Related papers: Remarks on nonlinear equations with measures

We study the monotonicity and one-dimensional symmetry of positive solutions to the problem $-\Delta_p u = f(u)$ in $\mathbb{R}^N_+$ under zero Dirichlet boundary condition, where $p>1$ and $f:(0,+\infty)\to\mathbb{R}$ is a locally…

Let (M,g) be a smooth connected compact Riemannian manifold of finite dimension n \geq 2 with a smooth boundary \partial M. We consider the problem -{\epsilon}^2\Delta_gu+u=|u|^{p-2}u, u>0 on M, \partial u/ \partial{\nu}=0 on \partial M…

This paper investigates the existence of positive solutions for regular discrete second-order single-variable boundary value problems with mixed boundary conditions, including a nonhomogeneous Dirichlet boundary condition, of the form:…

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…

In this paper, we study the solvability of the nonlinear Dirichlet problem with sum of the operators of independent non standard growths in a bounded domain $\Omega \subset \mathbb{R}^{n}$. We obtain sufficient conditions and show the…

We study the Dirichlet problem for the semi--linear partial differential equations ${\rm div}\,(A\nabla u)=f(u)$ in simply connected domains $D$ of the complex plane $\mathbb C$ with continuous boundary data. We prove the existence of the…

In the paper we consider elliptic equations of the form $-Au=u^{-\gamma}\cdot\mu$, where $A$ is the operator associated with a regular symmetric Dirichlet form, $\mu$ is a positive nontrivial measure and $\gamma>0$. We prove the existence…

In this article, we study the existence of positive solutions to elliptic equation (E1) $$(-\Delta)^\alpha u=g(u)+\sigma\nu \quad{\rm in}\quad \Omega,$$ subject to the condition (E2) $$u=\varrho\mu\quad {\rm on}\quad \partial\Omega\ \ {\rm…

We study the boundary behaviour of solutions $u$ of $-\Delta_{N}u+ |u|^{q-1}u=0$ in a bounded smooth domain $\Omega\subset\mathbb R^{N}$ subject to the boundary condition $u=0$ except at one point, in the range $q>N-1$. We prove that if…

Let \(\mu\) be a finite Borel measure on \((-\pi,\pi)\). Consider the one-dimensional Poisson equation \(-u''=\mu\), where equality holds in the sense of distributions, with Dirichlet boundary conditions \(u(\pm\pi)=0\). In this paper, we…

We study the Dirichlet problem for the stationary Schr\"odinger fractional Laplacian equation $(-\Delta)^s u + V u = f$ posed in bounded domain $ \Omega \subset \mathbb R^n$ with zero outside conditions. We consider general nonnegative…

In this paper, we investigate the symmetry properties of positive solutions $u$ to a semilinear elliptic equation under mixed Dirichlet-Neumann boundary conditions in symmetric domains. First, we establish a maximum principle tailored to…

We prove existence of a positive radial solution to the Choquard equation $$-\Delta u +V u=(I_\alpha\ast |u|^p)|u|^{p-2}u\qquad\text{in}\,\,\,\Omega$$ with Neumann or Dirichlet boundary conditions, when $\Omega$ is an annulus, or an…

Let $\Omega\subset\BBR^N$ be a bounded $C^2$ domain and $\CL_\gk=-\Gd-\frac{\gk}{d^2}$ the Hardy operator where $d=\dist (.,\prt\Gw)$ and $0<\gk\leq\frac{1}{4}$. Let $\ga_{\pm}=1\pm\sqrt{1-4\gk}$ be the two Hardy exponents, $\gl_\gk$ the…

We consider an inverse boundary value problem for the equation $\nabla\cdot(\sigma-i\omega\epsilon)\nabla u=0$ in a given bounded domain $\Omega$ at a fixed $\omega>0$. $\sigma$ and $\epsilon$ denote the conductivity and permittivity of the…

We study the periodic boundary value problem associated with the second order nonlinear differential equation $$ u" + c u' + \left(a^{+}(t) - \mu \, a^{-}(t)\right) g(u) = 0, $$ where $g(u)$ has superlinear growth at zero and at infinity,…

We consider the quasi-linear eigenvalue problem $-\Delta_p u = \lambda g(u)$ subject to Dirichlet boundary conditions on a bounded open set $\Omega$, where $g$ is a locally Lipschitz continuous functions. Imposing no further conditions on…

In this work we provide conditions for the existence of solutions to nonlinear boundary value problems of the form \begin{equation*} y(t+n)+a_{n-1}(t)y(t+n-1)+\cdots a_0(t)y(t)=g(t,y(t+m-1)) \end{equation*} subject to \begin{equation*}…

We study the Dirichlet problem $-\div(|\nabla u|^{p(x)-2} \nabla u) =0 $ in $\Omega$, with $u=f$ on $\partial \Omega$ and $p(x) = \infty$ in $D$, a subdomain of the reference domain $\Omega$. The main issue is to give a proper sense to what…

Let $\Omega\subset\mathbb{R}^N$ ($N\geq 3$) be a bounded $C^2$ domain and $\Sigma\subset\partial\Omega$ be a compact $C^2$ submanifold of dimension $k$. Denote the distance from $\Sigma$ by $d_\Sigma$. In this paper, we study positive…