We consider both a problem over ℝn and a boundary problem over an exterior subregion for a Douglis–Nirenberg system of differential operators under limited smoothness assumptions as well as under the assumption of parameter ellipticity in a closed sector Ꮭ in the complex plane with vertex at the origin. We pose each problem on an Lp Sobolev space setting, 1 < p < ∞, and denote by the operator induced in this setting by the problem over ℝn and by Ap the operator induced in this setting by the boundary problem under null boundary conditions. We then derive results pertaining to the Fredholm theory for each of these operators for values of the spectral parameter λ lying in Ꮭ as well as results for these values of λ pertaining to the invariance of their Fredholm domains with p.

We consider the weighted $L_p$ solvability for divergence and nondivergence form parabolic equations with partially bounded mean oscillation (BMO) coefficients and certain positive potentials. As an application, global regularity in Morrey spaces for divergence form parabolic operators with partially BMO coefficients on a bounded domain is established.

This paper deals with the existence of positive radial solutions for the quasilinear system $\text{div}(|\nabla u_i|^{p-2}\nabla u_i)+\lambda f^i(u_1,\dots,u_n)=0$, $|x|\lt1$, $u_i(x)=0$, on $|x|=1$, $i=1,\dots,n$, $p\gt1$, $\lambda>0$, $x\in\mathbb{R}^N$. The $f^i$, $i=1,\dots,n$, are continuous and non-negative functions. Let $\bm{u}=(u_1,\dots,u_n)$, $\|\bm{u}\|=\sum_{i=1}^n|u_i|$,

$$ f_0^i=\lim_{\|\bm{u}\|\to0}\frac{f^i(\bm{u})}{\|\bm{u}\|^{p-1}}, $$

$i=1,\dots,n$, $\bm{f}=(f^1,\dots,f^n)$, $\bm{f}_0=\sum_{i=1}^nf_0^i$. We prove that the problem has a positive solution for sufficiently small $\lambda>0$ if $\bm{f}_0=\infty$. Our methods employ a fixed-point theorem in a cone.