Related papers: On elliptic equations with N-independent stable op…
We investigate partial symmetry of solutions to semi-linear and quasi-linear elliptic problems with convex nonlinearities, in domains that are either axially symmetric or radially symmetric.
\noindent Using the techniques connected with the measure of noncompactness we investigate the neutral difference equation of the following form \begin{equation*} \Delta \left(r_{n}\left(\Delta \left(x_{n}+p_{n}x_{n-k}\right) \right)…
In this work we analyze the existence of solutions to the nonlinear elliptic system: \begin{equation*} \left\{ \begin{array}{rcll} -\Delta u & = & v^q+\a g & \text{in }\Omega , \\ -\Delta v& = &|\nabla u|^{p}+\l f &\text{in }\Omega , \\…
We are concerned with positive radial solutions of the inhomogeneous elliptic equation $\Delta u+K(|x|)u^p+\mu f(|x|)=0$ on $\mathbb{R}^N$, where $N\ge 3$, $\mu>0$ and $K$ and $f$ are nonnegative nontrivial functions. If $K(r)\sim…
We review the indefinite sublinear elliptic equation $-\Delta u=a(x)u^{q}$ in a smooth bounded domain $\Omega\subset\mathbb{R}^{N}$, with Dirichlet or Neumann homogeneous boundary conditions. Here $0<q<1$ and $a$ is continuous and changes…
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…
Several results about positive solutions -in a Lipschitz domain- of a nonlinear elliptic equation in a general form $ \Delta u(x)-g(x,u(x))=0$ are proved, extending thus some known facts in the case of $ g(x,t)=t^q$, $q>1$, and a smooth…
We consider positive solutions to semilinear elliptic problems with singular nonlinearities, under zero Dirichlet boundary condition. We exploit a refined version of the moving plane method to prove symmetry and monotonicity properties of…
In this paper we qualitatively study radial solutions of the semilinear elliptic equation $\Delta u + u^n = 0$ with $u(0)=1$ and $u'(0)=0$ on the positive real line, called the Emden-Fowler or Lane-Emden equation. This equation is of great…
We consider the standing-wave problem for a nonlinear Schr\"{o}dinger equation, corresponding to the semilinear elliptic problem \begin{equation*} -\Delta u+V(x)u=|u|^{p-1}u,\ u\in H^1(\mathbb{R}^2), \end{equation*} where $V(x)$ is a…
In this paper, we consider the following Kirchhoff type problem $$\left\{\aligned&-\biggl(a + b\int_{\mathbb{R}^N} |\nabla u|^2 dx \biggr) \Delta u + V(x) u = |u|^{p-2}u &\text{ in } \mathbb{R}^N,\cr &u\in H^1(\mathbb{R}^N),…
Our purpose is to find positive solutions $u \in D^{1,2}(\rz^N)$ of the semilinear elliptic problem $-\laplace u - \lambda V(x) u = h(x) u^{p-1}$ for $2<p$. The functions $V$ and $h$ may have an indefinite sign and the linearized operator…
In this paper we prove radial symmetry for solutions to a free boundary problem with a singular right hand side, in both elliptic and parabolic regime. More exactly, in the unit ball $B_1$ we consider a solution to the fully nonlinear…
In this paper, we focus our attention on the positive solutions to second-order nonlinear ordinary differential equations of the form $u''+q(t)g(u)=0$, where $q$ is a sign-changing weight and $g$ is a superlinear function. We exploit the…
We derive a priori bounds for positive supersolutions of $ - \Delta_{p} u = \rho(x) f(u) $, where $p>1$ and $\Delta_{p}$ is the $p$-Laplace operator, in a smooth bounded domain of $R^{N}$ with zero Dirichlet boundary conditions. We apply…
The present paper establishes the first result on the absolute continuity of elliptic measure with respect to the Lebesgue measure for a divergence form elliptic operator with non-smooth coefficients that have a BMO anti-symmetric part. In…
We establish quantitative asymptotic behavior of positive solutions of a family of nonlinear elliptic equations on the half cylinder near the end. This unifies the study of isolated singularities of some semilinear elliptic equations, such…
The paper proves Liouville-type results for stable solutions of semilinear elliptic PDEs with convex nonlinearity, posed on the entire Euclidean space. Extensions to solutions which are stable outside a compact set are also presented.
In the present paper we investigate the following semilinear singular elliptic problem: \begin{equation*} (\rm P)\qquad \left \{\begin{array}{l} -\Delta u = \dfrac{p(x)}{u^{\alpha}}\quad \text{in} \Omega \\ u = 0\ \text{on} \Omega,\ u>0…
We are concerned with the existence and boundary behaviour of positive radial solutions for the system \begin{equation*} \left\{ \begin{aligned} \Delta u&=g(|x|,v(x)) &&\quad\mbox{in}\ \Omega, \\ \Delta v&=f(|x|,|\nabla u(x)|)…