Related papers: Existence and uniqueness results for possibly sing…

This paper is concerned with the Dirichlet problem for an equation involving the 1--Laplacian operator $\Delta_1 u$ and having a singular term of the type $\frac{f(x)}{u^\gamma}$. Here $f\in L^N(\Omega)$ is nonnegative, $0<\gamma\le1$ and…

In this paper, we study the good-$\lambda$ type bounds for renormalized solutions to nonlinear elliptic problem: \begin{align*} \begin{cases} -\div(A(x,\nabla u)) &= \mu \quad \text{in} \ \ \Omega, \\ u &=0 \quad \text{on} \ \ \partial…

We study a singular elliptic problem driven by a mixed local-nonlocal operator of the form \begin{equation*} \begin{aligned} -\Delta_p u + (-\Delta_q)^s u &= \frac{\lambda}{u^{\delta}} + u^r \text{ in } \Omega\newline u > 0 \text{ in }…

In this paper we are concerned with a general singular Dirichlet boundary value problem whose model is the following $$ \begin{cases} -\Delta u = \frac{\mu}{u^{\gamma}} & \text{in}\ \Omega, u=0 &\text{on}\ \partial\Omega, u>0 &\text{on}\…

The authors of this paper deal with the existence and regularities of weak solutions to the homogenous $\hbox{Dirichlet}$ boundary value problem for the equation $-\hbox{div}(|\nabla u|^{p-2}\nabla u)+|u|^{p-2}u=\frac{f(x)}{u^{\alpha}}$.…

We deal with existence, uniqueness and regularity of nonnegative solutions to a Dirichlet problem for equations as \begin{equation*} \displaystyle -\operatorname{div}\left(\frac{|\nabla u|^{p-2}\nabla u}{(1+u)^{\theta(p-1)}}\right) = h(u)f…

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…

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 are concerned with the inverse boundary problem of determining anomalies associated with a semilinear elliptic equation of the form $-\Delta u+a(\mathbf x, u)=0$, where $a(\mathbf x, u)$ is a general nonlinear term that belongs to a…

We prove the existence of a solution of (--$\Delta$) s u + f (u) = 0 in a smooth bounded domain $\Omega$ with a prescribed boundary value $\mu$ in the class of positive Radon measures for a large class of continuous functions f satisfying a…

We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on a bounded smooth domain $\Omega\subset\R^N$ with Dirichlet boundary conditions on $\partial\Omega$. The operator $L$ is a uniformly elliptic linear operator of…

We study quasilinear degenerate singular elliptic equation of type -Delta_p u = \frac{u^{p^*(s)-1}}{|y|^t}$ in a smooth bounded domain \Omega in R^n=R^k \times R^{N-k}$, x=(y,z) in R^k \times R^{N-k}, 2 \leq k<N and N \geq 3, 1<p<2, 0\leq…

In this paper, we study the following singular nonlinear elliptic problem \begin{equation}\label{eq:1} \left\{ \begin{array}{ll} \displaystyle (-\Delta)^{\frac \alpha 2} u=\lambda |u|^{r-2}u+\mu\frac{|u|^{q-2}u}{|x|^{s}}\quad &{\rm in…

We prove the existence of a solution to a singular anisotropic elliptic equation in a bounded open subset $\Omega$ of $\mathbb R^N$ with $N\ge 2$, subject to a homogeneous boundary condition: \begin{equation} \label{eq0} \left\{…

We study the following quasilinear elliptic equation $$ -\Delta_p u + (\beta\Phi(x)-a(x)) u^{p-1} + b(x)g(u)=0 \quad \text{in } \mathbb{R}^N \quad \quad \quad (P_\beta) $$ where $p>1$, $\Phi \in L^\infty_{loc}(\mathbb{R}^N)$, $a,b \in…

We develop numerical algorithms to approximate positive solutions of elliptic boundary value problems with superlinear subcritical nonlinearity on the boundary of the form $-\Delta u + u = 0$ in $\Omega$ with $\frac{\partial u}{\partial…

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…

We study positive solutions of equation (E) $-\Delta u + u^p|\nabla u|^q= 0$ ($0<p$, $0\leq q\leq 2$, $p+q>1$) and other related equations in a smooth bounded domain $\Omega \subset {\mathbb R}^N$. We show that if $N(p+q-1)<p+1$ then, for…

We study the asymptotic behavior, as $\gamma$ tends to infinity, of solutions for the homogeneous Dirichlet problem associated to singular semilinear elliptic equations whose model is $$ -\Delta u=\frac{f(x)}{u^\gamma}\,\text{ in }\Omega,…

We study the boundary value problem $-{\rm div}(\log(1+ |\nabla u|^q)|\nabla u|^{p-2}\nabla u)=f(u)$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded domain in $\RR^N$ with smooth boundary. We distinguish the cases where…