  We study the asymptotic Dirichlet and Plateau problems on Cartan-Hadamard
manifolds satisfying the so-called Strict Convexity (abbr. SC) condition. The
main part of the paper consists in studying the SC condition on a manifold
whose sectional curvatures are bounded from above and below by certain
functions depending on the distance to a fixed point. In particular, we are
able to verify the SC condition on manifolds whose curvature lower bound can go
to -infinity and upper bound to 0 simultaneously at certain rates, or on some
manifolds whose sectional curvatures go to -infinity faster than any prescribed
rate. These improve previous results of Anderson, Borb\'ely, and Ripoll and
Telichevsky. We then solve the asymptotic Plateau problem for locally
rectifiable currents with Z_2-multiplicity in a Cartan-Hadamard manifold
satisfying the SC condition given any compact topologically embedded
(k-1)-dimensional submanifold of \partial_{\infty}M, 2\leq k\leq n-1, as the
boundary data. We also solve the asymptotic Plateau problem for locally
rectifiable currents with Z-multiplicity on any rotationally symmetric manifold
satisfying the SC condition given a smoothly embedded submanifold as the
boundary data. These generalize previous results of Anderson, Bangert, and
Lang. Moreover, we obtain new results on the asymptotic Dirichlet problem for a
large class of PDEs. In particular, we are able to prove the solvability of
this problem on manifolds with super-exponential decay (to -infinity) of the
curvature.
