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Register a new userHarnack inequality and principal eigentheory for general infinity Laplacian operators with gradient in $mathbb{R}^N$ and applications
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Under the lack of variational structure and nondegeneracy, we investigate three notions of textit{generalized principal eigenvalue} for a general infinity Laplacian operator with gradient and homogeneous term. A Harnack inequality and boundary Harnack inequality are proved to support our analysis. This is a continuation of our first work [3] and a substantial contribution in the development of the theory of textit{generalized principal eigenvalue} beside the works [8, 13, 12, 9, 29]. We use these notions to characterize the validity of maximum principle and study the existence, nonexistence and uniqueness of positive solutions of Fisher-KPP type equations in the whole space. The sliding method is intrinsically improved for infinity Laplacian to solve the problem. The results are related to the Liouville type results, which will be meticulously explained.
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We study the generalized eigenvalue problem in $mathbb{R}^N$ for a general convex nonlinear elliptic operator which is locally elliptic and positively $1$-homogeneous. Generalizing article of Berestycki and Rossi in [Comm. Pure Appl. Math. 68 (2015), no. 6, 1014-1065] we consider three different notions of generalized eigenvalues and compare them. We also discuss the maximum principles and uniqueness of principal eigenfunctions.
We prove, with a purely analytic technique, a one-side Liouville theorem for a class of Ornstein--Uhlenbeck operators ${mathcal L_0}$ in $mathbb{R}^N$, as a consequence of a Liouville theorem at $t=- infty$ for the corresponding Kolmogorov operators ${mathcal L_0} - partial_t$ in $mathbb{R}^{N+1}$. In turn, this last result is proved as a corollary of a global Harnack inequality for non-negative solutions to $({mathcal L_0} - partial_t) u = 0$ which seems to have an independent interest in its own right. We stress that our Liouville theorem for ${mathcal L_0}$ cannot be obtained by a probabilistic approach based on recurrence if $N>2$. We provide a self-contained proof of a Liouville theorem involving recurrent Ornstein--Uhlenbeck stochastic processes in the Appendix.
We establish the Trudinger-Moser inequality on weighted Sobolev spaces in the whole space, and for a class of quasilinear elliptic operators in radial form of the type $displaystyle Lu:=-r^{-theta}(r^{alpha}vert u(r)vert^{beta}u(r))$, where $theta, betageq 0$ and $alpha>0$, are constants satisfying some existence conditions. It worth emphasizing that these operators generalize the $p$- Laplacian and $k$-Hessian operators in the radial case. Our results involve fractional dimensions, a new weighted Polya-Szeg{o} principle, and a boundness value for the optimal constant in a Gagliardo-Nirenberg type inequality.
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 $f$.
We consider second-order linear parabolic operators in non-divergence form that are intrinsically defined on Riemannian manifolds. In the elliptic case, Cabre proved a global Krylov-Safonov Harnack inequality under the assumption that the sectional curvature of the underlying manifold is nonnegative. Later, Kim improved Cabres result by replacing the curvature condition by a certain condition on the distance function. Assuming essentially the same condition introduced by Kim, we establish Krylov-Safonov Harnack inequality for nonnegative solutions of the non-divergent parabolic equation. This, in particular, gives a new proof for Li-Yau Harnack inequality for positive solutions to the heat equation in a manifold with nonnegative Ricci curvature.
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