Liouville-type theorems and existence results for stable-at-infinity solutions of higher-order m-polyharmonic problems

We discuss the existence and nonexistence of stable-at-infinity solutions of the m-polyharmonic equation Delta(r)(m)u + lambda vertical bar u vertical bar(m-2)u = vertical bar u vertical bar(p-1)u + beta vertical bar u vertical bar(q-1)u in R-N, where m >= 2, N > mr, lambda and beta are nonnegative real parameters and m - 1 < p <= q (see the definition of the m-polyharmonic operator Delta(r)(m) in (1.2)). Precisely, we prove that this problem has no nontrivial stable-at-infinity solutions provided that one of the following conditions holds: lambda > 0 and m* - 1 <= p <= q, where m* = mN/N-rm. lambda = 0 and m - 1 < p < q <= m* - 1. lambda = 0, beta > 0 and p = m* - 1 < q. Also, we prove that the above problem has no nontrivial stable solutions in the following cases: lambda > 0 and m - 1 < p <= q. lambda = 0, m - 1 < p = m*- 1 and m - 1 < q. Finally, when lambda > 0 and m - 1 < p = q <= m* - 1, we establish the existence of infinitely many finite Morse index radial solutions. (C) 2021 Elsevier Inc. All rights reserved.