Related papers: Exact behavior around isolated singularity for sem…

We study the semilinear indefinite elliptic problem \[ -\Delta u = Q_\Omega |u|^{p-2}u \quad \text{in } \mathbb{R}^N, \] where $Q_\Omega = \chi_\Omega - \chi_{\mathbb{R}^N \setminus \Omega}$, $\Omega \subset \mathbb{R}^N$ is a bounded…

We will investigate the asymptotic behavior of positive solutions of the elliptic equation \Delta u+|x|^{l_{1}}u^{p}+|x|^{l_{2}}u^{q}=0 {in} R^{n}. We establish that for $n\geq 3$ and $q>p>1$, any positive radial solution of (0.1) has the…

In this paper we consider semilinear elliptic equations with singularities, whose prototype is the following \begin{equation*} \begin{cases} \displaystyle - div \,A(x) D u = f(x)g(u)+l(x)& \mbox{in} \; \Omega,\\ u = 0 & \mbox{on} \;…

In this paper, we concern the isolated singular solutions for semi-linear elliptic equations involving the Hardy-Leray potentials \begin{equation}\label{0} -\Delta u+\frac{\mu}{|x|^2} u=u^p\quad {\rm in}\quad \Omega\setminus\{0\},\qquad…

In this paper we investigate the isolated singularities of the Hartree type equation \begin{equation*} -\Delta u (x)= \left(\frac{1}{|x|^\alpha}*e^u\right)e^{u(x)}\quad \text{in } B_{1}\setminus\{0\} , \end{equation*} where $\alpha>0$,…

Given $B_1(0)$ the unit ball of $\mathbb{R}^n$ ($n\geq 3$), we study smooth positive singular solutions $u\in C^2(B_1(0)\setminus \{0\})$ to $-\Delta u=\frac{u^{2^\star(s)-1}}{|x|^s}-\mu u^q$. Here $0< s<2$, $2^\star(s):=2(n-s)/(n-2)$ is…

This paper deals with solutions to the equation \begin{equation*} -\Delta u = \lambda_+ \left(u^+\right)^{q-1} - \lambda_- \left(u^-\right)^{q-1} \quad \text{in $B_1$} \end{equation*} where $\lambda_+,\lambda_- > 0$, $q \in (0,1)$,…

We consider the semilinear elliptic equation $-\Delta u =\lambda f(u)$ in a smooth bounded domain $\Omega$ of $R^{n}$ with Dirichielt boundary condition, where $f$ is a $C^{1}$ positive and nondeccreasing function in $[0,\infty)$ such that…

The aim of this paper is to prove the existence of weak solutions to the equation $\Delta u + u^p = 0$ which are positive in a domain $\Omega \subset {\Bbb R}^N$, vanish at the boundary, and have prescribed isolated singularities. The…

We consider the following problem: \begin{eqnarray*} ( P)\qquad \displaystyle\left\{\begin{array} {ll} & \Delta^2 u = K(x)u^{-\alpha} \quad \mbox{ in }\,\Omega , \\ &u> 0\quad \mbox{ in }\,\Omega, \;\;u\vert_{\partial\Omega}=0, \,\Delta…

We study a nonlinear equation in the half-space $\{x_1>0\}$ with a Hardy potential, specifically \[-\Delta u -\frac{\mu}{x_1^2}u+u^p=0\quad\text{in}\quad \mathbb R^n_+,\] where $p>1$ and $-\infty<\mu<1/4$. The admissible boundary behavior…

We study the existence and multiplicity of nonnegative solutions, as well as the behaviour of corresponding parameter-dependent branches, to the equation $-\Delta u = (1-u) u^m - \lambda u^n$ in a bounded domain $\Omega \subset…

We prove the monotonicity of positive solutions to the problem $-\Delta u = f(u)$ in $\mathbb{R}^N_+ := \{(x',x_N)\in\mathbb{R}^N \mid x_N>0 \}$ under zero Dirichlet boundary condition with a possible singular nonlinearity $f$. In some…

We study the regularity of the extremal solution of the semilinear biharmonic equation $\beta \Delta^2 u-\tau \Delta u=\frac{\lambda}{(1-u)^2}$ on a ball $B \subset \R^N$, under Navier boundary conditions $u=\Delta u=0$ on $\partial B$,…

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…

We are interested in the following Dirichlet problem $$ \left\{ \begin{array}{ll} -\Delta u + \lambda u - \mu \frac{u}{|x|^2} - \nu \frac{u}{\mathrm{dist}\,(x,\mathbb{R}^N \setminus \Omega)^2} = f(x,u) & \quad \mbox{in } \Omega \\ u = 0 &…

We study the semilinear elliptic equation $\Delta u + g(x,u,Du) = 0$ in $\R^n$. The nonlinearities $g$ can have arbitrary growth in $u$ and $Du$, including in particular the exponential behavior. No restriction is imposed on the behavior of…

We consider positive solutions $u$ of the semilinear biharmonic equation $\Delta^2 u = |x|^{-\frac{n+4}{2}} g(|x|^\frac{n-4}{2} u)$ in $\mathbb R^n \setminus \{0\}$ with non-removable singularities at the origin. Under natural assumptions…

In this paper, we study the local behaviors of nonnegative local solutions of fractional order semi-linear equations $(-\Delta)^\sigma u=u^{\frac{n+2\sigma}{n-2\sigma}}$ with an isolated singularity, where $\sigma\in (0,1)$. We prove that…

We study existence and uniqueness of solutions of (E 1) --$\Delta$u + $\mu$ |x| ^{-2} u + g(u) = $\nu$ in $\Omega$, u = $\lambda$ on $\partial$$\Omega$, where $\Omega$ $\subset$ R N + is a bounded smooth domain such that 0 $\in$…