Asymptotically critical problems on higher-dimensional spheres

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July 2009, 23(3): 919-935. doi: 10.3934/dcds.2009.23.919

Asymptotically critical problems on higher-dimensional spheres

1.	Babeş-Bolyai University, Department of Economics, 400591 Cluj-Napoca

Received February 2008 Revised July 2008 Published November 2008

Abstract
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The paper is concerned with the equation $-\Delta_{h}u=f(u)$ on $S^d$ where $\Delta_{h}$ denotes the Laplace-Beltrami operator on the standard unit sphere $(S^d,h)$, while the continuous nonlinearity $f:\mathbb R\to \mathbb R$ oscillates either at zero or at infinity having an asymptotically critical growth in the Sobolev sense. In both cases, by using a group-theoretical argument and an appropriate variational approach, we establish the existence of $[{d}/{2}]+(-1)^{d+1}-1$ sequences of sign-changing weak solutions in $H_1^2(S^d)$ whose elements in different sequences are mutually symmetrically distinct whenever $f$ has certain symmetry and $d\geq 5$. Although we are dealing with a smooth problem, we are forced to use a non-smooth version of the principle of symmetric criticality (see Kobayashi-Ôtani, J. Funct. Anal. 214 (2004), 428-449). The $L^\infty$-- and $H_1^2$--asymptotic behaviour of the sequences of solutions are also fully characterized.

Keywords: asymptotically critical growth, sign-changing solution, Szulkin-type functional, non-smooth principle of symmetric criticality, oscillatory term, Laplacian on the sphere, symmetrically distinct solutions.

Mathematics Subject Classification: Primary: 35J65, 49J40.

Citation: Alexandru Kristály. Asymptotically critical problems on higher-dimensional spheres. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 919-935. doi: 10.3934/dcds.2009.23.919

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