Related papers: Sublinear equations and Schur's test for integral …

This paper investigates two classes of quasilinear and essentially nonlinear integral equations with a sum-difference kernel on the half-line. Such equations arise in various areas of physics, including the theory of radiative transfer in…

In this article we introduce a stochastic counterpart of the H\"ormander condtion on the kernel $K(r,t,x,y)$: there exists a pseudo-metric $\rho$ on $(0,\infty)\times R^d$ and a positive constant $C_0$ such that for $X=(t,x), Y=(s,y),…

In this paper, we consider the semilinear wave equation involving the nonlinear damping term $g(u_t) $ and nonlinearity $f(u)$. The well-posedness of the weak solution satisfying some additional regularity is achieved under the wider ranges…

In this paper, we study the existence of nonnegative weak solutions to (E) $ (-\Delta)^\alpha u+h(u)=\nu $ in a general regular domain $\Omega$, which vanish in $\R^N\setminus\Omega$, where $(-\Delta)^\alpha$ denotes the fractional…

Let $\mathscr G:= (V,E)$ be a weighted locally finite graph whose finite measure $\mu$ has a positive lower bound. Motivated by wide interest in the current literature, in this paper we study the existence of classical solutions for a class…

Let $\Omega $ be a bounded domain in $\mathbb{R} ^N $, and let $u\in C^1 (\overline{\Omega }) $ be a weak solution of the following overdetermined BVP: $-\nabla (g(|\nabla u|)|\nabla u|^{-1} \nabla u )=f(|x|,u)$, $ u>0 $ in $\Omega $ and…

For indices p and q, 1 < p <= q < infini and a linear operator L satisfying some weak-type boundedness conditions on suitable function spaces, we give in the Dunkl setting sufficient conditions on nonnegative pairs of weight functions to…

Let $\Omega$ be homogeneous of degree zero and have mean value zero on the unit sphere ${S}^{n-1}$, $T_{\Omega}$ be the convolution singular integral operator with kernel $\frac{\Omega(x)}{|x|^n}$. For $b\in{\rm BMO}(\mathbb{R}^n)$, let…

This paper deals with the asymptotic behavior as $t\rightarrow T<\infty$ of all weak (energy) solutions of a class of equations with the following model representative: \begin{equation*} (|u|^{p-1}u)_t-\Delta_p(u)+b(t,x)|u|^{\lambda-1}u=0…

We investigate the existence and multiplicity of positive solutions to the problem \begin{equation} \begin{cases} \begin{aligned} - \Delta_{\gamma} u &= \lambda u^{p} + u^{-\delta} &\quad \text{in } \Omega, \quad u &= 0 &\quad \text{on }…

In this paper, we investigate the regularity of weak solutions $u\colon\Omega\to\mathbb{R}$ to elliptic equations of the type \begin{equation*} \mathrm{div}\, \nabla \mathcal{F}(x,Du) = f\qquad\text{in $\Omega$}, \end{equation*} whose…

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…

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 &…

We are motivated by studying a boundary-value problem for a class of semilinear degenerate elliptic equations \begin{align}\tag{P}\label{P} \begin{cases} - \Delta_x u - |x|^{2\alpha} \dfrac{\partial^2 u}{\partial y^2} = f(x,y,u) &…

We study the interior regularity of solutions to the Dirichlet problem $Lu=g$ in $\Omega$, $u=0$ in $\R^n\setminus\Omega$, for anisotropic operators of fractional type $$ Lu(x)= \int_{0}^{+\infty}\,d\rho \int_{S^{n-1}}\,da(\omega)\, \frac{…

We consider the following class of mixed local-nonlocal equations: \begin{align}\label{abs}\tag{$\mathcal{P}$} -\Delta_p u + (-\Delta)_p^s u = V |u|^{p-2}u \text{ in } \Omega, \end{align} where $s \in (0,1), p \in (1, \infty)$, and the…

In this paper, we prove existence and regularity results for solutions of some nonlinear Dirichlet problems for an elliptic equation defined by a degenerate coercive operator and a singular right hand side. \begin{equation}\label{01}…

We consider Volterra-type integration operators $T_g$ between Bergman spaces induced by weights $\omega$ satisfying a doubling property. We derive estimates for the operator norms, essential and weak essential norms of $T_g: A_\omega^p \to…

This article is concerned with the existence of positive weak solutions for the following quasilinear Schr\"odinger Choquard equation: \begin{equation*} \begin{array}{cc} \displaystyle -div(g^2(u)\nabla u) + g(u)g'(u)\nabla u + a(x) u =…

We study existence and uniqueness of solutions of (E 1) --$\Delta$u + $\mu$ |x| ^{-2} u + g(u) = $\nu$ in $\Omega$, u = $\lambda$ on $\partial$$\Omega$, where $\Omega$ $\subset$ R N + is a bounded smooth domain such that 0 $\in$…