Related papers: Regularity and Symmetry for Semilinear Elliptic Eq…
The aim of this paper is investigating the existence of solutions of some semilinear elliptic problems on open bounded domains when the nonlinearity is subcritical and asymptotically linear at infinity and there is a perturbation term which…
In this paper, we study the boundary pointwise regularity for the divergence form elliptic boundary problem on domains with rough boundaries, specifically uniform domains. In general, it is not straightforward to define weak solutions for…
We study stable solutions to fractional semilinear equations $(-\Delta)^s u = f(u)$ in $\Omega \subset \mathbb{R}^n$, for convex nonlinearities $f$, and under the Dirichlet exterior condition $u=g$ in $\mathbb{R}^n \setminus \Omega$ with…
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…
The boundedness of stable solutions to semilinear (or reaction-diffusion) elliptic PDEs has been studied since the 1970's. In dimensions 10 and higher, there exist stable energy solutions which are unbounded (or singular). This note…
We propose a probabilistic definition of solutions of semilinear elliptic equations with (possibly nonlocal) operators associated with regular Dirichlet forms and with measure data. Using the theory of backward stochastic differential…
This article proves the existence and regularity of weak solutions for a class of mixed local-nonlocal problems with singular nonlinearities. We examine both the purely singular problem and perturbed singular problems. A central…
We investigate existence and nonexistence of stationary stable nonconstant solutions, i.e. patterns, of semilinear parabolic problems in bounded domains of Riemannian manifolds satisfying Robin boundary conditions. These problems arise in…
We study properties of the semilinear elliptic equation $\Delta u = 1/u$ on domains in $R^n$, with an eye toward nonnegative singular solutions as limits of positive smooth solutions. We prove the nonexistence of such solutions in low…
This paper deals with symmetry phenomena for solutions of the Dirichlet problem involving semilinear PDEs on Riemannian domains. We shall present a rather general framework where the symmetry problem can be formulated and provide some…
In the paper, a new {\it slightly supercritical} condition, providing {\it local} regularity of axially symmetric solutions to the non-stationary 3D Navier-Stokes equations, is discussed. It generalises almost all known results in the local…
This paper deals with existence and regularity of positive solutions of singular elliptic problems on a smooth bounded domain with Dirichlet boundary conditions involving the $\Phi$-Laplacian operator. The proof of existence is based on a…
In this article, we consider a combination of local and nonlocal Laplace equation with singular nonlinearities. For such mixed problems, we establish existence of at least one weak solution for a parameter dependent singular nonlinearity…
In a smooth bounded domain we obtain existence, uniqueness, regularity and boundary behavior for a class of singular quasi-linear elliptic equations.
In this paper we study the asymptotic behavior of solutions to an elliptic equation near the singularity of an inverse square potential with a coefficient related to the best constant for the Hardy inequality. Due to the presence of a…
These notes record the lectures for the CIME Summer Course taught by the first author in Cetraro during the week of June 19-23, 2017. The notes contain the proofs of several results on the classification of stable solutions to some…
It is establish regularity results for weak solutions of quasilinear elliptic problems driven by the well known $\Phi$-Laplacian operator given by \begin{equation*} \left\{\ \begin{array}{cl} \displaystyle-\Delta_\Phi u= g(x,u), &…
In this paper, we are concerned with stable solutions , possibly unbounded and sign-changing, of some semi-linear elliptic problem with mixed nonlinear boundary conditions. We establish the nonexistence of stable solutions, the main methods…
This is an expository paper on the theory of local regularity for weak solutions to the non-stationary 3D Navier-Stokes equations near the boundary of a domain.
We study a semilinear parabolic equation that possesses global bounded weak solutions whose gradient has a singularity in the interior of the domain for all $t>0$. The singularity of these solutions is of the same type as the singularity of…