Fourth order Schr\"odinger equation with mixed dispersion on certain Cartan-Hadamard manifolds

We study the fourth order Schr\"odinger equation with mixed dispersion on an $N$-dimensional Cartan-Hadamard manifold. At first, we focus on the case of the hyperbolic space. Using the fact that there exists a Fourier transform on this space, we prove the existence of a global solution to our equation as well as scattering for small initial data. Next, we obtain weighted Strichartz estimates for radial solutions on a large class of rotationally symmetric manifolds by adapting the method of Banica and Duyckaerts (Dyn. Partial Differ. Equ., 07). Finally, we give a blow-up result for a rotationally symmetric manifold relying on a localized virial argument.Comment: To appear in Journal of Dynamics and Differential Equation

p-Capacity and p-Hyperbolicity of Submanifolds

We use explicit solutions to a drifted Laplace equation in warped product model spaces as comparison constructions to show $p$-hyperbolicity of a large class of submanifolds for p>2. The condition for $p$-hyperbolicity is expressed in terms of upper support functions for the radial sectional curvatures of the ambient space and for the radial convexity of the submanifold. In the process of showing $p$-hyperbolicity we also obtain explicit lower bounds on the $p$-capacity of finite annular domains of the submanifolds in terms of the drifted $2$-capacity of the corresponding annuli in the respective comparison space

Solvability of Minimal Graph Equation Under Pointwise Pinching Condition for Sectional Curvatures

We study the asymptotic Dirichlet problem for the minimal graph equation on a Cartan-Hadamard manifold M whose radial sectional curvatures outside a compact set satisfy an upper bound K(P) and a pointwise pinching condition |K(P)| for some constants phi > 1 and C-K >= 1, where P and P ' a re any 2-dimensional subspaces of TxM containing the (radial) vector del(x) and r (x) = d(o,x) is the distance to a fixed point o. M. We solve the asymptotic Dirichlet problem with any continuous boundary data for dimensions n = dim M > 4/phi+ 1.Peer reviewe

Existence and non-existence of minimal graphic and p-harmonic functions

To appearWe prove that every entire solution of the minimal graph equation that is bounded from below and has at most linear growth must be constant on a complete Riemannian manifold M with only one end if M has asymptotically non-negative sectional curvature. On the other hand, we prove the existence of bounded non-constant minimal graphic and p-harmonic functions on rotationally symmetric Cartan-Hadamard manifolds under optimal assumptions on the sectional curvatures.Peer reviewe

Construction of a two bubbles blowing-up solution for the fourth order energy critical nonlinear Schr\"odinger equation

We construct a blowing-up solution for the energy critical focusing biharmonic nonlinear Schr\"odinger equation in infinite time in dimension $N\geq 13$. Our solution is radially symmetric and converges asymptotically to the sum of two bubbles. The scale of one of the bubble is of order $1$ whereas the other one is of order $|t|^{-\frac{2}{N-12}}$. Moreover, the phase between the two bubbles form a right angle