We prove the existence and give estimates of the fundamental solution (the heat kernel) for the equation $\partial_t ={\mathcal L}^{\kappa}$ for non-symmetric non-local operators$${\mathcal L}^{\kappa}f(x):= \int_{\mathbb R^d}( f(x+z)-f(x)- {\bf 1}_{|z| < 1} \langle z,\nabla f(x) \rangle )\kappa(x,z)J(z)\, dz\,,$$under broad assumptions on $\kappa$ and $J$.Of special interest is the case when the order of the operator ${\mathcal L}^{\kappa}$ is smaller than or equal to 1. Our approach rests on imposingsuitable cancellation conditions on the internal drift coefficient$$\int_{r\leq |z| < 1} z \kappa(x,z)J(z)dz\,,\qquad 0 < r \leq 1\,,$$which allows us to handle the non-symmetry of$z\mapsto \kappa(x,z)J(z)$.The results are new even for the $1$-stable Lévy measure $J(z)=|z|^{-d-1}$.

We deal with existence, uniqueness and regularity of nonnegative solutions to a Dirichlet problem for equations as \begin{equation*} -\text{div} \Big ( \frac{|\nabla u|^{p-2}\nabla u}{(1+u)^{\theta(p-1)}} \Big ) = h(u)f \quad\text{in }\Omega, \end{equation*}where$\Omega$ is an open bounded subset of $\mathbb{R}^N$ ($N\ge 2$), $p>1$, $\theta\ge 0$, $f\geq 0$ belongs to a suitable Lebesgue space and $h$ is a continuous, nonnegative function which may blow up at zero and it is bounded at infinity.

We consider a space of $L^2$ vector fields with bounded mean oscillation whose ``normal'' component to the boundary is well-controlled.In the case when the dimension $n \geq 3$, we establish its Helmholtz decomposition for an arbitrary uniformly $C^3$ domain in $\mathbf{R}^n$.

This paper analyzes the generalized spatially heterogeneous diffusive predator-prey model introduced by the authors in [24],whose interaction terms depend on a saturation coefficient $m(x)\gneq0$. As the amplitude of the saturation term, measured by $\|m\|_\infty$, blows up to infinity, the existence of, at least, two coexistence states, is established in the region of the parameters where the semitrivial positive solution is linearly stable, regardless the sizes and the shapes of the remaining function coefficients in the setting of the model. In some further special cases, an $S$-shaped component of coexistence states can be constructed, which causes the existence of, at least, three coexistence states, thoughthis multiplicity occurs within the parameter regions where the semitrivial positive solution is linearly unstable. Therefore, these multiplicity results inherit a rather different nature.