Related papers: On semilinear elliptic equations with diffuse meas…

There are several methods for proving the existence of the solution to the elliptic boundary problem $Lu=f \text{\,\, in\,\,} D,\quad u|_S=0,\quad (*)$. Here $L$ is an elliptic operator of second order, $f$ is a given function, and…

This paper concerns the existence of a nontrivial solution for the following problem \begin{equation} \left\{\begin{aligned} -\Delta u + V(x)u & \in \partial_u F(x,u)\;\;\mbox{a.e. in}\;\;\mathbb{R}^{N},\nonumber u \in…

We prove the existence, uniqueness, and sharp bilateral pointwise estimates for positive bounded solutions to the Lane--Emden type problem \[ \begin{cases} L u = \sum\limits_{i=1}^{m}\sigma_{i} u^{q_{i}}+\sigma_0, \quad u\geq0 & \text{in }…

We show that a general nonlinearity $a(x,u)$ is uniquely determined, possibly up to a gauge, in a neighborhood of a fixed solution from boundary measurements of the corresponding semilinear equation. The main theorems are low regularity…

We study radial solutions of the semilinear elliptic equation $\Delta u+f(u)=0$ under rather general growth conditions on $f$. We construct a radial singular solution and study the intersection number between the singular solution and a…

We consider the reaction-diffusion problem $-\Delta_g u = f(u)$ in $\mathcal{B}_R$ with zero Dirichlet boundary condition, posed in a geodesic ball $\mathcal{B}_R$ with radius $R$ of a Riemannian model $(M,g)$. This class of Riemannian…

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…

We establish a precise connection between two elliptic quasilinear problems with Dirichlet data in a bounded domain of $\mathbb{R}^{N}.$ The first one, of the form \[ -\Delta_{p}u=\beta(u)| \nabla u| ^{p}+\lambda f(x)+\alpha, \] involves a…

In this note, we study Liouville theorems for the stable and finite Morse index weak solutions of the quasilinear elliptic equation $-\Delta_p u= f(x) F(u) $ in $\mathbb{R}^n$ where $p\ge 2$, $0\le f\in C(\mathbb{R}^n)$ and $F\in…

We consider a semilinear elliptic equation in a bounded domain with zero boundary conditions. The nonlinearity is discontinuous and monotone, but it is not a Carath\'eodory's function. The existence theorem has been proved.

Let $(V,\mu)$ be an infinite, connected, locally finite weighted graph. We study the problem of existence or non-existence of positive solutions to a semi-linear elliptic inequality \begin{equation*} \Delta u+u^{\sigma}\leq0\quad…

We obtain a universal energy estimate up to the boundary for stable solutions of semilinear equations with variable coefficients. Namely, we consider solutions to $- L u = f(u)$, where $L$ is a linear uniformly elliptic operator and $f$ is…

Recently, several works have been carried out in attempt to develop a theory for linear or sublinear elliptic equations involving a general class of nonlocal operators characterized by mild assumptions on the associated Green kernel. In…

In this manuscript we deal with elliptic equations with superlinear first order terms in divergence form of the following type \[ -\mbox{div}(M(x)\nabla u)= -\mbox{div}(h(u)E(x))+f(x), \] where $M$ is a bounded elliptic matrix, the vector…

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 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…

This article discusses a unified convergence analysis of the semilinear time-dependent equation $\partial_t u + (-1)^\mathrm{m}\Delta^{\mathrm{m}}u + u^3 - u = f$ with $\mathrm{m} \in \{1,2\}$ and homogeneous Dirichlet boundary conditions.…

We give new criteria for the existence of weak solutions to an equation with a super linear source term \begin{align*}-\Delta u = u^q ~~\text{in}~\Omega,~~u=\sigma~~\text{on }~\partial\Omega\end{align*}where $\Omega$ is a either a bounded…

We study the Dirichlet boundary value problem for equations with absorption of the form $-\Delta u+g\circ u=\mu$ in a bounded domain $\Omega\subset R^N$ where $g$ is a continuous odd monotone increasing function. Under some additional…

We analyze the semilinear elliptic equation $\Delta u=\rho(x) f(u)$, $u>0$ in ${\mathbf R}^D$ $(D\ge3)$, with a particular emphasis put on the qualitative study of entire large solutions, that is, solutions $u$ such that…