Related papers: A family of sharp inequalities for Sobolev functio…

We show the existence of a nodal solution with two nodal domains for a generalized Kirchhoff equation of the type $$ -M\left(\displaystyle\int_\Omega \Phi(|\nabla u|)dx\right)\Delta_\Phi u = f(u) \ \ \mbox{in} \ \ \Omega, \ \ u=0 \ \…

We consider the equation $\e^{2}\Delta u=(u-a(x))(u^2-1)$ in $\Omega$, $\frac{\partial u}{\partial \nu} =0$ on $\partial \Omega$, where $\Omega$ is a smooth and bounded domain in $\R^n$, $\nu$ the outer unit normal to $\pa\Omega$, and $a$ a…

We study the existence of non-trivial unbounded domains of $\Omega \subset \mathbb{R}^2$ where the equation \begin{align} - \lambda u_{xx} -u_{tt} &= u \qquad \text{in $\Omega$,}\nonumber u &=0 \qquad \text{on $\partial \Omega$,}\nonumber…

For $\g=sl(n)$ we construct a two parametric $U_h(\g)$-invariant family of algebras, $(S\g)_{t,h}$, which defines a quantization of the function algebra $S\g$ on the coadjoint representation and in the parameter $t$ gives a quantization of…

We prove the existence of two smooth families of unbounded domains in $\mathbb{R}^{N+1}$ with $N\geq1$ such that \begin{equation} -\Delta u=\lambda u\,\, \text{in}\,\,\Omega, \,\, u=0,\,\,\partial_\nu…

Let $k,N \in \mathbb{N}$ with $1\le k\le N$ and let $\Omega=\Omega_1 \times \Omega_2$ be an open set in $\mathbb{R}^k \times \mathbb{R}^{N-k}$. For $p\in (1,\infty)$ and $q \in (0,\infty),$ we consider the following Hardy-Sobolev type…

This article establishes the existence and multiplicity of normalized solutions to the weighted nonlinear Schr\"odinger-type equation governed by the Caffarelli-Kohn-Nirenberg operator, $$ -\text{div}(|x|^{-2a}\nabla u)=\lambda…

Let $\Omega\subset \mathbb{R}^4$ be a smooth bounded domain, $W_0^{2,2}(\Omega)$ be the usual Sobolev space. For any positive integer $\ell$, $\lambda_{\ell}(\Omega)$ is the $\ell$-th eigenvalue of the bi-Laplacian operator. Define…

In this paper, we study the following fractional nonlocal Sobolev-type inequality \begin{equation*} C_{HLS}\bigg(\int_{\mathbb{R}^n}\big(|x|^{-\mu} \ast |u|^{p_s}\big)|u|^{p_s}…

Let $q>1$, $(1-\frac{1}{q})a\geq 1$ and let $\Omega\subset \mathbb{R}^2$ be Lipschitz domain. We show that planar mappings in the second order Sobolev space $f\in W^{2,q}(\Omega,\mathbb{R}^2)$ with $|J_f|^{-a}\in L^1(\Omega)$ are…

Consider the following $m-$polyharmonic Kirchhoff problem: \begin{eqnarray} \label{ea} \begin{cases} M\left(\int_{\O}|D_r u|^{m} +a|u|^m\right)[\Delta^r_m u +a|u|^{m-2}u]= K(x)f(u) &\mbox{in}\quad \Omega, \\ u=\left(\frac{\partial}{\partial…

Let $0<s<1$ and $p>1$ be such that $ps<N$. Assume that $\Omega$ is a bounded domain containing the origin. Staring from the ground state inequality by R. Frank and R. Seiringer we obtain: 1- The following improved Hardy inequality for $p\ge…

In this article we provide existence, uniqueness and regularity results of a degenerate singular elliptic boundary value problem whose prototype is given by \begin{gather*} \begin{cases} -\operatorname{div}(w(x)|\nabla u|^{p-2}\nabla…

In this work, we revisit the following estimate due to Dahlberg \cite{Dahl}. Let $\textit{\textbf x}_0$ a fixed point in a bounded Lipschitz domain $\Omega$. Then there exists a constant $C > 0$ such that if $u$ is a harmonic function in…

We exhibit a smoothly bounded domain $\Omega$ with the property that for suitable $K\subset\partial \Omega$ and $z\in \Omega$ the "Sadullaev boundary relative extremal functions" satisfy the inequality…

This paper studies eigenvalues of some Steklov problems. Among other things, we show the following sharp estimtes. Let $\Omega$ be a bounded smooth domain in an $n(\geq 2)$-dimensional Hadamard manifold an let $0=\lambda_0 < \lambda_1\leq…

We prove the following sharp Sobolev inequality on the circle $$\int_{\mathbb{S}^1} [4(v')^2 - v^2] \mathrm{d} \theta \geq - \frac{4\pi^2}{\int_{\mathbb{S}^1} v^{-2} \mathrm{d} \theta},$$ with the equality being achieved when $v^{-2}…

In this paper, we show the existence and multiplicity of positive solutions of the following fractional Kirchhoff system\\ \begin{equation} \left\{ \begin{array}{rllll} \mc L_M(u)&=\lambda f(x)|u|^{q-2}u+…

In this paper, we examine the boundary $L^2$ term of the sharp Sobolev trace inequality $\|u\|_{L^{q}(\pa M)}^2\leq S \|\nabla_g u\|_{L^2(M)}^2 +A(M,g)\|u\|^2_{L^2(\pa M)}$ on Riemannian manifolds $(M,g)$ with boundaries $\pa M$, where…

Let $\Omega \subset \mathbb{R}^n$ be a smooth bounded domain having zero in its interior $0 \in \Omega.$ We fix $0 < \alpha \le 2$ and $0 \le s <\alpha.$ We investigate a sufficient condition for the existence of a positive solution for the…