Nonlinear elliptic equations on \(S_n\) in the supercritical case
The authors study some elliptic nonlinear differential equations on the sphere \(S_n\) \((n\geq 3)\) with data invariant under the group \(G=O(m)\times O(k)\subset O(n+1)\) with \(m+k=n+1\) and \(k\geq m\geq 2.\) In the first part of the paper they establish the value of the best Sobolev constant of the embedding of \(H= \{ u\in H_1(S_n):u\) invariant under \(G \}\) into \(L_q\), \(q=2k/(k-2)\). Then they apply the result to prove the existence of a smooth positive solution for the equation \[ \Delta u+a(x)u=f(x)|u|^{p-1}u\text{ on } S_n, \] where \(a\) and \(f\) are smooth and \(G\)-invariant, \(\Delta+a\) is coercive and \(f\) is positive somewhere, \(n\geq 4\), \(p=q\). This extends previous results by \textit{A. Cotsiolis} and \textit{D. Iliopoulos} [C. R. Acad. Sci. Paris, Sér. I 313, No.~9, 607-609 (1991; Zbl 0747.58044); Bull. Sci. Math. 119, No.~5, 419-431 (1995; Zbl 0836.35044)]. The authors provide a similar existence result for the equation on \(S_3\) \[ \Delta+a(x)=f(x)e^u. \]
elliptic nonlinear differential equations on manifolds
scientific article; zbMATH DE number 1268640