Liouville Type Theorem Research Articles

On stable solutions of a weighted elliptic equation involving the fractional Laplacian

In this paper, we study the following fractional Choquard equation with weight where and is a positive weight function satisfying at infinity, for some . We establish, in this paper, a Liouville type theorem saying that if then the above equation has no nontrivial stable solution. Our result, in particular, extends the result in [Le, Phuong. Bull. Aust. Math. Soc. 102 (2020), no. 3, 471–478.] from the Laplace operator to the fractional Laplacian.

Stable solutions to double phase problems involving a nonlocal term

AbstractIn this paper, we study weak solutions, possibly unbounded and sign-changing, to the double phase problem \begin{equation*} -\text{div} (|\nabla u|^{p-2} \nabla u + w(x)|\nabla u|^{q-2} \nabla u) = \left(\frac{1}{|x|^{N-\mu}}*f|u|^r\right) f(x)|u|^{r-2}u \quad\text{in}\ \mathbb{R}^N, \end{equation*} where $q\ge p\ge2$, r > q, $0 \lt \mu \lt N$ and $w,f \in L^1_{\rm loc}(\mathbb{R}^N)$ are two non-negative functions such that $w(x) \le C_1|x|^a$ and $f(x) \ge C_2|x|^b$ for all $|x| \gt R_0$, where $R_0,C_1,C_2 \gt 0$ and $a,b\in\mathbb{R}$. Under some appropriate assumptions on p, q, r, µ, a, b and N, we prove various Liouville-type theorems for weak solutions which are stable or stable outside a compact set of $\mathbb{R}^N$. First, we establish the standard integral estimates via stability property to derive the non-existence results for stable weak solutions. Then, by means of the Pohožaev identity, we deduce the Liouville-type theorem for weak solutions which are stable outside a compact set.

Liouville type theorems for the generalized higher order Choquard-Pekar equation

We study the generalized higher order Choquard-Pekar equation(⁎)(−Δ)mu=(∫RNΨ(|x−y|)up(y)dy)uqin RN, where N,m≥1 are integers, p,q>0. Under suitable assumption on Ψ, when N>2m the Liouville theorems in the ranges(i)(p≥1 or p+q≥2) and ∫|y|>1|y|−p(N−2m)Ψ(|y|)dy=∞; or(ii)p+q≥2 and limsupr→∞r2N−(N−2m)(p+q)Ψ(r)>0 for nonnegative classical solutions of (⁎) were got in previous studies. By using ‘‘Bootstrap’’ iteration procedure, we extend the above results by establishing the Liouville theorems for nonnegative classical solutions of (⁎) with N>2m in the ranges p≥1 and p+q<NN−2m. Moreover, we also consider the higher order Choquard-Pekar equation, i.e., the problem (⁎) with Ψ(r)=r−σ and 0<σ<N. By using the method of scaling sphere we also establish the Liouville theorems for nonnegative classical solutions with N>2m in the ranges 1≤p≤2N−σN−2m, 0<q≤N+2m−σN−2m, and p+q<3N+2m−2σN−2m, which also extend the recent studies on higher order Choquard-Pekar equation. Extensions to the integral representation of (⁎) with more general assumption p>0 are also included.

Liouville-type results for some quasilinear anisotropic elliptic equations

We prove some Liouville-type theorems for stable solutions (and solutions stable outside a compact set) of quasilinear anisotropic elliptic equations. Our results cover the particular case of the pure Finsler p-Laplacian.

On Liouville-type theorems for the stationary nematic liquid crystal equations

On Liouville-type theorems for the stationary nematic liquid crystal equations

On Liouville type theorems for the self-similar solutions to the generalized Euler equations

We study the backward discretely self-similar solutions to the generalized Euler equations, where usual Biot-Savart kernel representing the velocity in terms of the vorticity is replaced by various power of the laplacian. Under milder sufficient conditions than the previous results on the Euler equations we show that Liouville type theorems hold for the time periodic solutions to the profile equations, which means that there exists no backward discretely self-similar solutions to the generalized Euler system having such profile.

Liouville-type theorem for one-dimensional porous medium systems with sources

Liouville-type theorem for one-dimensional porous medium systems with sources

Liouville type theorems for Kirchhoff sub-elliptic equations involving $\Delta_\lambda$-operators

In this paper, we study the Kirchhoff elliptic equations of the form $$ -M(\|\nabla_\lambda u\|^2)\Delta_\lambda u=w(x)f(u) \quad \mbox{in }\mathbb R^{N}, $$ where $M$ is a smooth monotone function, $w$ is a weight function and $f(u)$ is of the form $u^p, e^u$ or $-u^{-p}$. The operator $\Delta_\lambda$ is strongly degenerate and given by $$ \Delta_\lambda=\sum_{j=1}^N \frac{\partial}{\partial x_j}\bigg(\lambda_j^2(x)\frac{\partial }{\partial x_j}\bigg). $$ We shall prove some classifications of stable solutions to the equation above under general assumptions on $M$ and $\lambda_j$, $j=1,\ldots,N$.

Differential Harnack inequalities for semilinear parabolic equations on Riemannian manifolds I: Bakry-Émery curvature bounded below

In this paper, we present a unified method for deriving differential Harnack inequalities for positive solutions of the semilinear parabolic equation∂tu=ΔVu+H(u) on complete Riemannian manifolds with Bakry-Émery curvature bounded below. This method transforms the problem of deriving differential Harnack inequalities into solving a related ODE system. As its application, on the one hand, we obtain new and improved estimates for logarithmic type equation and Yamabe type equation. On the other hand, we establish some sharp estimates for them under non-negative Bakry-Émery curvature condition. As their natural corollary, we also obtain some sharp Harnack inequalities and Liouville type theorems.

A Liouville-type theorem in conformally invariant equations

A Liouville-type theorem in conformally invariant equations

Liouville type theorem for a class quasilinear p-Laplace type equation on the sphere

Liouville type theorem for a class quasilinear p-Laplace type equation on the sphere

Liouville type theorems and periodic solutions for the nonhomogeneous parabolic systems

Liouville type theorems and periodic solutions for the nonhomogeneous parabolic systems

A Liouville-type theorem for the 3D primitive equations

The 3D primitive equations are used in most geophysical fluid models to approximate the large scale oceanic and atmospheric dynamics. We prove that there do not exist smooth stationary solutions to the 3D primitive equations with compact support, independently of the presence of the Coriolis rotation term or the viscosity. This result is in strong contrast with the recently established existence of compactly supported smooth solutions to the incompressible 3D Euler equations.

Liouville-type theorems for steady MHD and Hall-MHD equations in [formula omitted

In this paper, we study the Liouville-type theorems for three-dimensional stationary incompressible MHD and Hall-MHD systems in a slab with periodic boundary condition. We show that, under the assumptions that (uθ,bθ) or (ur,br) is axisymmetric, or (rur,rbr) is bounded, any smooth bounded solution to the MHD or Hall-MHD system with local Dirichlet integral growing as an arbitrary power function must be constant. This improves the result of Theorem 1.2 in Pan (2021) [33], where the Dirichlet integral of u is assumed to be finite. Inspired by Bang et al. (2022) [3], our proof relies on establishing Saint-Venant's estimates associated with our problem, and the result in the current paper extends that for stationary Navier-Stokes equations shown by [3] to MHD and Hall-MHD equations. To achieve this, more intricate estimates are needed to handle the terms involving b properly.

Uniqueness of subsolutions to the heat equation on Finsler measure spaces

AbstractLet $(M, F, m)$ be a forward complete Finsler measure space. In this paper, we prove that any nonnegative global subsolution in $L^p(M)(p&gt;1)$ to the heat equation on $\mathbb R^+\times M$ is uniquely determined by the initial data. Moreover, we give an $L^p(0&lt;p\leq 1)$ Liouville-type theorem for nonnegative subsolutions u to the heat equation on $\mathbb R\times M$ by establishing the local $L^p$ mean value inequality for u on M with Ric $_N\geq -K(K\geq 0)$ .

Asymptotic behavior and Liouville-type theorems for axisymmetric stationary Navier-Stokes equations outside of an infinite cylinder with a periodic boundary condition

We study the asymptotic behavior of solutions to the steady Navier-Stokes equations outside of an infinite cylinder in R3. We assume that the flow is periodic in x3-direction and has no swirl. This problem is closely related with two-dimensional exterior problem. Under a condition on the generalized finite Dirichlet integral, we give a pointwise decay estimate of the vorticity at the spatial infinity. This reveals the relation between the integrability of ∇v and the decay rate of ω near the spatial infinity. Moreover, we prove a Liouville-type theorem only from the condition of the generalized finite Dirichlet integral.

A Liouville-type theorem for the stationary MHD equations

We establish a new Liouville-type theorem for solutions of the stationary MHD equations imposing asymmetric oscillation growth conditions on the tensor-valued functions for the velocity and the magnetic field.

Liouville theorems for semilinear differential inequalities on sub-Riemannian manifolds

In this paper, we generalize Liouville type theorems for some semilinear partial differential inequalities to sub-Riemannian manifolds satisfying a nonnegative generalized curvature-dimension inequality introduced by Baudoin and Garofalo in [5]. In particular, our results apply to all Sasakian manifolds with nonnegative horizontal Webster-Tanaka-Ricci curvature. The key ingredient is to construct a class of “good” cut-off functions. We also provide some upper bounds for lifespan to parabolic and hyperbolic inequalities.

Liouville type theorem and Morse indices of a semilinear elliptic equation with nonlocal nonlinearities

In this study, we study the solution of a semilinear elliptic equation with nonlocal nonlinearities. By using the mountain pass theorem, Hölder’s inequality, and Sobolev embedding theorem, we obtain the existence result, establish the Liouville type theorem, and consider Morse indices of the equation.

Stable minimal hypersurfaces in ℝ^N+1+ℓ with singular set an arbitrary closed K⊂{0}×ℝ^ℓ

With respect to a $C^\infty$ metric which is close to the standard Euclidean metric on $\mathbb{R}^{N+1+\ell}$, where $N\ge 7$ and $\ell\ge 1$ are given, we construct a class of embedded $(N+\ell)$-dimensional hypersurfaces (without boundary) which are minimal and strictly stable, and which have singular set equal to an arbitrary preassigned closed subset $K\subset \{0\}\times \mathbb{R}^{\ell}$. Thus the question is settled, with a strong affirmative, as to whether there can be " gaps" or even fractional dimensional parts in the singular set. Such questions, for both stable and unstable minimal submanifolds, remain open in all dimensions in the case of real analytic metrics and, in particular, for the standard Euclidean metric. The construction used here involves the analysis of solutions $u$ of the symmetric minimal surface equation on domains $\Omega\subset\mathbb{R}^{n}$ whose symmetric graphs (i.e., $\{(x,\xi)\in\Omega\times \mathbb{R}^{m}: |\xi| = u(x)\}$) lie on one side of a cylindrical minimal cone including, in particular, a Liouville type theorem for complete solutions (i.e., the case $\Omega=\mathbb{R}^{n}$).