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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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