Best constants in second-order Sobolev inequalities on Riemannian manifolds and applications.
Let \(M\) be a compact Riemannian manifold of dimension \(n\geq 3\) and let \(1<p<n/2\). Set \(p^*=np/(n-2p)\). If \(M\) has no boundary look at the space \(E=H^{2,p}(M)\), if \(M\) has nonempty boundary let \(E=H^{2,p}_0(M)\) or \(E=H^{2,p}(M)\cap H^{1,p}_0(M)\). By the Sobolev embedding theorem the inclusion \(E \subset L^{p^*}(M)\) is continuous. The authors study the problem of best constants for \[ \| u\| ^p_{L^{p^*}(M)} \leq A\| \Delta u\| ^p_{L^{p}(M)} + B \| u\| ^p_{L^{p}(M)}. \] The results are then applied to construct solutions to certain nonlinear elliptic fourth order equations.
scientific article; zbMATH DE number 1985984
scientific article