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Register a new userLiouville type theorems for conformal Gaussian curvature equation
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In this note, we study Liouville type theorem for conformal Gaussian curvature equation (also called the mean field equation) $$ -Delta u=K(x)e^u, in R^2 $$ where $K(x)$ is a smooth function on $R^2$. When $K(x)=K(x_1)$ is a sign-changing smooth function in the real line $R$, we have a non-existence result for the finite total curvature solutions. When $K$ is monotone non-decreasing along every ray starting at origin, we can prove a non-existence result too. We use moving plane method and moving sphere method.
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In this paper, we prove several Liouville type results for a nonlinear equation involving infinity Laplacian with gradient of the form $$Delta^gamma_infty u + q(x)cdot abla{u} | abla{u}|^{2-gamma} + f(x, u),=,0quad text{in}; mathbb{R}^d,$$ where $gammain [0, 2]$ and $Delta^gamma_infty$ is a $(3-gamma)$-homogeneous operator associated with the infinity Laplacian. Under the assumptions $liminf_{|x|toinfty}lim_{sto0}f(x,s)/s^{3-gamma}>0$ and $q$ is a continuous function vanishing at infinity, we construct a positive bounded solution to the equation and if $f(x,s)/s^{3-gamma}$ decreasing in $s$, we also obtain the uniqueness. While, if $limsup_{|x|toinfty}sup_{[delta_1,delta_2]}f(x,s)<0$, then nonexistence result holds provided additionally some suitable conditions. To this aim, we develop new technique to overcome the degeneracy of infinity Laplacian and nonlinearity of gradient term. Our approach is based on a new regularity result, the strong maximum principle, and Hopfs lemma for infinity Laplacian involving gradient and potential. We also construct some examples to illustrate our results. We further study the related Dirichlet principal eigenvalue of the corresponding nonlinear operator $$Delta^gamma_infty u + q(x)cdot abla{u} | abla{u}|^{2-gamma} + c(x)u^{3-gamma},$$ in smooth bounded domains, which may be considered as of independent interest. Our results could be seen as the extension of Liouville type results obtained by Savin [48] and Ara{u}jo et. al. [1] and a counterpart of the uniqueness obtained by Lu and Wang [39,40] for sign-changing $f$.
In this paper, we derive a priori interior Hessian estimates for Lagrangian mean curvature equation if the Lagrangian phase is supercritical and has bounded second derivatives.
This note is devoted to investigating Liouville type properties of the two dimensional stationary incompressible Magnetohydrodynamics equations. More precisely, under smallness conditions only on the magnetic field, we show that there are no non-trivial solutions to MHD equations either the Dirichlet integral or some $L^p$ norm of the velocity-magnetic fields are finite. In particular, these results generalize the corresponding Liouville type properties for the 2D Navier-Stokes equations, such as Gilbarg-Weinberger cite{GW1978} and Koch-Nadirashvili-Seregin-Sverak cite{KNSS}, to the MHD setting.
We prove Liouville theorems for Dirac-harmonic maps from the Euclidean space $R^n$, the hyperbolic space $H^n$ and a Riemannian manifold $mathfrak{S^n}$ ($ngeq 3$) with the Schwarzschild metric to any Riemannian manifold $N$.
In this paper, we solve the Dirichlet problem with continuous boundary data for the Lagrangian mean curvature equation on a uniformly convex, bounded domain in $mathbb{R}^n$.
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