Supercritical Hénon-type equation with a forcing term

Abstract

Abstract This article is concerned with the structure of solutions to the elliptic problem for a Hénon-type equation with a forcing term: − Δ u = α ( x ) u p + κ μ , in R N , u > 0 , in R N , ( P κ ) \hspace{11.3em}-\Delta u=\alpha \left(x){u}^{p}+\kappa \mu ,\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{1.0em}u\gt 0,\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{13.0em}\left({{\rm{P}}}_{\kappa }) where N ≥ 3 N\ge 3 , p > 1 p\gt 1 , κ > 0 \kappa \gt 0 , and α \alpha is a positive continuous function in R N \ { 0 } {{\mathbb{R}}}^{N}\setminus \left\{0\right\} , and μ \mu is a nonnegative Radon measure in R N {{\mathbb{R}}}^{N} . Under suitable assumptions on the exponent p p , the coefficient α \alpha , and the forcing term μ \mu , we give a complete classification of the existence/nonexistence of solutions to problem ( P κ {{\rm{P}}}_{\kappa } ) with respect to κ \kappa .