Related papers: Weighted Sobolev Spaces and an Eigenvalue Problem …

We study the nonlinear eigenvalue problem $-{\rm div}(a(|\nabla u|)\nabla u)=\lambda|u|^{q(x)-2}u$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded open set in $\RR^N$ with smooth boundary, $q$ is a continuous function,…

This paper is the second part of a work devoted to the study of elliptic systems involving multiple Hardy-Sobolev critical exponents: $$\begin{cases} -\Delta u-\lambda \frac{|u|^{2^*(s_1)-2}u}{|x|^{s_1}}=\kappa\alpha…

We study elliptic and parabolic problems governed by the singular elliptic operators \begin{align*} \mathcal L=y^{\alpha_1}\mbox{Tr }\left(QD^2_xu\right)+2y^{\frac{\alpha_1+\alpha_2}{2}}q\cdot \nabla_xD_y+\gamma y^{\alpha_2}…

Let $\Omega$ be a bounded and smooth domain of $\mathbb{R}^{N}$, $N\geq2$, and consider the eigenvalue problem: $-\Delta_{p}u=\lambda\left| u\right| _{L^{q}(\Omega)}^{p-q}\left| u\right| ^{q-2}u$ in $\Omega,$ $u=0$ on $\partial\Omega,$…

We study the operator $\mathcal{A}$ of multiplication by an independent variable in a matrix Sobolev space $W^2(M)$. In the cases of finite measures on $[a,b]$ with $(2\times 2)$ and $(3\times 3)$ real continuous matrix weights of full rank…

In this paper, first we introduce the $s(.,.)$-fractional Musielak-Sobolev spaces $W^{s(x,y)}L_{\varPhi_{x,y}}(\Omega)$. Next, by means of Ekeland's variational principal, we show that there exists $\lambda_*>0$ such that any $\lambda\in(0,…

In this paper we discuss compactness estimates for the $\bar \partial $-Neumann problem in the setting of weighted $L^2$-spaces on $\mathbb{C}^n.$ For this purpose we use a version of the Rellich - Lemma for weighted Sobolev spaces.

In this work we study the existence of solutions $u \in W^{1,p}_0(\Omega)$ to the implicit elliptic problem $ f(x, u, \nabla u, \Delta_p u)= 0$ in $ \Omega $, where $ \Omega $ is a bounded domain in $ \mathbb R^N $, $ N \ge 2 $, with smooth…

We establish the unique solvability in weighted mixed-norm Sobolev spaces for a class of degenerate parabolic and elliptic equations in the upper half space. The operators are in nondivergence form, with the leading coefficients given by…

Bounded and unbounded weighted composition operators on $L^2$ spaces over $\sigma$-finite measure spaces are investigated. A variety of questions related to seminormality of such operators are discussed.

By adapting the mass transportation technique of Cordero-Erausquin, Nazaret and Villani, we obtain a family of sharp Sobolev and Gagliardo-Nirenberg (GN) inequalities on the half space $\mathbf{R}^{n-1}\times\mathbf{R}_+$, $n\geq 1$…

The main problem considered in this article is the following: if $\mathbf{F}$, $\mathbf{E}$ are normed spaces of continuous functions over topological spaces $X$ and $Y$ respectively, and $\omega:Y\to\mathbb{C}$ and $\Phi:Y\to X$ are such…

We study higher-order compact Sobolev embeddings on a domain $\Omega \subseteq \mathbb R^n$ endowed with a probability measure $\nu$ and satisfying certain isoperimetric inequality. Given $m\in \mathbb N$, we present a condition on a pair…

We obtain the inequality $$\int_{\Omega}|\nabla u(x)|^ph(u(x))dx\leq C(n,p)\int_{\Omega} \left( \sqrt{ |\Delta u(x)||{\cal T}_{h,C}(u(x))|}\right)^{p}h(u(x))dx,$$ where $\Omega\subset \mathbf{R}^n$ is a bounded Lipschitz domain, $u\in…

We study some non-local functionals on the Sobolev space $W^{1,p}_0(\Omega)$ involving a double integral on $\Omega\times\Omega$ with respect to a measure $\mu$. We introduce a suitable notion of convergence of measures on product spaces…

In this paper, we consider the following quasilinear elliptic problem with potential $$(P) \begin{cases} -\mbox{div}(\phi(x,|\nabla u|)\nabla u)+ V(x)|u|^{q(x)-2}u= f(x,u) & \ \ \mbox{ in }\Omega, u=0 & \ \ \mbox{ on } \partial\Omega,…

In this article we introduce weighted Sobolev spaces that are well suited to treat initial data for multiple black hole systems. We prove general results for elliptic operators on these spaces and give a simple proof of existence of a class…

We study nonuniform Sobolev spaces, i.e., spaces of functions whose partial derivatives lie in possibly different Lebesgue spaces. Although standard proofs do not apply, we show that nonuniform Sobolev spaces share similar properties as the…

Let $V\subset \mathbb{C}\mathbb{P}^n$ be an irreducible complex projective variety of complex dimension $v$ and let $g$ be the K\"ahler metric on $\reg(V)$, the regular part of $V$, induced by the Fubini Study metric of…

In this paper we study weighted Hardy-Sobolev spaces of vector valued functions analytic on double-napped cones of the complex plane. We introduce these spaces as a tool for complex scaling of linear ordinary differential equations with…