Related papers: A classification result for the quasi-linear Liouv…

Nonexistence results for positive supersolutions of the equation $$-Lu=u^p\quad\text{in $\mathbb R^N_+$}$$ are obtained, $-L$ being any symmetric and stable linear operator, positively homogeneous of degree $2s$, $s\in(0,1)$, whose spectral…

We introduce and study an approximate solution of the p-Laplace equation, and a linearlization $L_{\epsilon}$ of a perturbed p-Laplace operator. By deriving an $L_{\epsilon}$-type Bochner's formula and a Kato type inequality, we prove a…

We are concerned with some extensions of the classical Liouville theorem for bounded harmonic functions to solutions of more general equations. We deal with entire solutions of periodic and almost periodic parabolic equations including the…

We give a short and rigorous proof of the existence and uniqueness of the solution of Liouville equation with sources, both elliptic and parabolic, on the sphere and on all higher genus compact Riemann surfaces.

In the present paper we prove Liouville-type theorems: non-existence theorems for some complete Riemannian almost product manifolds and special mappings of complete Riemannian manifolds which generalize similar results for compact…

We will report some results concerning the Yamabe problem and the Nirenberg problem. Related topics will also be discussed. Such studies have led to new results on some conformally invariant fully nonlinear equations arising from geometry.…

We give necessary and sufficient conditions for the existence of positive radial solutions for a class of fully nonlinear uniformly elliptic equations posed in the complement of a ball in $\mathbb R^N$, and equipped with homogeneous…

Liouville theorems for scaling invariant nonlinear elliptic systems (saying that the system does not possess nontrivial entire solutions) guarantee a priori estimates of solutions of related, more general systems. Assume that $p=2q+3>1$ is…

We establish Liouville type theorems in the whole space and in a half-space for parabolic problems without scale invariance. To this end, we employ two methods, respectively based on the corresponding elliptic Liouville type theorems and…

We prove several classification results for $p$-Laplacian problems on bounded and unbounded domains, and deal with qualitative properties of sign-changing solutions to $p$-Laplacian equations on $\mathbb R^N$ involving critical…

We prove a global compactness result for Palais-Smale sequences associated with a class of quasi-linear elliptic equations on exterior domains.

We obtain a new Liouville comparison principle for entire weak solutions $(u,v)$ of semilinear parabolic second-order partial differential inequalities of the form $$ u_t -{\mathcal L}u- |u|^{q-1}u\geq v_t -{\mathcal L}v- |v|^{q-1}v (*) $$…

We obtain Liouville-type results for solutions to the CR Yamabe equation in $\mathbb{H}^n$, which extend a result obtained by Jerison and Lee for solutions in $L^{2+2/n}(\mathbb{H}^n)$. We obtain our results under either pointwise…

For $n\geq2,$ we obtain Liouville type theorems for minimal surface equations in half space $\mathbf R^n_+$ with affine Dirichlet boundary value or constant Neumann boundary value.

In this note, we obtain a classification result for positive solutions to the critical p-Laplace equation in $\mathbb{R}^n$ with $n\ge4$ and $p>p_n$ for some number $p_n\in\left(\frac{n}{3},\frac{n+1}{3}\right)$ such that…

We consider classical solutions to $-\Delta u = f(u)$ in half-spaces, under homogeneous Dirichlet boundary conditions. We prove that any positive solution is strictly monotone increasing in the direction orthogonal to the boundary, provided…

We prove a Liouville type classification theorem in half-spaces for infinite boundary value problems related to fully nonlinear, uniformly elliptic operators. We then apply the result in order to obtain gradient boundary blow up rates for…

We clarify how close a second order fully nonlinear equation can come to uniform ellipticity, through counting large eigenvalues of the linearized operator. This suggests an effective and novel way to understand the structure of fully…

We classify positive solutions to a class of quasilinear equations with Neumann or Robin boundary conditions in convex domains. Our main tool is an integral formula involving the trace of some relevant quantities for the problem. Under a…

Liouville theorems for scaling invariant nonlinear parabolic problems in the whole space and/or the halfspace (saying that the problem does not posses positive bounded solutions defined for all times $t\in(-\infty,\infty)$) guarantee…