Related papers: Strong maximum principle for fully nonlinear nonlo…
This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity: \begin{equation} \tag{$\mathcal E$} (-\Delta)^s u = a(x)…
Our purpose of this paper is to study the nonexistence of nonnegative very weak solutions of \begin{equation}\label{eq 0.1} \displaystyle (-\Delta)^\alpha u = u^p+\nu\quad {\rm in}\quad \Omega,\qquad\ u=g\quad {\rm in}\quad \mathbb{…
In this paper, we study the following class of weighted Choquard equations \begin{align*} -\Delta u =\lambda u + \Bigg(\displaystyle\int\limits_\Omega \frac{Q(|y|)F(u(y))}{|x-y|^\mu}dy\Bigg) Q(|x|)f(u) ~~\textrm{in}~~ \Omega~~ \text{and}~~…
The purpose of this paper is to study nonlinear singular parabolic equations with $p(x)$- Laplacian. Precisely, we consider the following problem and discuss the existence of a non-negative weak solution. \begin{align*} \frac{\partial…
We are concerned with nonexistence results for a class of quasilinear parabolic differential problems with a potential in $\Omega\times(0,+\infty)$, where $\Omega$ is a bounded domain. In particular, we investigate how the behavior of the…
This paper is devoted to the study of the existence of positive solutions for a problem related to a higher order fractional differential equation involving a nonlinear term depending on a fractional differential operator,…
Let $\Omega$ be a bounded domain in $\mathbb{R}^2$ with smooth boundary, we study the following anisotropic elliptic Neumann problem with Hardy-H\'{e}non weight $$ \begin{cases} -\nabla(a(x)\nabla u)+a(x)u=a(x)|x-q|^{2\alpha}u^p,\,\,\,\,…
The focus of this study is on exploring some qualitative properties of solutions to a class of semilinear elliptic problems in bounded domains, where the boundary conditions depend non-locally on the unknown solution at specified interior…
We establish a Liouville-type theorem for nonnegative weak supersolutions to $\mathcal{L}_K u = u^q$ in $\mathbb{R}^n$, where $\mathcal{L}_K$ is a translation-invariant integro-differential operator of order $2s$ with $s \in (0,1)$. The…
We establish the asymptotic behaviour of the least energy solutions of the following nonlocal Neumann problem: \begin{align*} \left\{\begin{array}{l l} { d(-\Delta)^{s}u+ u= \abs{u}^{p-1}u } \text{ in $\Omega,$ } { \mathcal{N}_{s}u=0 }…
Even without a variational background, a multiplicity result of positive solutions with ordered $L^{p}(\Omega)$-norms is provided to the following boundary value problem \begin{equation*} \left \{ \begin{array}{ll}…
In this paper we consider the model semilinear Neumann system $$\left\{ \begin{array}{lll} -\Delta u+a(x)u=\lambda c(x) F_u(u,v)& {\rm in} & \Omega,\\ -\Delta v+b(x)v=\lambda c(x) F_v(u,v)& {\rm in} & \Omega,\\ \frac{\partial u}{\partial…
We study the periodic and the Neumann boundary value problems associated with the second order nonlinear differential equation \begin{equation*} u'' + c u' + \lambda a(t) g(u) = 0, \end{equation*} where $g \colon…
We consider elliptic equations with non-Lipschitz nonlinearity $$ -\Delta u = \lambda |u|^{\beta-1}u-|u|^{\alpha-1}u$$ in a smooth bounded domain $\Omega \subset \mathbb{R}^n$, $n\geq 3$, with Dirichlet boundary conditions; here…
In this paper we survey some results on the Dirichlet problem \[\left\{ \begin{array}{rcll} L u &=&f&\textrm{in }\Omega \\ u&=&g&\textrm{in }\mathbb R^n\backslash\Omega \end{array}\right.\] for nonlocal operators of the form…
We study the periodic boundary value problem associated with the second order nonlinear equation \begin{equation*} u'' + ( \lambda a^{+}(t) - \mu a^{-}(t) ) g(u) = 0, \end{equation*} where $g(u)$ has superlinear growth at zero and sublinear…
We study existence of nonnegative solutions to a nonlinear parabolic boundary value problem with a general singular lower order term and a nonnegative measure as nonhomogeneous datum, of the form $$ \begin{cases} \displaystyle u_t -…
In this paper we consider the following Dirichlet problem for the $p$-Laplacian in the positive parameters $\lambda$ and $\beta$: [{{array} [c]{rcll}% -\Delta_{p}u & = & \lambda h(x,u)+\beta f(x,u,\nabla u) & \text{in}\Omega u & = & 0 &…
This work is devoted to the study of the existence of at least one weak solution to nonlocal equations involving a general integro-differential operator of fractional type. As a special case, we derive an existence theorem for the…
In this paper we develop a numerical scheme based on quadratures to approximate solutions of integro-differential equations involving convolution kernels, $\nu$, of diffusive type. In particular, we assume $\nu$ is symmetric and…