Session S02 - Diverse Aspects of Elliptic PDEs and Related Problems

Wednesday, July 14, 18:30 ~ 19:00 UTC-3

Non-local equation with critical growth on compact Riemannian manifolds.

Nicolas Saintier

Universidad de Buenos Aires, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

On a compact Riemannian manifold $(M,g)$ of dimension $n$ we consider a non-local equation of the form \[ \mathcal{L}_K u + hu = f |u|^{2^*-2} u \] where the operator $\mathcal{L}_K u$ is given in weak form by \[ (\mathcal{L}_K u,v) = \iint_{M\times M} (u(x)-u(y))(v(x)-v(y))K(x,y;g)\,dv_g(x)dv_g(x). \] The kernel $K(x,y;g)$ essentially behaves like $1/d_g(x,y)^{n+2s}$, $s\in (0,1)$, and satisfies some useful properties when the metric is blown-up at a point. The exponent $2^*:=2n/(n-2s)$ is critical from the point of view of Sobolev embedding. This kind of non-local equation on $\mathbb{R}^n$ has been the subject of an intense research activity in the past years.

We first study the associated functional spaces $\widetilde{W}^{s,2}(M) $ of the functions $u\in L^2(M)$ such that $ [u]_{s,2}<\infty$ where \[ u_{s,2}^2:= \int_{M\times M} \frac{|u(x)-u(y)|^2}{d_g(x,y)^{n+2s}}\,dv_g(x)dv_g(y). \] We then establish an optimal Sobolev inquality of the form \[ \Big( \int_M |u|^{2^*} \,dv_g\Big)^\frac{2}{2^*} \le (A(n,s,2)+\varepsilon) \iint_{M\times M} |u(x)-u(y)|^2 K(x,y;g)\,dv_g(x)dv_g(y) + C_\varepsilon \int_M u^2\,dv_g \] valid for any $\varepsilon>0$. Here $A$ is the least constant such that this inequality holds for any $u$, and is given by \[ A(n,s,2)^{-1} = \inf_u \frac{ \displaystyle \iint_{\mathbb{R}^n\times \mathbb{R}^n} \frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dxdy}{\|u\|_{2^*}^2}. \] This inequlity then allows to state in a standard way a sufficient existence condition.

Joint work with Carolina Rey (Univ. Buenos Aires).

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