Related papers: Positive solutions of quasilinear elliptic equatio…

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…

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 $\mathbb{R}^{N}$…

Optimal estimates on asymptotic behaviors of weak solutions both at the origin and at the infinity are obtained to the following quasilinear elliptic equations \begin{eqnarray*} -\Delta_{p}u-\frac{\mu}{|x|^{p}}|u|^{p-2}u+m|u|^{p-2}u=f(u), &…

Consider the problem \begin{eqnarray*} -\Delta u_\e &=& v_\e^p \quad v_\e>0\quad {in}\quad \Omega, -\Delta v_\e &=& u_\e^{q_\e}\quad u_\e>0\quad {in}\quad \Omega, u_\e&=&v_\e\:\:=\:\:0 \quad {on}\quad \partial \Omega, \end{eqnarray*} where…

Optimal estimates on asymptotic behaviors of weak solutions both at the origin and at the infinity are obtained to the following quasilinear elliptic equations \begin{eqnarray*} -\Delta_{p}u-\frac{\mu}{|x|^{p}}|u|^{p-2}u+m|u|^{p-2}u=f(u), &…

We consider the uniqueness of the following positive solutions of anisotropic elliptic equation: \begin{equation} \nonumber \left\{ \begin{aligned} -\Delta^F_p u&=u^q \quad \text{in} \quad \Omega, u&=0 \quad \text{on} \quad \partial \Omega,…

We study the inequality $$ {\rm div}\big(|x|^{-\alpha}|\nabla u|^{m-2}\nabla u\big)\geq (I_\beta\ast u^p)u^q \quad\mbox{ in } B_1\setminus\{0\}\subset {\mathbb R}^N, $$ where $\alpha>0$, $N\geq 1$, $m>1$, $p, q>m-1$ and $I_\beta$ denotes…

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

Let $\Omega$ be a domain in $\mathbb{R}^d$, $d\geq 2$, and $1<p<\infty$. Fix $V\in L_{\mathrm{loc}}^\infty(\Omega)$. Consider the functional $Q$ and its G\^{a}teaux derivative $Q^\prime$ given by $$Q(u):= \frac{1}{p}\int_\Omega. (|\nabla…

In this paper, we study the existence and uniqueness of positive solutions for the following nonlinear fractional elliptic equation: \begin{eqnarray*} (-\Delta)^\alpha u=\lambda a(x)u-b(x)u^p&{\rm in}\,\,\R^N, \end{eqnarray*} where $…

We examine equations of the form {eqnarray*} \{{array}{lcl} \hfill \HA u &=& \lambda g(x) f(u) \qquad \text{in}\ \Omega \hfill u&=& 0 \qquad \qquad \qquad \text{on}\ \pOm, {array}. {eqnarray*} where $ \lambda >0$ is a parameter and $…

We consider positive solutions of the problem \begin{equation} \left\{\begin{array}{l}-\mbox{div}(x_{n}^{a}\nabla u)=0\qquad \mbox{in}\;\;\mathbb{R}_+^n,\\ \frac{\partial u}{\partial \nu^a}=u^{q} \qquad \mbox{on}\;\;\partial…

This paper is concerned with the study of multiple positive solutions to the following elliptic problem involving a nonhomogeneous operator with nonstandard growth of $p$-$q$ type and singular nonlinearities \begin{equation*} \left\{…

In this paper, we prove some qualitative properties for the positive solutions to some degenerate elliptic equation given by \[-\operatorname{div}(w|\nabla u|^{p-2}\nabla u)=f(x,u);\;\;w\in \mathcal{A}_p\] on smooth domain and for varying…

We deal with existence and uniqueness of positive solutions of an elliptic boundary value problem modeled by \begin{equation*} \begin{cases} \displaystyle -\Delta_p u= \frac{f}{u^\gamma} + g u^q & \mbox{in $\Omega$,} \\ u = 0 & \mbox{on…

In this paper, we study isolated singular positive solutions for the following Kirchhoff--type Laplacian problem: \begin{equation*} -\left(\theta+\int_{\Omega} |\nabla u| dx\right)\Delta u =u^p \quad{\rm in}\quad \Omega\setminus…

We study the local properties of positive solutions of the equation $-\Delta u=u^p-m|\nabla u|^q$ in a punctured domain $\Omega\setminus\{0\}$ of $\mathbb{R}^N$ or in a exterior domain $\mathbb{R}^N\setminus B_{r_0}$ in the range…

Optimal estimates on the asymptotic behaviors of weak solutions both at the origin and at the infinity are obtained to the following quasilinear elliptic equations…

Let $\Omega \subset {\mathbb R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $\delta$ be the distance to $\partial \Omega$. We study positive solutions of equation (E) $-L_\mu u+ g(|\nabla u|) = 0$ in $\Omega$ where $L_\mu=\Delta +…

In this paper we consider nonlinear elliptic PDEs of the type $$-\Delta_p u+a(x)|u|^{p-2}u=|u|^{p^*-2}u \qquad \mbox{ in }\Omega,$$ where $1<p<N$ and $p^*=Np/(N-p)$ is the critical Sobolev exponent, and allowing the asymptotic behavior of…