Liouville Type Theorem Research Articles

On problems with weighted elliptic operator and general growth nonlinearities

This article establishes existence, non-existence and Liouville-type theorems for nonlinear equations of the form \begin{document}$ \begin{equation*} -div (|x|^{a} D u ) = f(x,u), \; u > 0,\, \mbox{ in } \Omega, \end{equation*} $\end{document} where \begin{document}$ N \geq 3 $\end{document} , \begin{document}$ \Omega $\end{document} is an open domain in \begin{document}$ \mathbb{R}^N $\end{document} containing the origin, \begin{document}$ N-2+a > 0 $\end{document} and \begin{document}$ f $\end{document} satisfies structural conditions, including certain growth properties. The first main result is a non-existence theorem for boundary-value problems in bounded domains star-shaped with respect to the origin, provided \begin{document}$ f $\end{document} exhibits supercritical growth. A consequence of this is the existence of positive entire solutions to the equation for \begin{document}$ f $\end{document} exhibiting the same growth. A Liouville-type theorem is then established, which asserts no positive solution of the equation in \begin{document}$ \Omega = \mathbb{R}^N $\end{document} exists provided the growth of \begin{document}$ f $\end{document} is subcritical. The results are then extended to systems of the form \begin{document}$ \begin{equation*} -div (|x|^{a} D u_1) \! = \! f_{1}(x,u_1,u_2), -div (|x|^{a} D u_2) \! = \! f_{2}(x,u_1,u_2), u_1, u_2 \!>\! 0,\, \mbox{ in } \Omega, \end{equation*} $\end{document} but after overcoming additional obstacles not present in the single equation. Specific cases of our results recover classical ones for a renowned problem connected with finding best constants in Hardy-Sobolev and Caffarelli-Kohn-Nirenberg inequalities as well as existence results for well-known elliptic systems.

Classification of Nonnegative Solutions to Static Schrödinger--Hartree--Maxwell Type Equations

In this paper, we are mainly concerned with the physically interesting static Schr\"{o}dinger-Hartree-Maxwell type equations \begin{equation*} (-\Delta)^{s}u(x)=\left(\frac{1}{|x|^{\sigma}}\ast |u|^{p}\right)u^{q}(x) \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\, \mathbb{R}^{n} \end{equation*} involving higher-order or higher-order fractional Laplacians, where $n\geq1$, $0<s:=m+\frac{\alpha}{2}<\frac{n}{2}$, $m\geq0$ is an integer, $0<\alpha\leq2$, $0<\sigma<n$, $0<p\leq\frac{2n-\sigma}{n-2s}$ and $0<q\leq\frac{n+2s-\sigma}{n-2s}$. We first prove the super poly-harmonic properties of nonnegative classical solutions to the above PDEs, then show the equivalence between the PDEs and the following integral equations \begin{equation*} u(x)=\int_{\mathbb{R}^n}\frac{R_{2s,n}}{|x-y|^{n-2s}}\left(\int_{\mathbb{R}^{n}}\frac{1}{|y-z|^{\sigma}}u^p(z)dz\right)u^{q}(y)dy. \end{equation*} Finally, we classify all nonnegative solutions to the integral equations via the method of moving spheres in integral form. As a consequence, we obtain the classification results of nonnegative classical solutions for the PDEs. Our results completely improved the classification results in \cite{CD,DFQ,DL,DQ,Liu}. In critical and super-critical order cases (i.e., $\frac{n}{2}\leq s:=m+\frac{\alpha}{2}<+\infty$), we also derive Liouville type theorem.

The Liouville type theorem for the stationary magnetohydrodynamic equations in weighted mixed-norm Lebesgue spaces

The Liouville type theorem for the stationary magnetohydrodynamic equations in weighted mixed-norm Lebesgue spaces

Liouville type theorems and regularity of solutions to degenerate or singular problems part I: even solutions

We consider a class of equations in divergence form with a singular/degenerate weight Under suitable regularity assumptions for the matrix A and f (resp. F) we prove Hölder continuity of solutions which are even in and possibly of their derivatives up to order two or more (Schauder estimates). In addition, we show stability of the and a priori bounds for approximating problems in the form as Finally, we derive and bounds for inhomogenous Neumann boundary problems as well. Our method is based upon blow-up and appropriate Liouville type theorems.

Default functions and Liouville type theorems based on symmetric diffusions

Default functions appear when one discusses conditions which ensure that a local martingale is a true martingale. We show vanishing of default functions of Dirichlet processes enables us to obtain Liouville type theorems for subharmonic functions and holomorphic maps.

Notes on Liouville type theorems for the stationary compressible Navier–Stokes equations

In this paper, we investigate the three dimensional stationary compressible Navier–Stokes equations, and obtain Liouville type theorems if a smooth solution (ρ,u) satisfies some suitable conditions. In particular, our results improve and generalize the corresponding result of Li and Yu (2014).

Liouville theorems for nonlinear elliptic equations on Riemannian manifolds

Motivated by the joint work of the first author with Khanh and Ngo, and a recent work regarding to a Liouville property of p-Lichnerowicz equation by Zhao, in this paper, we study the following elliptic equation on smooth metric measure spaces (M,g,e−fdv) with a weighted Poincaré inequalityΔp,fv+h(v)=0. We obtain a Liouville type theorem for this equation by using the approach in [11] by the first author and Seo. As a consequence, we are able to point out a Liouville result which is a refinement and improvement of those by Zhao [31].

On stable and finite Morse index solutions of the fractional Toda system

We develop a monotonicity formula for solutions of the fractional Toda system(−Δ)sfα=e−(fα+1−fα)−e−(fα−fα−1)inRn, when 0<s<1, α=1,⋯,Q, f0=−∞, fQ+1=∞ and Q≥2 is the number of equations in this system. We then apply this formula, technical integral estimates, classification of stable homogeneous solutions, and blow-down analysis arguments to establish Liouville type theorems for finite Morse index (and stable) solutions of the above system when n>2s andΓ(n2)Γ(1+s)Γ(n−2s2)Q(Q−1)2>Γ2(n+2s4)Γ2(n−2s4). Here, Γ is the Gamma function. When Q=2, the above equation is the classical (fractional) Gelfand-Liouville equation.

Liouville type theorems for fractional and higher-order fractional systems

In this paper, we first establish decay estimates for the fractional and higher-order fractional H\'enon-Lane-Emden systems by using a nonlocal average and integral estimates, which deduce a result of non-existence. Next, we apply the method of scaling spheres introduced in \cite{DQ2} to derive a Liouville type theorem. We also construct an interesting example on super $\frac{\alpha}{2}$-harmonic functions (Proposition 1.2).

Gradient estimates for a weighted nonlinear parabolic equation and applications

Abstract This paper derives elliptic gradient estimates for positive solutions to a nonlinear parabolic equation defined on a complete weighted Riemannian manifold. Applications of these estimates yield Liouville-type theorem, parabolic Harnack inequalities and bounds on weighted heat kernel on the lower boundedness assumption for Bakry-Émery curvature tensor.

Classification results for a sub‐elliptic system involving the Δλ‐Laplacian

In this paper, we study a system of the form where and Δλ is a sub‐elliptic operator defined by Under some general hypotheses of the functions λi, , we first prove that the system has no positive super‐solution when p ≤ 1. In the case p &gt; 1, we establish a Liouville type theorem for the class of stable positive solutions. This result is an extension of some result in Hajlaoui et al. (Pacific J Math. 2014;270(1):79–93) for the case of Laplace operator.

Some Liouville theorems for Hénon type equations in half-space with nonlinear boundary value conditions and finite Morse indices

In this paper, we are concerned with Liouville-type theorems for Henon type equations in half-space with nonlinear boundary value conditions. We prove Liouville-type theorems for solutions belonging to one of the following classes: stable solutions and finite Morse index solutions (whether positive or sign-changing). Our proof is based on a combination of the integral estimates, the Pohozaev-type identity and the monotonicity formula of solutions and blowing down sequence.

Liouville-type theorems for sub-elliptic systems involving Δλ-Laplacian

We study the Liouville-type theorems for the sub-elliptic inequality and for the corresponding system where , and is a strongly degenerate operator of the form Here the functions satisfy certain conditions such that is homogeneous of degree 2 with respect to a group of dilations in . We first prove that the scalar inequality has no positive solution provided where Q is the homogeneous dimension of associated to the operator . Then, we establish the nonexistence of positive solutions of the corresponding system in one of following cases: or , p, q>0 and , p, q>0, pq>1 and . Our proofs are based on a maximum principle argument combined with a refinement of Souto's reduction.

Liouville-type Theorem for Fractional Kirchhoff Equations with Weights

In this paper, we prove a Liouville type theorem for stable solutions to fractional Kirchhoff equations with polynomial nonlinearities and weights.

Liouville type theorems for degenerate parabolic systems with advection terms

We study a parabolic equation of the form $$\begin{aligned} u_t - \Delta _\lambda u+a\cdot \nabla _\lambda u=u^p \hbox { in}\,\,{\mathbb {R}}^{{N}}\times {\mathbb {R}}, \end{aligned}$$ and a parabolic system $$\left\{ {\begin{array}{*{20}l} {u_{t} - \Delta _{\lambda } u + a \cdot \nabla _{\lambda } u = v^{p} } \hfill \\ {u_{t} - \Delta _{\lambda } v + a \cdot \nabla _{\lambda } v = u^{q} } \hfill \\ \end{array} } \right.{\text{in}}\;\mathbb{R}^{N} \times \mathbb{R},$$ where p is a real number, a(x) is a smooth vector field, $$\Delta _\lambda$$ is a strongly degenerate operator given by $$\begin{aligned} \Delta _\lambda =\sum _{i=1}^N\partial _{x_i}\left( \lambda _i^2\partial _{x_i}\right) \end{aligned}$$ and $$\nabla _\lambda$$ is the gradient operator associated to $$\Delta _\lambda$$ . Under some general condition of $$\lambda _i$$ introduced in Kogoj and Lanconelli [Nonlinear Anal 75: 4637–4649, 2012], we establish a Liouville type theorem for positive supersolutions to the problems above. In particular, we compute explicitly the critical exponent depending on both the homogeneous dimension of $${\mathbb {R^N}}$$ associated to $$\Delta _\lambda$$ and the behavior of the advection term a at infinity. This critical exponent is sharp in the case of Laplace operator without advection term according to [15].

Liouville theorems for fractional and higher-order Hénon–Hardy systems on ℝn

ABSTRACT In this paper, we are concerned with the Hénon–Hardy type systems on : where , , or . We prove Liouville theorems (i.e. non-existence of nontrivial nonnegative solutions) for the above Hénon–Hardy systems. The arguments used in our proof is the method of scaling spheres developed in [Dai W, Qin GLiouville type theorems for fractional and higher-order Hénon–Hardy type equations via the method of scaling spheres. preprint, submitted for publication, arXiv: 1810.02752.]. Our results generalize the Liouville theorems for single Hénon–Hardy equation on in Bidaut-Véron and Pohozaev [Nonexistence results and estimates for some nonlinear elliptic problems. J Anal Math. 2001;84:1.49], Chen et al. [Liouville type theorems, a priori estimates and existence of solutions for critical order Hardy–Hénon equations in . preprint, submitted, arXiv: 1808.06609], Dai et al. [Liouville type theorems, a priori estimates and existence of solutions for non-critical higher-order Lane–Emden–Hardy equations. preprint, submitted for publication, arXiv: 1808–10771], Dai and Qin [Liouville type theorems for Hardy–Hénon equations with concave nonlinearities. Math Nachrichten. 2020;293(6):1084–1093. https://doi.org/10.1002/mana.201800532; Liouville type theorems for fractional and higher-order Hénon–Hardy type equations via the method of scaling spheres. preprint, submitted for publication, arXiv: 1810.02752], Guo and Liu [Liouville-type theorems for polyharmonic equations in and in Liouville-type theorems for. Proc Roy Soc Edinburgh Sect A. 2008;138(2):339–359], and Phan and Souplet [Liouville-type theorems and bounds of solutions of Hardy–Hénon equations. J Diff Equ. 2012;252:2544–2562] to systems.

Research in the Field of Geometric Analysis at Volgograd State University

This article discusses the main directions of research in geometric analysis, which were conducted and are being carried out by the scientific mathematical school of Volgograd State University. The results of the founder of the scientific school, Doctor of Physics and Mathematics, Professor Vladimir Mikhailovich Miklyukov and his students are summarized. These results concern the solution of a number of problems in the field of quasiconformal flat mappings and mappings with bounded distortion of surfaces and Riemannian manifolds, the theory of minimal surfaces and surfaces of prescribed mean curvature, surfaces of zero mean curvature in Lorentz spaces, as well as problems associated with the study of the stability of such surfaces. In addition, the results of the study of various classes of triangulations — an object that appears at the junction of research in the field of geometric analysis and computational mathematics — are noted. Besides, this review discusses papers that use the Fourier decomposition method for solutions of the Laplace — Beltrami equations and the stationary Schr¨odinger equation with respect to the eigenfunctions of the corresponding boundary value problems. In particular, the authors give the results on finding capacitive characteristics that allowed for the first time to formulate and prove the criteria for the fulfillment of various theorems of Liouville type and the solvability of boundary value problems on model and quasimodel Riemannian manifolds. The role of the equivalent function method is also indicated in the study of such problems on manifolds of a fairly general form. In addition to this, the article gives an overview of the results concerning estimates of calculating error integral functionals and convergence of piecewise polynomial solutions of nonlinear variational type equations: minimal surface equations, equilibrium equations capillary surface and equations of biharmonic functions.

Liouville type theorems and regularity of solutions to degenerate or singular problems part II: odd solutions

We consider a class of equations in divergence form with a singular/degenerate weight $$ -\mathrm{div}(|y|^a A(x,y)\nabla u)=|y|^a f(x,y)+\textrm{div}(|y|^aF(x,y))\;. $$ Under suitable regularity assumptions for the matrix $A$, the forcing term $f$ and the field $F$, we prove H\"older continuity of solutions which are odd in $y\in\mathbb{R}$, and possibly of their derivatives. In addition, we show stability of the $C^{0,\alpha}$ and $C^{1,\alpha}$ a priori bounds for approximating problems in the form $$ -\mathrm{div}((\varepsilon^2+y^2)^{a/2} A(x,y)\nabla u)=(\varepsilon^2+y^2)^{a/2} f(x,y)+\textrm{div}((\varepsilon^2+y^2)^{a/2}F(x,y)) $$ as $\varepsilon\to 0$. Our method is based upon blow-up and appropriate Liouville type theorems.

Some results on the complex Monge-Amp&amp;egrave;re equation

The complex Monge-Ampere equation has been the subject ofintensive studies in the past fifty years, since its significant applications in complex analysis, complex geometry and Kahler geometry. In this paper, we will introduce our recent work on the Liouville type theorem, $C^{1,~\\alpha}$ and $C^{2,~\\alpha~}$ estimates of the complex Monge-Ampere equation and its applications in Sasakian geometry. These works are joint with S. Dinew, P. F. Guan, C. Li, J. Y. Li and X. W. Zhang.

On isolated singularities of fractional semi-linear elliptic equations

In this paper, we study the local behavior of nonnegative solutions of fractional semi-linear equations (−Δ)σu=up with an isolated singularity, where σ∈(0,1) and nn−2σ<p<n+2σn−2σ. We first use the blow up method and a Liouville type theorem to derive an upper bound. Then we establish a monotonicity formula and a sufficient condition for removable singularity to give a classification of the isolated singularities. When σ=1, this classification result has been proved by Gidas and Spruck (1981) [23], Caffarelli et al. (1989) [7].