Liouville theorems for stable solutions of biharmonic problem

Abstract

Abstract. We prove some Liouville type results for stable solutions to the biharmonic problem ∆ 2 u = u q , u > 0 in R n where 1 < q < ∞. For example, for n ≥ 5, we show that there are no stable classical solution in R n when n+4 n−4 < q ≤ n−8 n −1 + .

Liouville theorems for stable Lane–Emden systems and biharmonic problems

“…It was shown there exists nontrivial finite Morse index solutions of (14) if and only if N ≥ 11 and θ ≥ θ JL . In [51] the nonexistence of stable solutions of (5) was examined. It was shown that there is no positive stable solutions of (5) provided either: N ≤ 8 or N ≥ 9 and 1 < θ < N N −8 + ε N where ε N is some positive, but unknown parameter.…”

“…Critical to our approach in the following result: Lemma 6. [51] Suppose that (u, v) is a stable smooth positive solution of (4) with p = 1. Then by Lemma 7, u is a positive stable solution of (5) and then the results of [51] imply there is some C < ∞ such that…”

On the classification of stable solutions to biharmonic problems in large dimensions

“…where we used (2.3) in the last step. For the second term on the right hand side of (2.20), applying the estimate (2.3) from [15], i.e., (∆u) 2 ≥ 2 p+1 u p+1 , and the fact that ∆u < 0 from [14] or [16], we have…”

On the Lazer-McKenna conjecture and its applications

“…We use the idea of Moser and multiply (1.7) by uη 2 where η ∈ C ∞ 0 (R N ); see [16] and [17]. Note that as u ∈ L ∞ (R N ), by the regularity theory u ∈ C 4 (R N ).…”