## Asymptotic behavior of least energy solutions to the Lane-Emden system near the critical hyperbola.(English. French summary)Zbl 1437.35065

The following Lane-Emden system $-\Delta u(x) = v(x)^p,\;\; -\Delta v(x) = u(x)^q, \ u(x)>0,\; v(x) >0, \; x \in \Omega, \;\; u|_{\partial \Omega} = v|_{\partial \Omega} = 0,$ where $$\Omega \subset \mathbb{R}^n$$ is a smooth bounded domain ($$n \geq 3$$) and $$0 < p < q < \infty$$ is considered. A general result on the asymptotic behavior of least energy solutions for the above system near the critical hyperbola is proved in this paper. For $$p\geq 1$$ and the domain $$\Omega$$ is convex, this kind of result was obtained by I. A. Guerra [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 25, No. 1, 181–200 (2008; Zbl 1136.35025)]. In this paper, the author allows $$p<1$$ and $$\Omega$$ being arbitrary smooth bounded domains, which also answers a conjecture proposed by I. A. Guerra.

### MSC:

 35B40 Asymptotic behavior of solutions to PDEs 35B33 Critical exponents in context of PDEs 35J47 Second-order elliptic systems 35J61 Semilinear elliptic equations

### Keywords:

non-convex domain; exponent less than one

Zbl 1136.35025
Full Text:

### References:

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