The existence of a nontrivial solution to p‐Laplacian equations in R N with supercritical growth

In this paper, we consider the p ‐Laplacian equations inR Nwith supercritical growth− △ p u + V ( x ) | u | p − 2 u = f ( u ) ,x ∈R N ,u ∈W 1 , p(R N) ,where △  p u  = div( | ∇  u  |  p  − 2  ∇  u ),1 <  p  <  N is the p ‐Laplacian operator. Under certain assumptions on V ( x ) and f ( u ) that will be given in Section 1, we prove that the problem has at least a nontrivial solution by using variational methods combined with perturbation arguments. The solutions to subcritical p‐Laplacian equations are estimated applying the L  ∞  norm. Copyright © 2012 John Wiley & Sons, Ltd.