A Pohozaev identity for the fractional Hénon system
Abstract: In this paper, we study the Pohozaev identity associated with a Hénon-Lane-Emden system involving the fractional Laplacian:in a star-shaped and bounded domain Ω for s ∈ (0, 1). As an application of our identity, we deduce the nonexistence of positive solutions in the critical and supercritical cases.
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Cited by 6 publications
References 19 publications
In this paper, we aim to develop the (direct) method of scaling spheres, its integral forms, and the method of scaling spheres in a local way. As applications, we investigate Liouville properties of nonnegative solutions to fractional and higher-order Hénon–Hardy type equations $$ \begin{align*}& (-\Delta)^{\frac{\alpha}{2}}u(x)=f(x,u(x)) \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\, \mathbb{R}^{n}, \,\,\, \mathbb{R}^{n}_{+} \,\,\, \text{or} \,\,\, B_{R}(0) \end{align*}$$with $n>\alpha $, $0<\alpha <2$ or $\alpha =2m$ with $1\leq m<\frac {n}{2}$. We first consider the typical case $f(x,u)=|x|^{a}u^{p}$ with $a\in (-\alpha ,\infty )$ and $0<p<p_{c}(a):=\frac {n+\alpha +2a}{n-\alpha }$. By using the method of scaling spheres, we prove Liouville theorems for the above Hénon–Hardy equations and equivalent integral equations (IEs). In $\mathbb {R}^{n}$, our results improve the known Liouville theorems for some especially admissible subranges of $a$ and $1<p<\min \big \{\frac {n+\alpha +a}{n-\alpha },p_{c}(a)\big \}$ to the full range $a\in (-\alpha ,\infty )$ and $p\in (0,p_{c}(a))$. In particular, when $a>0$, we covered the gap $p\in \big [\frac {n+\alpha +a}{n-\alpha },p_{c}(a)\big )$. For bounded domains (i.e., balls), we also apply the method of scaling spheres to derive Liouville theorems for super-critical problems. Extensions to PDEs and IEs with general nonlinearities $f(x,u)$ are also included (Theorem 1.31). In addition to improving most of known Liouville type results to the sharp exponents in a unified way, we believe the method of scaling spheres developed here can be applied conveniently to various fractional or higher order problems with singularities or without translation invariance or in the cases the method of moving planes in conjunction with Kelvin transforms do not work.
In the present article, we investigate the following Hénon-Lane-Emden elliptic system: − Δ u = ∣ x ∣ a v p , x ∈ R N , − Δ v = ∣ x ∣ b u q , x ∈ R N , \left\{\begin{array}{ll}-\Delta u={| x| }^{a}{v}^{p},& x\in {{\mathbb{R}}}^{N},\\ -\Delta v={| x| }^{b}{u}^{q},& x\in {{\mathbb{R}}}^{N},\end{array}\right. where N ≥ 2 N\ge 2 , p p , q > 0 q\gt 0 , a a , b ∈ R b\in {\mathbb{R}} . We partially prove the Hénon-Lane-Emden conjecture in the case of four and five dimensions. More specifically, we show that there is no nonnegative nontrivial classical solution for the Hénon-Lane-Emden elliptic system when a a , b > − 2 b\gt -2 and the parameter pair ( p , q p,q ) meets p q > 1 , N + a p + 1 + N + b q + 1 > N − 2 , pq\gt 1,\hspace{1.0em}\frac{N+a}{p+1}+\frac{N+b}{q+1}\gt N-2, and additionally p , q < 4 / 3 p,q\lt 4\hspace{0.1em}\text{/}\hspace{0.1em}3 if N = 4 N=4 or p , q < 10 / 9 p,q\lt 10\hspace{0.1em}\text{/}\hspace{0.1em}9 if N = 5 N=5 .
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