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Liouville type theorems, a priori estimates and existence of solutions for critical and super-critical order Hardy–Hénon type equations in \(\mathbb {R}^{n}\)

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Abstract

In this paper, we first consider the critical order Hardy–Hénon type equations and inequalities

$$\begin{aligned} (-\Delta )^{\frac{n}{2}}u(x)\ge \frac{u^{p}(x)}{|x|^{a}}, \,\,\,\,\,\,\,\,\,\,\, x \, \in \,\, \mathbb {R}^{n}, \end{aligned}$$

where \(n\ge 4\) is even, \(-\infty<a<n\), and \(1<p<+\infty \). We prove Liouville theorems (Theorems 1.1 and 1.3), that is, the unique nonnegative solution is \(u\equiv 0\). Then as an immediate application, by applying method of moving planes in a local way, blowing-up techniques and the Leray–Schauder fixed point theorem, we derive a priori estimates and hence existence of positive solutions to critical order Lane–Emden equations in bounded domains (Theorems 1.4 and 1.6). Extensions to super-critical order Hardy–Hénon type equations and inequalities will also be included (Theorems 1.8 and 1.11). In critical and super-critical order Hardy–Hénon type inequalities, there are no growth conditions on the nonlinearity f(x, u) w.r.t. u and hence the nonlinear term can grow exponentially (or even faster) on u (Theorems 1.3, 1.8 and Remark 1.10).

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Acknowledgements

The authors are grateful to the referees for their careful reading and valuable comments and suggestions that improved the presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Yeshiva University, New York, NY, USA

    Wenxiong Chen

  2. School of Mathematical Sciences, Beihang University (BUAA), Beijing, 100191, People’s Republic of China

    Wei Dai

  3. LAGA, UMR 7539, Institut Galilée, Université Sorbonne Paris Nord, 93430, Villetaneuse, France

    Wei Dai

  4. Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Guolin Qin

  5. University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China

    Guolin Qin

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  1. Wenxiong Chen
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Correspondence to Wei Dai.

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W. Chen is partially supported by the Simons Foundation Collaboration Grant for Mathematicians 245486. W. Dai is supported by the NNSF of China (Nos. 12222102, 11971049 and 11501021), the Fundamental Research Funds for the Central Universities and the State Scholarship Fund of China (No. 201806025011).

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Chen, W., Dai, W. & Qin, G. Liouville type theorems, a priori estimates and existence of solutions for critical and super-critical order Hardy–Hénon type equations in \(\mathbb {R}^{n}\). Math. Z. 303, 104 (2023). https://doi.org/10.1007/s00209-023-03265-y

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