\(L^q\) estimates of functions in the kernel of an elliptic operator and applications
A natural question in Riemannian geometry is whether for a given function \(K: \mathcal S^n\to\mathbb R\) there exists a metric \(g\) conformally related to the standard metric \(\delta_{ij}\) on the unit sphere \(\mathcal S^n\) such that \(K\) is the scalar curvature of \(\mathcal S^n\) with respect to \(g\). It is known that it is equivalent to the problem of finding a positive smooth solution \(u: \mathcal S^n\to\mathbb R\) of the equation NEWLINE\[NEWLINE \Delta u-\frac{n(n-2)}{4}u+\frac{n-2}{4(n-1)}Ku^{\frac{n+2}{n-2}}=0. NEWLINE\]NEWLINE Then \(g=u^{\frac{4}{n-2}}\delta_{ij}\). The exponent \(\frac{n+2}{n-2}\) of \(u\) is critical, and so it is natural to start with subcritical problems NEWLINE\[NEWLINE \Delta u_p-\frac{n(n-2)}{4}u_p+\frac{n-2}{4(n-1)}Ku_p^{p}=0, \quad p\in\big(1,\tfrac{n+2}{n-2}\big), \tag{\(*\)} NEWLINE\]NEWLINE and then consider the limit of solutions \(u_p\) when \(p\to\tfrac{n+2}{n-2}\).NEWLINENEWLINEMotivated by the papers [\textit{R. Schoen} and \textit{D. Zhang}, Calc. Var. Partial Differ. Equ. 4, No. 1, 1--25 (1996; Zbl 0843.53037); \textit{J. F. Escobar} and \textit{G. Garcia}, J. Funct. Anal. 211, No. 1, 71--152 (2004; Zbl 1056.53026)], the authors study the equation NEWLINE\[NEWLINE \Delta u-\frac{n(n-2)}{4}u+\frac{n(n-2)}{4}\text{vol}(\mathcal S^n) \big(\overline{J_p}(y)\big)^{-1}Ku^{p}=0, \tag{\(**\)} NEWLINE\]NEWLINE where \(K: \mathcal S^n\to\mathbb R\) is a Morse function such that \(\Delta K\neq0\) at its critical point, \(\overline{J_p}=J_p\circ\Phi\) with \(J_p\) being (the reciprocal of) the Yamabe quotient in the subcritical case and \(\Phi(y)=|(F_y^{-1})'|^{(n-2)/2}\) considered on \(\mathcal S^n\) (here, for \(y\) on the closed unit ball \(\overline{B^{n+1}}\), \(F_y\) denotes the centered dilation that maps \(0\) to \(y\)). Using a conformal transformation, equation \((**)\) can be replaced by a corresponding one on the orthogonal complement of the first eigenspace of \(\mathcal S^n\). Its solutions \(\eta_y\) are then obtained by the Inverse Function Theorem, and \(L^p\)-estimates for these solutions are established. These solutions lead to solutions of equation \((*)\).
scientific article; zbMATH DE number 6651177