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The prescribed scalar curvature problem for polyharmonic operator. (English) Zbl 1465.35210

Summary: We consider the following prescribed curvature problem involving polyharmonic operator: \[ \begin{aligned}D_mu=Q(|y'|,y'')u^{m^*-1},u> 0,u \in\mathcal{H}^m(\mathbb{S}^N),\end{aligned} \] where \(m^*=\frac{2N}{N-2m}\), \(N\ge 4m+1\), \(m\in\mathbb{N}_+\), \((y',y'')\in\mathbb{R}^2 \times\mathbb{R}^{N-2}\), and \(Q(|y'|,y'')\) is a bounded nonnegative function in \(\mathbb{R}^+ \times\mathbb{R}^{N-2}\). \(\mathbb{S}^N\) is the unit sphere with induced Riemannian metric \(g\), \(D_m\) is the polyharmonic operator given by \(D_m=\prod_{k=1}^m(-\Delta_g+\frac{1}{4}(N-2k)(N+2k-2))\), where \(\Delta_g\) is the Laplace-Beltrami operator on \(\mathbb{S}^N\). By using a finite reduction argument and local Pohozaev-type identities for polyharmonic operator, we prove that if \(N\ge 4m+1\) and \(Q(r,y'')\) has a stable critical point \((r_0,y_0'')\), then the above problem has infinitely many solutions, whose energy can be arbitrarily large.

MSC:

35J60 Nonlinear elliptic equations
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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