Minimal energy solutions to the fractional Lane-Emden system: existence and singularity formation.(English)Zbl 1418.35358

Summary: In this paper, we study asymptotic behavior of minimal energy solutions to the fractional Lane-Emden system in a smooth bounded domain $$\Omega$$ $(-\Delta)^s u = v^p,\ (-\Delta)^s v = u^q,\ u, v > 0\ \mathrm{in}\ \Omega \quad \mathrm{and} \quad u = v = 0\ \mathrm{on}\ \partial \Omega$ for $$0 < s < 1$$ under the assumption that $$(-\Delta)^s$$ is the spectral fractional Laplacian and the subcritical pair $$(p,q)$$ approaches to the critical Sobolev hyperbola. If $$p = 1$$, the above problem is reduced to the subcritical higher-order fractional Lane-Emden equation with the Navier boundary condition $(-\Delta)^s u = u^{\frac{n+2s}{n-2s}-\epsilon},\ u > 0\ \mathrm{in}\ \Omega \quad \mathrm{and} \quad u = (-\Delta)^{s/2} u = 0\ \mathrm{on}\ \partial \Omega$ for $$1 < s < 2$$. The main objective of this paper is to deduce the existence of minimal energy solutions, and to examine their (normalized) pointwise limits provided that $$\Omega$$ is convex, generalizing the work of Guerra that studied the corresponding results in the local case $$s = 1$$. As a by-product of our study, a new approach for the existence of an extremal function for the Hardy-Littlewood-Sobolev inequality is provided.

MSC:

 35R11 Fractional partial differential equations 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35B33 Critical exponents in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs 35J47 Second-order elliptic systems
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