The nonlocal boundary value problems for linear and nonlinear degenerate abstract differential equations of arbitrary order are studied. The equations have the variable coefficients and small parameters in principal part. The separability properties for linear problem, sharp coercive estimates for resolvent, discreetness of spectrum and completeness of root elements of the corresponding differential operator are obtained. Moreover, optimal regularity properties for nonlinear problem is established....

Let $N$ be a compact Riemannian manifold. A quasi-harmonic sphere on $N$ is a harmonic map from $({ℝ}^m,e^{{\left|x\right|}^2/2(m-2)}/d{s}_0^2)$ to $N$ ($m\ge 3$) with finite energy ([LnW]). Here $ds{2}_0$ is the Euclidean metric in ${ℝ}^m$. Such maps arise from the blow-up analysis of the heat flow at a singular point. In this paper, we prove some kinds of Liouville theorems for the quasi-harmonic spheres. It is clear that the Liouville theorems imply the existence of the heat flow to the target $N$. We also derive gradient estimates and Liouville theorems for positive...

We establish a local Lipschitz regularity result for local minimizers of asymptotically convex variational integrals.

We establish a local Lipschitz regularity result for local minimizers of asymptotically convex variational integrals.

In this note we are going to address the question of when a second order differential operator is controlled by a subelliptic second order differential operator.

We deal with maximum principles for a class of linear, degenerate elliptic differential operators of the second order. In particular the Weak and Strong Maximum Principles are shown to hold for this class of operators in bounded domains, as well as a Hopf type lemma, under suitable hypothesis on the degeneracy set of the operator. We derive, as consequences of these principles, some generalized maximum principles and an a priori estimate on the solutions of the Dirichlet problem for the linear equation....

We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group. We interpret the p-mean curvature not only as the tangential sublaplacian of a defining function, but also as the curvature of a characteristic curve, and as a quantity in terms of calibration geometry. As a differential equation, the p-minimal surface equation...

We present some monotonicity and symmetry results for positive solutions of the equation $-\text{div}\left({\left|Du\right|}^{p-2}Du\right)=f\left(u\right)$ satisfying an homogeneous Dirichlet boundary condition in a bounded domain $\mathrm{\Omega }$. We assume 1 < p < 2 and $f$ locally Lipschitz continuous and we do not require any hypothesis on the critical set of the solution. In particular we get that if $\mathrm{\Omega }$ is a ball then the solutions are radially symmetric and strictly radially decreasing.

We show that, under appropriate structure conditions, the quasilinear Dirichlet problem $$\left\{\begin{array}{ccc}-div\left(\right|\nabla u{|}^{p-2}\nabla u)=f(x,u),\hfill & x\in \Omega ,u=0,\hfill & x\in \partial \Omega ,\end{array}\right.$$ where $\Omega$is a bounded domain in ${ℝ}^n$, $1<p<+\infty$, admits two positive solutions $u_0$, $u_1$ in $W_0^{1,p}\left(\Omega \right)$ such that $0<u_0\le u_1$ in $\Omega$, while $u_0$ is a local minimizer of the associated Euler-Lagrange functional.

In this paper, we study the multiplicity of solutions for a class of noncooperative p-Laplacian operator elliptic system. Under suitable assumptions, we obtain a sequence of solutions by using the limit index theory.