Supercritical Hénon-type equation with a forcing term

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