Related papers: Removable singularity of the polyharmonic equation

We propose and study a concept of renormalized solution to the problem $\Delta_p u=0$ in $\mathbb{R}^N_+$, $|\nabla u|^{p-2}u_{\nu} + g(u) = \mu$ on $\partial\mathbb{R}^N_+$, where $1<p\leq N$, $N\geq 2$,…

In this paper, we prove the existence of multiple solutions for a nonlinear nonlocal elliptic PDE involving a singularity which is given as \begin{eqnarray} (-\Delta_p)^s u&=& \frac{\lambda}{u^\gamma}+u^q~\text{in}~\Omega,\nonumber…

We study the system $-\Delta \mathbf{u}=| \mathbf{u}|^{\alpha-1} \mathbf{u}$ with $1<\alpha\leq\frac{n+2}{n-2}$, where $ \mathbf{u}=(u_1,\dots,u_m)$, $m\geq 1$, is a $C^2$ nonnegative function that develops an isolated singularity in a…

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…

We investigate the following mixed local and nonlocal quasilinear equation with singularity given by \begin{eqnarray*} \begin{split} -\Delta_pu+(-\Delta)_q^s u&=\frac{f(x)}{u^{\delta}}\text { in } \Omega, \\u&>0 \text{ in } \Omega,\\u&=0…

This paper deals with the existence of positive solution for the singular quasilinear Schr\"odinger equation $-\Delta u -\Delta (u^{2})u=h(x) u^{-\gamma} + f(x,u)~\mbox{in} ~ \Omega,$ where $\gamma > 1$, $\Omega \subset \mathbb{R}^{N},…

We study existence and stability of solutions of (E 1) --$\Delta$u + $\mu$ |x| 2 u + g(u) = $\nu$ in $\Omega$, u = 0 on $\partial$$\Omega$, where $\Omega$ is a bounded, smooth domain of R N , N $\ge$ 2, containing the origin, $\mu$ $\ge$ --…

In this paper we prove the existence of multiple nontrivial solutions of the following equation. \begin{align*} \begin{split} -\Delta_{p}u & = \lambda |u|^{q-2}u+f(x,u)+\mu\,\,\mbox{in}\,\,\Omega, u & = 0\,\, \mbox{on}\,\, \partial\Omega;…

We study the following singular problem involving the p$(x)$-Laplace operator $\Delta_{p(x)}u= div(|\nabla u|^{p(x)-2}\nabla u)$, where $p(x)$ is a nonconstant continuous function, \begin{equation} \nonumber {{(\rm P_\lambda)}}…

We give a partial uniqueness result concerning comparable renormalized solutions of the nonlinear elliptic problem $-\diw(\aop(x,Du))=\mu$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\mu$ is a Radon measure with bounded variation on…

We study a minimizing problem associated with the singular problem \[ \left\{ \begin{array} [c]{ll} -\operatorname{div}\left( \left\vert \nabla u\right\vert ^{p-2}\nabla u\right) =\lambda u^{-1} & \mathrm{in\ }\Omega\\ u>0 & \mathrm{in\…

Via a constrained minimization, we find a solution $(\lambda,u)$ to the problem \begin{equation*} \begin{cases} (-\Delta)^m u+\frac{\mu}{|x|^{2m}}u + \lambda u = \eta u^3 + g(u)\\ \int_{\mathbb{R}^{2m}} u^2 \, dx = \rho \end{cases}…

We consider the problem $$ (P_\lambda)\quad -\Delta_{p}u=\lambda u^{p-1}+a(x)u^{q-1},\quad u\geq0\quad\mbox{ in }\Omega $$ under Dirichlet or Neumann boundary conditions. Here $\Omega$ is a smooth bounded domain of $\mathbb{R}^{N}$…

We study the problem of removable singularities for degenerate elliptic equations. Let F be a fully nonlinear second-order partial differential subequation of degenerate elliptic type on a manifold X. We study the question: Which closed…

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…

We consider a continuous function $f$ on a domain in $\mathbf C^n$ satisfying the inequality that $|\bar \partial f|\leq |f|$ off its zero set. The main conclusion is that the zero set of $f$ is a complex variety. We also obtain removable…

Let $n\geq2$ and $ \Omega\subset \mathbb{R}^{n+1}$ be a Lipschitz wedge- like domain . We construct positive weak solutions of the problem $$\Delta u + u^p = 0 \quad\hbox{in}\, \Omega,$$ which vanish in a suitable trace sense on…

In this paper, we establish the uniqueness of positive solutions to the following fractional nonlinear elliptic equation with harmonic potential \begin{align*} (-\Delta)^s u+ \left(\omega+|x|^2\right) u=|u|^{p-2}u \quad \mbox{in}\,\, \R^n,…

We consider the uniqueness of the following positive solutions of $m$-Laplacian equation: \begin{equation} \left\{ \begin{aligned} -\Delta _m u&=\lambda u^{m-1}+u^{p-1} \quad \text{in} \quad \Omega\\ u&=0 \quad \text{on} \quad \partial…

This paper studies the existence of positive normalized solutions to the singular elliptic equation \[ -\Delta u + \lambda u = u^{-r} + u^{p-1} \quad \text{in } \Omega, \] with the Dirichlet boundary condition $u=0$ on $\partial\Omega$ and…