ASYMPTOTICALLY CRITICAL PROBLEMS ON HIGHER-DIMENSIONAL SPHERES

作者：Kristaly Alexandru^*

来源：Discrete and Continuous Dynamical Systems, 2009, 23(3): 919-935.

DOI：10.3934/dcds.2009.23.919

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

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 : R -> 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 symmertry and d >= 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 Kobavashi-Otani, J. Funct. Anal. 214 (2004), 428-449). The L-infinity- and H-1(2)-asymptotic behaviour of the sequences of solutions are also fully characterized.

• 出版日期2009-3