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

We consider the following equation−div(K(x)∇u)=λK(x)|x|β|u|q−2u+Q(x)K(x)|u|2⁎−2u,x∈RN, where N≥3, 20 is a parameter, K(x)=exp⁡(|x|α/4), α≥2, β=(α−2)(2⁎−q)(2⁎−2) and 0≤Q(x)∈C(RN). 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.