On the existence of nontrivial solutions for p-harmonic equations on unbounded domains

Abstract

In this paper, we consider the existence of nontrivial solutions for the p -harmonic problem { Δ ( | Δ u | p − 2 Δ u ) + a ( x ) | u | p − 2 u = f ( x , u ) , u ∈ W 2 , p ( R N ) , where p ≥ 2 is a constant, N > 2 p , p < q < p ∗ , p ∗ = N p N − 2 p is the critical Sobolev exponent, W 2 , p ( R N ) is a standard Sobolev space; Δ = ∑ i = 1 n ∂ 2 ∂ x i 2 denotes the N -dimensional Laplacian; f ( x , u ) is a given function satisfying some assumptions. It is verified that local Palais–Smale condition holds by using the concentration–compactness principle on R N . Based on this fact, existence results are established via a min–max type theorem for the subcritical case. Meanwhile, we obtain a nonexistence result for a class of p -harmonic equations by proving the corresponding Pohozaev identity in R N .