Critical exponents and multiple critical dimensions for polyharmonic operators
For integers \(k\), \(n\) with \(n > 2k \geq 4\), let \(\Delta^k\) denote the \(k\)-th iterate of the \(n\)-dimensional Laplacian, and let \(B\) denote the unit ball in \(\mathbb{R}^n\). The semilinear polyharmonic Dirichlet problem under consideration is \[ (- \Delta)^ku = \lambda u + u |u |^{p - 1} \text{ in } B, \quad \nabla^j u = 0 \text{ on } \partial B, \quad j = 0,1, \ldots, k - 1, \tag{1} \] where \(p + 1 = 2n/(n - 2k)\) is the critical Sobolev exponent and \(\lambda\) is a real parameter. Theorem: If \(2k < n < 2k + 4\), a necessary condition for (1) to have a nontrivial radial solution \(u \in C^{2k} (\overline B)\) is \[ \lambda > {1 \over 4} (n^2 - 4k^2) (2k + 4 - n) (n + 2k - 4) \Lambda_{k - 2}, \tag{2} \] where \(\Lambda_j\) is the lowest (positive) Dirichlet eigenvalue of \((- \Delta)^j\) in \(B\). In the terminology of \textit{P. Pucci} and \textit{J. Serrin} [J. Math. Pures Appl., IX. Sér. 69, No. 1, 55-83 (1990; Zbl 0717.35032)], this result shows that the integers \(n = 2k + 1\), \(2k + 2\), and \(2k + 3\) are all critical dimensions for (1), i.e., the lower bound for \(\lambda\) is positive. Accordingly, the present paper constitutes considerable progress toward resolution of the Pucci-Serrin conjecture that the critical dimensions for (1) (and generalizations) are precisely the \(2k - 1\) integers \(n \in (2k, 4k)\). For \(k = 1\) this conjecture is true by a celebrated result of \textit{H. Brezis} and \textit{L. Nirenberg} [Commun. Pure Appl. Math. 36, 437-477 (1983; Zbl 0541.35029)], and for \(k = 2\) it is Theorem 3 of Pucci and Serrin, loc. cit., corresponding to (2) above for \(k = 2\).
semilinear polyharmonic Dirichlet problem
scientific article; zbMATH DE number 756572