Finite Morse index solutions of weighted elliptic equations and the critical exponents

Abstract

We study the behavior of finite Morse index solutions to the weighted elliptic equation $$\begin{aligned} -\mathrm{div} \left( |x|^\theta \nabla v\right) =|x|^l |v|^{p-1} v \quad {\mathrm{in} \quad \Omega \subset \mathbb {R}^N \; (N \ge 2)}, \end{aligned}$$ where $$\Omega $$ may stand for $$\mathbb {R}^N$$ , $$\mathbb {R}^N\!\setminus \!\{0\}$$ , a punctured ball $$B_R(0)\!\setminus \!\{0\}$$ or an exterior domain $$\mathbb {R}^N\!\setminus \!B_R(0)$$ , the constants p, $$\theta $$ and l satisfy $$\begin{aligned} p>1,\; N':=N+\theta >2,\; \tau :=l-\theta >-2. \end{aligned}$$ We investigated this problem recently in Dancer et al. (J Differ Equ 250:3281–3310, 2011), Du and Guo (Adv Differ Equ 18:737–768, 2013), Du et al. (Calc. Var. PDEs, 2014), with the best results obtained in Du et al. (Calc. Var. PDEs, 2014). In this paper, we show that the main results of Du et al. (Calc. Var. PDEs, 2014) continue to hold when p takes the critical exponent $$p_s:=\frac{N'+2+2\tau }{N'-2}$$ . We also improve the Liouville theorem in Du et al. (Calc. Var. PDEs, 2014) and discuss some related questions. Our results in this paper suggest that the critical exponent $$p_s$$ does not play a significant role in the class of stable solutions, but $$p=p_s$$ becomes an exceptional case in the class of finite Morse index solutions. It is conjectured that $$p_s$$ plays a dividing role in the class of nonnegative solutions.