On conformally invariant equation \((-\Delta)^p u-K(x)u^{\frac{N+2p}{N-2p}}=0\) and its generalizations
The authors prove several existence and nonexistence results for positive entire solutions of conformally invariant equations of the form \[ (-\Delta)^p u - K(x) u^q = 0 \quad\text{in}\quad \mathbb{R}^N, \] where \(N\geq 2p+1\), \(q>1\), and \(K\) is a Hölder continuous function. They prove the existence of infinitely many positive solutions if \(K\) decays sufficiently fast at infinity, and the nonexistence of positive solutions if \(K\) grows sufficiently fast at infinity. They also establish a Kazdan-Warner type condition for nonexistence of solutions decaying at infinity. The results are similar in spirit to those obtained by \textit{W.-M. Ni} [Indiana Univ. Math. J. 31, 493-529 (1982; Zbl 0496.35036)] and many subsequent authors for the second-order case.
scientific article; zbMATH DE number 1993717