Asymptotically critical problems on higher-dimensional spheres

Abstract

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-Otani, 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.