpaa.2005.4.143
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