Uniform Equicontinuity for a family of Zero Order operators approaching the fractional Laplacian

In this paper we consider a smooth bounded domain $\Omega \subset \R^N$ and a parametric family of radially symmetric kernels $K_\epsilon: \R^N \to \R_+$ such that, for each $\epsilon \in (0,1)$, its $L^1-$norm is finite but it blows up as $\epsilon \to 0$. Our aim is to establish an $\epsilon$ independent modulus of continuity in ${\Omega}$, for the solution $u_\epsilon$ of the homogeneous Dirichlet problem \begin{equation*} \left \{ \begin{array}{rcll} - \I_\epsilon [u] \&=\& f \& \mbox{in} \ \Omega. \\ u \&=\& 0 \& \mbox{in} \ \Omega^c, \end{array} \right . \end{equation*} where $f \in C(\bar{\Omega})$ and the operator \I_\epsilon has the form \begin{equation*} \I_\epsilon[u](x) = \frac12\int \limits_{\R^N} [u(x + z) + u(x - z) - 2u(x)]K_\epsilon(z)dz \end{equation*} and it approaches the fractional Laplacian as $\epsilon\to 0$. The modulus of continuity is obtained combining the comparison principle with the translation invariance of \I_\epsilon, constructing suitable barriers that allow to manage the discontinuities that the solution $u_\epsilon$ may have on $\partial \Omega$. Extensions of this result to fully non-linear elliptic and parabolic operators are also discussed

Highly oscillatory solutions of a Neumann problem for a pp-laplacian equation

We deal with a boundary value problem of the form $-\epsilon(\phi_p(\epsilon u'))'+a(x)W'(u)=0,\quad u'(0)=0=u'(1),$ where $\phi_p(s) = \vert s \vert^{p-2} s$ for $s \in \mathbb{R}$ and $p>1$, and $W:[-1,1] \to {\mathbb R}$ is a double-well potential. We study the limit profile of solutions when $\epsilon \to 0^+$ and, conversely, we prove the existence of nodal solutions associated with any admissible limit profile when $\epsilon$ is small enough

Solvability of nonlinear elliptic equations with gradient terms

We study the solvability in the whole Euclidean space of coercive quasi-linear and fully nonlinear elliptic equations modeled on $\Delta u\pm g(|\nabla u|)= f(u)$, $u\ge0$, where $f$ and $g$ are increasing continuous functions. We give conditions on $f$ and $g$ which guarantee the availability or the absence of positive solutions of such equations in $\R^N$. Our results considerably improve the existing ones and are sharp or close to sharp in the model cases. In particular, we completely characterize the solvability of such equations when $f$ and $g$ have power growth at infinity. We also derive a solvability statement for coercive equations in general form

Elliptic equations involving general subcritical source nonlinearity and measures

In this article, we study the existence of positive solutions to elliptic equation (E1) $$(-\Delta)^\alpha u=g(u)+\sigma\nu \quad{\rm in}\quad \Omega,$$ subject to the condition (E2) $$u=\varrho\mu\quad {\rm on}\quad \partial\Omega\ \ {\rm if}\ \alpha=1\qquad {\rm or\ \ in}\ \ \Omega^c \ \ {\rm if}\ \alpha\in(0,1),$$ where $\sigma,\varrho\ge0$, $\Omega$ is an open bounded $C^2$ domain in $\mathbb{R}^N$, $(-\Delta)^\alpha$ denotes the fractional Laplacian with $\alpha\in(0,1)$ or Laplacian operator if $\alpha=1$, $\nu,\mu$ are suitable Radon measures and $g:\mathbb{R}_+\mapsto\mathbb{R}_+$ is a continuous function. We introduce an approach to obtain weak solutions for problem (E1)-(E2) when $g$ is integral subcritical and $\sigma,\varrho\ge0$ small enough