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

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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.

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Notes

  1. Indeed, with \(Q_v(\psi )\) defined in (1.12), we have

    $$\begin{aligned} Q_{V_\infty }(\psi )=\int \left( |x|^\theta |\nabla \psi |^2-pC_0^{p-1}|x|^{\theta -2}\psi ^2\right) . \end{aligned}$$

    To stress the dependence of \(V_\infty \) on p, we write \(V_\infty =V_{\infty , p}\). Using \(pC_0^{p-1}=f(p)\) and \(f'(p_s)=\frac{(N'-2)^2}{4}>0\), we see that

    $$\begin{aligned} Q_{V_{\infty , p_s}}(\psi )\le Q_{V_{\infty , p_s-\epsilon }}(\psi ) \quad \text { for small} \quad \epsilon >0. \end{aligned}$$

    It follows that the Morse index of \(V_{\infty , p_s}\) is no less than that of \(V_{\infty , p_s-\epsilon }\). But the latter is known to be infinity by Theorem 1.2. Hence the Morse index of \(V_{\infty , p_s}\) is infinity.

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Authors and Affiliations

  1. School of Science and Technology, University of New England, Armidale, NSW, 2351, Australia

    Yihong Du

  2. Department of Mathematics, Henan Normal University, Xinxiang, 453007, China

    Zongming Guo

Authors
  1. Yihong Du
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  2. Zongming Guo
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Correspondence to Yihong Du.

Additional information

Communicated by C. S. Lin.

The research of Y. Du is supported by the Australian Research Council. The research of Z. Guo is supported by NSFC (11171092) and Innovation Scientists and Technicians Troop Construction Projects of Henan Province (114200510011).

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Du, Y., Guo, Z. Finite Morse index solutions of weighted elliptic equations and the critical exponents. Calc. Var. 54, 3161–3181 (2015). https://doi.org/10.1007/s00526-015-0897-z

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