Related papers: Some properties of solutions to weakly hypoellipti…

It is establish regularity results for weak solutions of quasilinear elliptic problems driven by the well known $\Phi$-Laplacian operator given by \begin{equation*} \left\{\ \begin{array}{cl} \displaystyle-\Delta_\Phi u= g(x,u), &…

We prove weighted $L^p$-Liouville theorems for a class of second order hypoelliptic partial differential operators $\mathcal{L}$ on Lie groups $\mathbb{G}$ whose underlying manifold is $n$-dimensional space. We show that a natural weight is…

In this paper, we apply the moving plane method to the following high order degenerate elliptic equation,\begin{equation*} (-A)^p u=u^\alpha\text{ in } \mathbb R^{n+1}_+,n\geq 1, \end{equation*}where the operator…

We obtain conditions for the differentiability of weak solutions for a second-order uniformly elliptic equation in divergence form with a homogeneous co-normal boundary condition. The modulus of continuity for the coefficients is assumed to…

This paper is devoted to the study of $L_p$ Lyapunov-type inequalities for linear systems of equations with Neumann boundary conditions and for any constant $p \geq 1$. We consider ordinary and elliptic problems. The results obtained in the…

We consider the linear elliptic systems or equations in divergence form with periodically oscillating coefficients. We prove the large-scale boundary Lipschitz estimate for the weak solutions in domains satisfying the so-called…

A system of quasilinear elliptic equations on an unbounded domain is considered. The existence of a sequence of radially symmetric weak solutions is proved via variational methods.

We present necessary and sufficient conditions to have global hypoellipticity and global solvability for a class of vector fields defined on a product of compact Lie groups. In view of Greenfield's and Wallach's conjecture, about the…

This work is devoted to study the existence of infinitely many weak solutions to nonlocal equations involving a general integrodifferential operator of fractional type. These equations have a variational structure and we find a sequence of…

We obtain global analytic hypoellipticity for a class of differential operators that can be expressed as a zero-order perturbation of a sum of squares of vector fields with real-analytic coefficients on compact Lie groups. The key…

We prove H\"older continuity of weak solutions of the uniformly elliptic and parabolic equations %$\Delta u-\frac{A}{|x|^{2+\beta}}u=0,\,\,(\beta\geq 0)$, and variable second order term coefficients case $%% \begin{equation}\label{01}…

We consider a one-Laplace equation perturbed by $p$-Laplacian with $1<p<\infty$. We prove that a weak solution is continuously differentiable ($C^{1}$) if it is convex. Note that similar result fails to hold for the unperturbed one-Laplace…

We establish existence, uniqueness and optimal regularity results for very weak solutions to certain nonlinear elliptic boundary value problems. We introduce structural asymptotic assumptions of Uhlenbeck type on the nonlinearity, which are…

We prove nonexistence results of Liouville type for nonnegative viscosity solutions of some equations involving the fully nonlinear degenerate elliptic operators ${\cal P}^\pm_k$, defined respectively as the sum of the largest and the…

We show some examples for uniformly monotone operators arising in weak formulation of nonlinear elliptic and parabolic problems. Besides the classical $p$-Laplacian some other less known examples are given which might be of interest because…

In this paper, we consider the existence and uniqueness of weak solutions of a nonlinear elliptic equation with a variable exponent, a monotonic type operator and a convection term. With the topological degree theory, we prove the existence…

We give necessary and sufficient conditions for the existence of weak solutions to the model equation $$-\Delta_p u=\sigma \, u^q \quad \text{on} \, \, \, \R^n,$$ in the case $0<q<p-1$, where $\sigma\ge 0$ is an arbitrary locally integrable…

Consider an operator equation $F(u)=0$ in a real Hilbert space. Let us call this equation ill-posed if the operator $F'(u)$ is not boundedly invertible, and well-posed otherwise. If $F$ is monotone $C^2_{loc}(H)$ operator, then we construct…

We establish the uniqueness of large solutions to the non-cutoff Boltzmann equation with moderate soft potentials. Specifically, the weak solution $F=\mu+\mu^{\frac{1}{2}}f$ is unique as long as it has finite energy, in the sense that the…

Every absolutely summing linear operator is weakly compact. However, for strongly summing multilinear operators and polynomials - one of the most natural extensions of the linear case to the non linear framework - weak compactness does not…