Related papers: Generalized Harnack inequality for semilinear elli…

We give a direct analytic proof of the classical Boundary Harnack inequality for solutions to linear uniformly elliptic equations in either divergence or non-divergence form.

We study inhomogeneous semilinear parabolic equations with source term f independent of time u_{t}={\Delta}u+u^{p}+f(x) on a metric measure space, subject to the conditions that f(x)\geq 0 and u(0,x)=\phi(x)\geq 0. By establishing…

In this paper, we establish a scale invariant Harnack inequality for some inhomogeneous parabolic equations in a suitable intrinsic geometry dictated by the nonlinearity. The class of equations that we consider correspond to the parabolic…

We study the semilinear elliptic inequality $-\Delta u\geq\varphi(\delta_K(x))f(u)$ in $R^N\setminus K,$ where $\varphi, f$ are non-negative and continuous functions, $K\subset R^N$ $(N\geq 2)$ is a compact set and $\delta_K(x)={\rm…

In this paper we prove some symmetry results for entire solutions to the semilinear equation $-\Delta u=f(u)$, with $f$ nonincreasing in a right neighbourhood of the origin. We consider solutions decaying only in some directions and we give…

We prove Harnack inequality and local regularity results for weak solutions of a quasilinear degenerate equation in divergence form under natural growth conditions. The degeneracy is given by a suitable power of a strong $A_\infty$ weight.…

On a bounded smooth domain we study solutions of a semilinear elliptic equation with an exponential nonlinearity and a Hardy potential depending on the distance to the boundary of the domain. We derive global a priori bounds of the…

We are mainly concerned with equations of the form $-Lu=f(x,u)+\mu$, where $L$ is an operator associated with a quasi-regular possibly nonsymmetric Dirichlet form, $f$ satisfies the monotonicity condition and mild integrability conditions,…

We prove an elliptic Harnack's inequality for a general form of a parabolic equation that generalizes both the standard parabolic $p$-Laplace equation and the normalized version that has been proposed in stochastic game theory. This version…

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…

In this paper, we discuss the more general Hessian inequality $\sigma_{k}^{\frac{1}{k}}(\lambda (D_i (A\left(|Du|\right) D_j u)))\geq f(u)$ including the Laplacian, p-Laplacian, mean curvature, Hessian, k-mean curvature operators, and…

We study the semilinear elliptic system \[ \Delta u = p(|x|)\,g(v), \qquad \Delta v = q(|x|)\,f(u), \qquad x \in \mathbb{R}^n,\; n \geq 3, \] under new Keller--Osserman-type integral conditions on the nonlinearities $f,g$ and decay…

We consider nonnegative solutions $u:\Omega\longrightarrow \mathbb{R}$ of second order hypoelliptic equations \begin{equation*} \mathscr{L} u(x) =\sum_{i,j=1}^n \partial_{x_i} \left(a_{ij}(x)\partial_{x_j} u(x) \right) + \sum_{i=1}^n b_i(x)…

We consider semilinear equation of the form $-Lu=f(x,u)+\mu$, where $L$ is the operator corresponding to a transient symmetric regular Dirichlet form ${\mathcal E}$, $\mu$ is a diffuse measure with respect to the capacity associated with…

This work proposes to generalize certain results regarding some semilinear elliptic systems.

Let $f :\R\to\R$ be a continuous function. We prove that under some additional assumptions on $f$ and $A:\R\to\R_{+}$, weak $\Cuno$ solutions of the differential inequality $-\diver(A(\abs{\nabla u})\nabla u)\ge f(u)$ on $\RN$ are…

We derive a matrix version of Li \& Yau--type estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative sectional curvatures and parallel Ricci tensor, similarly to what R.~Hamilton did…

We study asymptotic behavior of sub-solutions to non-uniformly elliptic equations with nonstandard growth. In particular, Harnack type inequalities are proved. Our approach gives new results for the cases with (p,q) nonlinearity and…

A logarithmic type Harnack inequality is established for the semigroup of solutions to a stochastic differential equation in Hilbert spaces with non-additive noise. As applications, the strong Feller property as well as the entropy-cost…

We study the existence of weak solutions of (E) $ (-\Delta)^\alpha u+g(u)=\nu $ in a bounded regular domain $\Omega$ in $\R^N (N\ge2)$ which vanish on $\R^N\setminus\Omega$, where $(-\Delta)^\alpha$ denotes the fractional Laplacian with…