Existence and multiplicity results for singular quasilinear elliptic equation on unbounded domain

In this paper, we are interested in the existence and multiplicity results of solutions for the singular quasilinear elliptic problem with concave–convex nonlinearities 0.1− div ( | x | − ap | ∇ u | p − 2 ∇ u ) = h ( x ) | u | s − 2 u + λH ( x ) | u | r − 2 u , x ∈ Ω ,u ( x ) = 0 , x ∈ ∂Ω ,where Ω ⊂ R N( N ≥ 3 ) is an unbounded exterior domain with smooth boundary ∂ Ω, 1 <  p  <  N ,0 ≤  a  < ( N  −  p ) ∕  p , λ  > 0,1 <  s  <  p  <  r  <  q  =  pN  ∕ ( N  −  pd ), d  =  a  + 1 −  b , a  ≤  b  <  a  + 1. By the variational methods, we prove that problem admits a sequence of solutions u k under the appropriate assumptions on the weight functions H ( x ) and H ( x ). For the critical case, s  =  q , h ( x ) = |  x  |   −  bq , we obtain that problem has at least a nonnegative solution with p  <  r  <  q and a sequence of solutions u k with 1 <  r  <  p  <  q and J ( u k ) → 0 as k  → ∞ , where J ( u ) is the energy functional associated to problem . Copyright © 2013 John Wiley & Sons, Ltd.