Related papers: Weighted integrability of polyharmonic functions

We consider the operator $L=-{\rm div}(A\nabla)$, where the $n\times n$ matrix $A$ is real-valued, elliptic, with the symmetric part of $A$ in $L^\infty(\mathbb{R}^n)$, and the anti-symmetric part of $A$ only belongs to the space…

Let $D$ be a smooth domain in $\mathbb{R}^N$, $N\geq 3$ and let $f$ be a positive continuous function on $\partial D$. Under some assumptions on $\varphi$, it is shown that the problem $\Delta u=2\varphi(u)$ in $D$ and $u=f$ on $\partial…

We establish sharp $L^p$ integral mean estimates for $(\alpha,\beta)$-harmonic functions on the unit disk. Explicit bounds for the functions and their partial derivatives are obtained in terms of boundary data, by means of the associated…

We derive a necessary condition for compactness of the weighted $\overline\partial$-Neumann operator on the space $L^2(\mathbb C^n,e^{-\varphi})$, under the assumption that the corresponding weighted Bergman space of entire functions has…

We establish interior $C^{1,\alpha}$ regularity estimates for some $\alpha > 0$, for solutions of the fractional $p$-Laplace equation $(-\Delta_p)^s u = 0$ when $p$ is in the range $p \in [2,2/(1-s))$.

In this paper, we investigate further the weighted $p(x)$-Hardy inequality with the additional term of the form \[ \int_\Omega |\xi|^{p(x)}\mu_{1,\beta} (dx) \leqslant \int_\Omega |\nabla \xi|^{p(x)}\mu_{2,\beta} (dx)+\int_\Omega…

We study the zero Dirichlet problem for the equation $-\Delta_p u -\Delta_q u = \alpha |u|^{p-2}u+\beta |u|^{q-2}u$ in a bounded domain $\Omega \subset \mathbb{R}^N$, with $1<q<p$. We investigate the relation between two critical curves on…

We decompose $p$ - integrable functions on the boundary of a simply connected Lipschitz domain $\Omega \subset \mathbb C$ into the sum of the boundary values of two, uniquely determined holomorphic functions, where one is holomorphic in…

We are concerned with the existence of solution of the problem $ -\Delta ^H_pu+|u|^{p-2}u=\lambda|u|^{q-2}u+ |u|^{p^*-2}u\quad \mbox{in}\quad\Omega,$ $u>0\quad \mbox{in}\quad\Omega,$ $a(\nabla u)\cdot \nu =0\quad \mbox{on}\quad\partial…

The purpose of this article is to give a complete study of the weak solutions of the fractional elliptic equation \begin{equation}\label{00} \arraycolsep=1pt \begin{array}{lll} (-\Delta)^{\alpha} u+u^p=0\ \ \ \ &\ {\rm in}\ \…

In this work, we study the existence of weak solution to the following quasi linear elliptic problem involving the fractional $p$-Laplacian operator, a Hardy potential and multiple critical Sobolev nonlinearities with singularities,…

We consider, for $a,l\geq1,$ $b,s,\alpha>0,$ and $p>q\geq1,$ the homogeneous Dirichlet problem for the equation $-\Delta_{p}u=\lambda u^{q-1}+\beta u^{a-1}\left\vert \nabla u\right\vert ^{b}+mu^{l-1}e^{\alpha u^{s}}$ in a smooth bounded…

We study the existence and non-existence of classical solutions for inequalities of type $$ \pm \Delta^m u \geq \big(\Psi(|x|)*u^p\big)u^q \quad\mbox{ in } {\mathbb R}^N (N\geq 1). $$ Here, $\Delta^m$ $(m\geq 1)$ is the polyharmonic…

Denote by $C^{\alpha}(\mathbb{D})$ the space of the functions $f$ on t}he unit disk $\mathbb{D}$ which are H\"older continuous with the exponent $\alpha$, and denote by $C^{1, \alpha}(\mathbb{D})$ the space which consists of differentiable…

We consider the problem of uniqueness of positive solutions to boundary value problems containing the equation: -\Delta_p u =K(|x|)f(u), p>1. f is positive, is locally Lipschitz and satisfies some superlinear growth condition after u_0, a…

The aim of this paper is to establish properties of the solutions to the $\alpha$-harmonic equations: $\Delta_{\alpha}(f(z))=\partial{z}[(1-{|{z}|}^{2})^{-\alpha} \overline{\partial}{z}f](z)=g(z)$, where…

Let $u: \Omega \subseteq \mathbb{R}^n \longrightarrow \mathbb{R}^N$ be a smooth map and $n,N \in \mathbb{N}$. The $\infty$-Laplacian is the PDE system \[ \tag{1} \label{1} \Delta_\infty u \, :=\, \Big(Du \otimes Du + |Du|^2[Du]^\bot\!…

In this paper, we mainly consider nonnegative weak solution to the $D^{1,p}(\R^{N})$-critical quasi-linear static Schr\"{o}dinger-Hartree equation with $p$-Laplacian $-\Delta_{p}$ and nonlocal nonlinearity: \begin{align*} -\Delta_p u…

We give necessary and sufficient conditions for the existence of weak solutions to the model equation $$-\Delta_p u=\sigma \, u^q \quad \text{on} \, \, \, \R^n,$$ in the case $0<q<p-1$, where $\sigma\ge 0$ is an arbitrary locally integrable…

In this paper, we establish some Schwarz type lemmas for mappings $\Phi$ satisfying the inhomogeneous biharmonic Dirichlet problem $ \Delta (\Delta(\Phi)) = g$ in $\mathbb{D}$, $\Phi=f$ on $\mathbb{T}$ and $\partial_n \Phi=h$ on…