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The Brezis-Nirenberg result for the fractional Laplacian. (English) Zbl 1323.35202

The authors enhance a well-known result on the existence of a solution of a non-linear differential equation. Precisely, for \(n>2s\) such that \(s\in(0,1)\), they consider the following Dirichlet (in)homogeneous problem in a bounded Lipschitz open set \(\Omega\): \[ (*)_f:\quad \mathcal{L}_Ku-\lambda u=|u|^{\frac{4s}{n-2s}}u+f(x,u)\text{ in }\Omega,\quad u=0\text{ on }\mathbb R^n\setminus \Omega, \] where \(\mathcal{L}_K\) stands for an explicit integro-differential operator of order \(s\), \(K\) is a positive function (which is part of the \(\mathcal{L}_K\) integrand) satisfying meaningful conditions, and \(\lambda\) is a positive factor. Accordingly, by assuming satisfying that \(f\) is a Carathéodory function satisfying some conditions, that \(\lambda\) is bounded by the first eigenvalue of \(\mathcal{L}_K\), that the functional associated to \((*)_f\) is bounded by a function of the fractional critical Sobolev constant-associated to the embedding map \(X_0\hookrightarrow L_{\frac{2n}{n-2s}}(\mathbb R^n)\) such that \(X_0\) is a suitable Sobolev space-, then (through a polarization) they show that \((*)_f\) (resp. \((*)_0\)) has a non-zero solution in \(X_0\) (resp. in \(H^s(\mathbb R^n)\), the fractional Sobolev space (Theorem 1)) (resp. Theorem 2). Then, thanks to the embedding continuous map \(X_0\hookrightarrow H^s(\mathbb R^n)\) (Lemma 7), they apply the previous results for the pseudo-differential operator \((-\Delta)^s\) (Theorems 3 and 4).

MSC:

35R11 Fractional partial differential equations
35R09 Integro-partial differential equations
49J45 Methods involving semicontinuity and convergence; relaxation
35A15 Variational methods applied to PDEs
35S15 Boundary value problems for PDEs with pseudodifferential operators
47G20 Integro-differential operators
45G05 Singular nonlinear integral equations
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