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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 this paper, we study the Dirichlet problem for superlinear equation (E) ${mathbb L} u = u^p +lambda mu$ in a bounded domain $Omega$ with homogeneous boundary or exterior Dirichlet condition, where $p>1$ and $lambda>0$. The operator ${mathbb L}$ belongs to a class of nonlocal operators including typical types of fractional Laplacians and the datum $mu$ is taken in the optimal weighted measure space. The interplay between the operator ${mathbb L}$, the source term $u^p$ and the datum $mu$ yields substantial difficulties and reveals the distinctive feature of the problem. We develop a new unifying technique based on a fine analysis on the Green kernel, which enables us to construct a theory for semilinear equation (E) in measure frameworks. A main thrust of the paper is to provide a fairly complete description of positive solutions to the Dirichlet problem for (E). In particular, we show that there exist a critical exponent $p^*$ and a threshold value $lambda^*$ such that the multiplicity holds for $1<p<p^*$ and $0<lambda<lambda^*$, the uniqueness holds for $1<p<p^*$ and $lambda=lambda^*$, and the nonexistence holds in other cases. Various types of nonlocal operator are discussed to exemplify the wide applicability of our theory.
Let $Omega subset {mathbb R}^N$ ($N geq 3$) be a $C^2$ bounded domain and $delta$ be the distance to $partial Omega$. We study positive solutions of equation (E) $-L_mu u+ g(| abla u|) = 0$ in $Omega$ where $L_mu=Delta + frac{mu}{delta^2} $, $mu in (
Given a smooth domain $OmegasubsetRR^N$ such that $0 in partialOmega$ and given a nonnegative smooth function $zeta$ on $partialOmega$, we study the behavior near 0 of positive solutions of $-Delta u=u^q$ in $Omega$ such that $u = zeta$ on $partialOm
A characterization of a semilinear elliptic partial differential equation (PDE) on a bounded domain in $mathbb{R}^n$ is given in terms of an infinite-dimensional dynamical system. The dynamical system is on the space of boundary data for the PDE. Thi
This paper continues the development of regularity results for quasilinear measure data problems begin{align*} begin{cases} -mathrm{div}(A(x, abla u)) &= mu quad text{in} Omega, quad quad qquad u &=0 quad text{on} partial Omega, end{cases} end{a
In the present paper, we investigate the regularity and symmetry properties of weak solutions to semilinear elliptic equations which are locally stable.