We study the quasilinear elliptic equation \begin{equation*} -Qu=e^u \ \ \text{in} \ \ Ω\subset \mathbb{R}^{N} \end{equation*} where the operator $Q$, known as Finsler-Laplacian (or anisotropic Laplacian), is defined by $$Qu:=\sum_{i=1}^{N}\frac{\partial}{\partial x_{i}}(F(\nabla u)F_{ξ_{i}}(\nabla u)),$$ where $F_{ξ_{i}}=\frac{\partial F}{\partialξ_{i}}$ and $F: \mathbb{R}^{N}\rightarrow[0,+\infty)$ is a convex function of $ C^{2}(\mathbb{R}^{N}\setminus\{0\})$, that satisfies certain assumptions. For bounded domain $Ω$ and for a stable weak solution of the above equation, we prove that the Hausdorff dimension of singular set does not exceed $N-10$. For the entire space, we apply Moser iteration arguments, established by Dancer-Farina and Crandall-Rabinowitz in the context, to prove Liouville theorems for stable solutions and for finite Morse index solutions in dimensions $N<10$ and $2<N<10$, respectively. We also provide an explicit solution that is stable outside a compact set in $N=2$. In addition, we provide similar Liouville theorems for the power-type nonlinearities.
In this paper, we prove a boundary pointwise regularity for fully nonlinear elliptic equations on cones. In addition, based on this regularity, we give simple proofs of the Liouville theorems on cones.
In this paper, we investigate the 3D inhomogeneous Navier-Stokes flows with vacuum, and obtain regularity criteria and Liouville type theorems in the Lorentz space if a smooth solution $(ρ, \mathbf{u})$ satisfies suitable conditions.
We analyze stable weak solutions to the fractional Gel'fand problem \begin{equation*} (-Δ)^su=e^u\quad\mathrm{in}\quad Ω\subset\mathbb{R}^n. \end{equation*} We prove that the dimension of the singular set is at most $n-10s.$
We consider the large-scale regularity of solutions to second-order linear elliptic equations with random coefficient fields. In contrast to previous works on regularity theory for random elliptic operators, our interest is in the regularity at the boundary: We consider problems posed on the half-space with homogeneous Dirichlet boundary conditions and derive an associated $C^{1,α}$-type large-scale regularity theory in the form of a corresponding decay estimate for the homogenization-adapted tilt-excess. This regularity theory entails an associated Liouville-type theorem. The results are base…
We study a class of second-order boundary-degenerate elliptic equations in two dimensions with minimal regularity assumptions. We prove a maximum principle and a Harnack inequality at the degenerate boundary, and assuming local boundedness, we prove continuity. On globally defined non-negative solutions we provide strong constraints on behavior at infinity, and prove a Liouville-type theorem for entire solutions on the closed half-plane. The class of PDE in question includes many from mathematical finance, Keldysh- and Tricomi-type PDE, and the 2nd order reduction of the fully non-linear 4th o…
We classify stable and finite Morse index solutions to general semilinear elliptic equations posed in Euclidean space of dimension at most 10, or in some unbounded domains.
We obtain an entire Liouville type theorem to the classical semilinear subcritical elliptic equation on Heisenberg group. A pointwise estimate near the isolated singularity was also proved. The soul of the proofs is an a priori integral estimate, which deduced from a generalized formula of that found by Jerison and Lee.
Small-energy regularity for stationary weak solutions of the Liouville equation
School of Mathematical Sciences, Peking University（北京大学数学科学学院）；School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）；School of Mathematics, Yunnan Normal University（云南师范大学数学学院）
In this paper, we establish Liouville theorems for the following system of elliptic differential inequalities $$ Δ_{\mathbb H}u^{m_1}+|η|_{\mathbb H}^{γ_1}|v|^p\leq0,$$ $$ Δ_{\mathbb H}v^{m_2}+|η|_{\mathbb H}^{γ_2}|u|^q\leq0,$$ on different unbounded open domains of Heisenberg group $\mathbb H$, including the whole space, and half space of $\mathbb H$. Here $p>m_2>0$, $q>m_1>0$.
We study nonexistence and existence of nontrivial positive solutions to the following semilinear elliptic inequality involving gradient terms \[ Δu+u^p\left|\nabla u\right|^q\leq0, \] on weighted graphs, where $(p,q)\in\mathbb{R}^2$. We give a complete classification of $(p,q)$ under which sharp volume growth assumptions are established.
In this paper, we are concerned with stable solutions to the fractional elliptic equation $$ (-Δ)^s u=e^u\mbox{ in }\mathbb R^{N}, $$ where $(-Δ)^s$ is the fractional Laplacian with $0<s<1$. We establish the nonexistence of stable solutions provided that $N<10s$. This result is optimal when $s\uparrow 1$.
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 smallest $k$ eigenvalues of the Hessian matrix. For the operator ${\cal P}^+_k$ we obtain results analogous to those which hold for the Laplace operator in space dimension $k$. Whereas, owing to the stronger degeneracy of the operator ${\cal P}^-_k$, we get totally different results.
We prove some Liouville properties for sub- and supersolutions of fully nonlinear degenerate elliptic equations in the whole space. Our assumptions allow the coefficients of the first order terms to be large at infinity, provided they have an appropriate sign, as in Ornstein- Uhlenbeck operators. We give two applications. The first is a stabilization property for large times of solutions to fully nonlinear parabolic equations. The second is the solvability of an ergodic Hamilton-Jacobi-Bellman equation that identifies a unique critical value of the operator.
A version of Liouville's theorem is proved for solutions of some degenerate elliptic equations defined in $\mathbb{R}^n\backslash K$, where $K$ is a compact set, provided the structure of this equation and the dimension $n$ are related. This result is a correction of a previous one established by Serrin, since some additional hypotheses are necessary. Theoretical and numerical examples are given. Furthermore, a comparison result and the uniqueness of solution are obtained for such equations in exterior domains.
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 Hölder continuity of solutions which are odd in $y\in\mathbb{R}$, and possibly of their derivatives. In addition, we show stability of the $C^{0,α}$ and $C^{1,α}$ 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}(…
Let $P$ be a linear, second-order, elliptic operator with real coefficients defined on a noncompact Riemannian manifold $M$ and satisfies $P1=0$ in $M$. Assume further that $P$ admits a minimal positive Green function in $M$. We prove that there exists a smooth positive function $ρ$ defined on $M$ such that $M$ is stochastically incomplete with respect to the operator $ P_ρ := ρ\, P $, that is, \[ \int_{M} k_{P_ρ}^{M}(x, y, t) \ {\rm d}y < 1 \qquad \forall (x, t) \in M \times (0, \infty), \] where $k_{P_ρ}^{M}$ denotes the minimal positive heat kernel associated with $P_ρ$. Moreover, $M$ is $L…
We consider semilinear elliptic second-order partial differential inequalities of the form Lu +|u|q-1u < and = Lv +|v|q-1v (*) in the whole space Rn, where n > and = 2, q > 0 and L is a linear elliptic second-order partial differential operator in divergence form. We assume that the coefficients of the operator L are defined, measurable and locally bounded in Rn, and that the quadratic form associated with the operator L is symmetric and non-negative definite. We obtain a Liouville comparison principle in terms of a capacity associated with the operator L for solutions of (*), which are define…
In this article we study local and global properties of positive solutions of $-Δ_mu=|u|^{p-1}u+M|\nabla u|^q$ in a domain $Ω$ of $\mathbb R^N$, with $m>1$, $p,q>0$ and $M\in\mathbb R$. Following some ideas used in \cite{BV,Vron1}, and by using a direct Bernstein method combined with Keller-Osserman's estimate, we obtain several a priori estimates as well as Liouville type theorems. Moreover, we prove a local Harnack inequality with the help of Serrin's classical results.
This note studies local integral gradient bounds for distributional solutions of a large class of partial differential inequalities with diffusion in divergence form and power-like first-order terms. The applications of these estimates are two-fold. First, we show the (sharp) global Hölder regularity of distributional semi-solutions to this class of diffusive PDEs with first-order terms having supernatural growth and right-hand side in a suitable Morrey class posed on a bounded and regular open set $Ω$. Second, we provide a new proof of entire Liouville properties for inequalities with superli…
In this paper, we prove some Liouville theorem for the following elliptic equations involving nonlocal nonlinearity and nonlocal boundary value condition $$ \left\{ \begin{array}{ll} \displaystyle -Δu(y)=\intpr \frac{ F(u(x',0))}{|(x',0)-y|^{N-α}}dx'g(u(y)), &y\in\R, \\ \\ \displaystyle \frac{\partial u}{\partial ν}(x',0)=\intr \frac{G(u(y))}{|(x',0)-y|^{N-α}}\,dy f(u(x',0)), &(x',0)\in\partial \mathbb R_+^N, \end{array} \right. $$ where $\mathbb R_+^N=\{x\in \mathbb R^N:x_N>0\}$, $f,g,F,G$ are some nonlinear functions. Under some assumptions on the nonlinear functions $f,g,F,G$, we will show …
In this article we investigate the existence and regularity of 1-d steady state fractional order diffusion equations. Two models are investigated:the Riemann-Liouville fractional diffusion equation, and the Riemann-Liouville-Caputo fractional diffusion equation. For these models we explicitly show how the regularity of the solution depends upon the right hand side (rhs) function. We also establish for which Dirichlet and Neumann boundary conditions the models are well posed.
We consider a class of equations in divergence form with a singular/degenerate weight $$-\mathrm{div}(|y|^a A(x,y)\nabla u)=|y|^a f(x,y)\; \quad\textrm{or} \ \textrm{div}(|y|^aF(x,y))\;.$$ Under suitable regularity assumptions for the matrix $A$ and $f$ (resp. $F$) we prove Hölder continuity of solutions which are even in $y\in\mathbb{R}$, and possibly of their derivatives up to order two or more (Schauder estimates). In addition, we show stability of the $C^{0,α}$ and $C^{1,α}$ a priori bounds for approximating problems in the form $$-\mathrm{div}((\varepsilon^2+y^2)^{a/2} A(x,y)\nabla u)=(\v…
In this paper, first we study carefully the positive solutions to $Δu+λ_{1}u\ln u +λ_{2}u^{b+1}=0$ defined on a complete noncompact Riemannian manifold $(M, g)$ with $Ric(g)\geq -Kg$, which can be regarded as Lichnerowicz-type equations, and obtain the gradient estimates of positive solutions to these equations which do not depend on the bounds of the solutions and the Laplacian of the distance function on $(M, g)$. Then, we extend our techniques to a class of more general semilinear elliptic equations $Δu(x)+uh(\ln u)=0$ and obtain some similar results under some suitable analysis conditions …
Liouville theorems for scaling invariant nonlinear elliptic systems (saying that the system does not possess nontrivial entire solutions) guarantee a priori estimates of solutions of related, more general systems. Assume that $p=2q+3>1$ is Sobolev subritical, $n\le3$ and $β\in{\mathbb R}$. We first prove a Liouville theorem for the system $$\left.\begin{aligned} -Δu &=|u|^{2q+2}u+β|v|^{q+2}|u|^q u, \\ -Δv &=|v|^{2q+2}v+β|u|^{q+2}|v|^q v, \end{aligned}\ \right\} \quad\hbox{in}\quad {\mathbb R}^n,$$ in the class of radial functions $(u,v)$ such that the number of nodal domains of $u,v,u-v,u+v$ i…
We consider the elliptic quasilinear equation --$Δ$ m u = u p |$\nabla$u| q in R N with q $\ge$ m and p > 0, 1 < m < N. Our main result is a Liouville-type property, namely, all the positive C 1 solutions in R N are constant. We also give their asymptotic behaviour : all the solutions in an exterior domain R N \B r0 are bounded. The solutions in B r0 \ {0} can be extended as a continuous functions in B r0. The solutions in R N \ {0} has a finite limit l $\ge$ 0 as |x| $\rightarrow$ $\infty$. Our main argument is a Bernstein estimate of the gradient of a power of the solution, combined with a p…
We prove that the Dirichlet problem for the Lane-Emden equation in a half-space has no positive solutions which grow at most like the distance to the boundary to a power given by the natural scaling exponent of the equation; in other words, we rule out {\it type~I grow-up} solutions. Such a nonexistence result was previously available only for bounded solutions, or under a restriction on the power in the nonlinearity. Instrumental in the proof are local pointwise bounds for the logarithmic gradient of the solution and its normal derivative, which we also establish.
