Multiple positive and sign-changing solutions of an elliptic equation with fast increasing weight and critical growth

Abstract We consider the following equation − div ( K ( x ) ∇ u ) = λ K ( x ) | x | β | u | q − 2 u + Q ( x ) K ( x ) | u | 2 ⁎ − 2 u , x ∈ R N , where N ≥ 3 , 2 q 2 ⁎ = 2 N / ( N − 2 ) , λ > 0 is a parameter, K ( x ) = exp ⁡ ( | x | α / 4 ) , α ≥ 2 , β = ( α − 2 ) ( 2 ⁎ − q ) ( 2 ⁎ − 2 ) and 0 ≤ Q ( x ) ∈ C ( R N ) . Using variational methods and delicate estimates, we establish some existence and multiplicity of positive and sign-changing solutions for the problem, provided that the maximum of Q ( x ) is achieved at different points.