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• W2963506522 abstract "We consider Liouville-type theorems for the following Hénon-Lane-Emden systembegin{eqnarray*} left{ begin{array}{lcl} -Delta u&=& |x|^{a}v^p in mathbb{R}^n, -Delta v&=& |x|^{b}u^q in mathbb{R}^n,end{array}right. end{eqnarray*}when $p,q ge 1,$ $pqneq1$, $a,bge0$. The main conjecture states that there is no non-trivial non-negative solution whenever $(p,q)$ is under the critical Sobolev hyperbola, i.e.$ frac{n+a}{p+1}+frac{n+b}{q+1}>{n-2}$. We show that this is indeed the case in dimension $n=3$ provided the solution is also assumed to be bounded, extending a result established recently by Phan-Souplet in the scalar case. Assuming stability of the solutions, we could then prove Liouville-type theorems in higher dimensions. For the scalar cases, albeit of second order ($a=b$ and $p=q$) or of fourth order ($age 0=b$ and $p>1=q$), we show that for all dimensions $nge 3$ in the first case (resp., $nge 5$ in the second case), there is no positive solution with a finite Morse index, whenever $p$ is below the corresponding critical exponent, i.e $ 1< p < frac{n+2+2a}{n-2}$ (resp., $ 1< p < frac{n+4+2a}{n-4}$). Finally, we show that non-negative stable solutions of the full Hénon-Lane-Emden system are trivial provided begin{equation*}label{sysdim00} n < 2 + 2 (frac{p(b+2)+a+2}{pq-1}) (sqrt{frac{pq(q+1)}{p+1}} + sqrt{ frac{pq(q+1)}{p+1} - sqrt{frac{pq(q+1)}{p+1}}}). end{equation*}" @default.
• W2963506522 created "2019-07-30" @default.
• W2963506522 creator A5004440950 @default.
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• W2963506522 date "2014-01-01" @default.
• W2963506522 modified "2023-10-18" @default.
• W2963506522 title "On the Hénon-Lane-Emden conjecture" @default.
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• W2963506522 doi "https://doi.org/10.3934/dcds.2014.34.2513" @default.
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