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Asymptotically critical problems on higher-dimensional spheres. (English) Zbl 1154.35051

Summary: 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 J. Kobayashi, M. Ôtani, J. Funct. Anal. 214, No. 2, 428–449 (2004; Zbl 1077.49016)]. The \(L^\infty -\) and \(H_1^2-\)asymptotic behaviour of the sequences of solutions are also fully characterized.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
49J40 Variational inequalities

Citations:

Zbl 1077.49016
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