Related papers: Stability of Hardy-Sobolev Inequality

Let $\Omega$ be a $C^1$ open bounded domain in $\R^N$ ($N\geq 3$) with $0\in \partial \Omega$. Suppose that $\partial\Omega$ is $C^2$ at $0$ and the mean curvature of $\partial\Omega$ at $0$ is negative. Consider the following perturbed PDE…

Let $\Omega\subset\mathbb{R}^N$ ($N\geq 3$) be a bounded $C^2$ domain and $\Sigma\subset\partial\Omega$ be a compact $C^2$ submanifold of dimension $k$. Denote the distance from $\Sigma$ by $d_\Sigma$. In this paper, we study positive…

In this paper, we investigate the following fractional Choquard type equation: \[ (- \Delta)_p^s\, u = \lambda\frac{|u|^{r-2}u}{|x|^\alpha}\,+\gamma \big(\int_\Omega \frac{|u|^q}{|x-y|^\mu}dy\big) |u|^{q-2}u \ \ \text{in } \Omega,\ \ u = 0…

We prove a quantitative Sobolev inequality in cones of Bianchi-Egnell type, which implies a stability property. Our result holds for any cone as long as the minimizers of the Sobolev quotient are nondegenerate, which is the case of most…

Let $N \ge 4$, $\Omega$ be a bounded domain in $\mathbb{R}^N$, and let $\Sigma \subset \Omega$ be a smooth closed submanifold of dimension $k$ with $2 \le k \le N-2$. We study the existence of positive solutions $u \in H_0^1(\Omega)$ to the…

Recently, Dolbeault-Esteban-Figalli-Frank-Loss [20] established the optimal stability of the first-order $L^2$-Sobolev inequality with dimension-dependent constant. Subsequently, Chen-Lu-Tang [18] obtained the optimal stability for the…

In this article, we study the following Hardy-Sobolev-Maz'ya type equation: \begin{equation} -\Delta u - \mu \frac{u}{|z|^2} = \frac{|u|^{q-2}u}{|z|^t}, \quad u \in D^{1,2} (\mathbb{R}^n), \end{equation} where $x = (y,z) \in \mathbb{R}^h…

In this paper, we consider the following magnetic nonlinear Choquard equation \[-(\nabla+iA(x))^2u+ V(x)u = \left(\frac{1}{|x|^{\alpha}}*|u|^{2_{\alpha}^*}\right) |u|^{2_{\alpha}^*-2} u + \lambda f(u)\ \textrm{ in }\ \R^N,\] where…

In this paper, we are interested in some problems related to the following biharmonic hartree equation \begin{equation*} \Delta^{2} u=(|x|^{-\alpha}\ast |u|^{p})u^{p-1},\sp \text{in}\quad\R^N. \end{equation*} where…

In this paper, we obtain necessary conditions and sufficient conditions on the initial data for the local-in-time solvability of the Cauchy problem \[ \partial_t u +(-\Delta)^\frac{\theta}{2} u=|x|^{-\gamma} u^p ,\quad x\in{\bf R}^N, t>0,…

In this paper, we study the following singular nonlinear elliptic problem \begin{equation}\label{eq:1} \left\{ \begin{array}{ll} \displaystyle (-\Delta)^{\frac \alpha 2} u=\lambda |u|^{r-2}u+\mu\frac{|u|^{q-2}u}{|x|^{s}}\quad &{\rm in…

We apply a duality method to prove an optimal stability theorem for the logarithmic Hardy-Littlewood-Sobolev inequality, and we apply it to the estimation of the rate of approach to equilibrium for the critical mass Keller-Segel system.

Obtaining explicit stability estimates in classical functional inequalities like the Sobolev inequality has been an essentially open question for 30 years, after the celebrated but non-constructive result of G. Bianchi and H. Egnell in…

We consider the second best constant in the Hardy-Sobolev inequality on a Riemannian manifold. More precisely, we are interested with the existence of extremal functions for this inequality. This problem was tackled by Djadli-Druet [5] for…

Let (M,g) be a compact Riemmanian manifold with non-empty boundary. Consider the second order hyperbolic initial-boundary value problem (\delta_t^2 + P(x,D))u = 0 in (0,T) x M, u(0,x) = \delta_t u(0,x) = 0 for x in M, u(t,x) = f(t,x) on…

In this paper, we first prove some propositions of Sobolev spaces defined on a locally finite graph $G=(V,E)$, which are fundamental when dealing with equations on graphs under the variational framework. Then we consider a nonlinear…

The aim of this paper is to study the critical elliptic equations with Stein-Weiss type convolution parts $$ \displaystyle-\Delta u =\frac{1}{|x|^{\alpha}}\left(\int_{\mathbb{R}^{N}}\frac{|u(y)|^{2_{\alpha,…

In this paper, we deal with the boundary value problem $-\Delta u= |u|^{4/(n-2)}u/[\ln (e+|u|)]^\varepsilon$ in a bounded smooth domain $\Omega$ in $\mathbb{R}^n$, $n\geq 3$ with homogenous Dirichlet boundary condition. Here…

We study the higher-order Schr\"odinger equation with critical Sobolev exponent on the hyperbolic space $\mathbb{H}^n$: $$P_m u + a(x)\,u = |u|^{q-2}u, \quad u \in D^{m,2}(\mathbb{H}^n),$$ where $P_m$ is the GJMS operator of order $2m$, $q…

We prove a Hardy inequality on convex sets, for fractional Sobolev-Slobodecki\u{\i} spaces of order $(s,p)$. The proof is based on the fact that in a convex set the distance from the boundary is a superharmonic function, in a suitable…