## 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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