Abstract
We discuss the existence and nonexistence of stable-at-infinity solutions of the m-polyharmonic equationΔmru+λ|u|m−2u=|u|p−1u+β|u|q−1uin RN, where m≥2, N>mr, λ and β are nonnegative real parameters and m−1<p≤q (see the definition of the m-polyharmonic operator Δmr in (1.2)). Precisely, we prove that this problem has no nontrivial stable-at-infinity solutions provided that one of the following conditions holds:•λ>0 and m⁎−1≤p≤q, where m⁎=mNN−rm.•λ=0 and m−1<p<q≤m⁎−1.•λ=0, β>0 and p=m⁎−1<q. Also, we prove that the above problem has no nontrivial stable solutions in the following cases:•λ>0 and m−1<p≤q.•λ=0, m−1<p≤m⁎−1 and m−1<q. Finally, when λ>0 and m−1<p≤q<m⁎−1, we establish the existence of infinitely many finite Morse index radial solutions.