The aim of this paper is to prove a qualitative property, namely the preservation of positivity, for Schrödinger-type operators acting on $L^p$ functions defined on (possibly incomplete) Riemannian manifolds. A key assumption is a control of the behaviour of the potential of the operator near the Cauchy boundary of the manifolds. As a by-product, we establish the essential self-adjointness of such operators, as well as its generalization to the case $p\neq 2$, i.e. the fact that smooth compactly supported functions are an operator core for the Schrödinger operator in $L^p$.

We consider semilinear elliptic problems on two-dimensional hyperbolic space. A model problem of our study is

where H 1(𝔹2) denotes the Sobolev space on the disc model of the hyperbolic space and f(x, t) denotes the function of critical growth in dimension 2. We first establish the Palais–Smale (PS) condition for the functional corresponding to the above equation, and using the PS condition we obtain existence of solutions. In addition, using a concentration argument, we also explore existence of infinitely many sign-changing solutions.

We show that a class of semilinear Laplace–Beltrami equations on the unit sphere in ${{\mathbb{R}}^{n}}$ has infinitely many rotationally symmetric solutions. The solutions to these equations are the solutions to a two point boundary value problem for a singular ordinary differential equation. We prove the existence of such solutions using energy and phase plane analysis. We derive a Pohozaev-type identity in order to prove that the energy to an associated initial value problem tends to infinity as the energy at the singularity tends to infinity. The nonlinearity is allowed to grow as fast as ${{\left| s \right|}^{p-1}}s$ for $\left| s \right|$ large with $1\,<\,p\,<\,\left( n+5 \right)/\left( n-3 \right)$.

In this paper we establish the Nehari manifold on edge Sobolev spaces and study some of their properties. Furthermore, we use these results and the mountain pass theorem to get non-negative solutions of a class of edge-degenerate elliptic equations on singular manifolds under different conditions.

In this paper, we discuss the isometric embedding problem in hyperbolic space with nonnegative extrinsic curvature. We prove a priori bounds for the trace of the second fundamental form $H$ and extend the result to $n$-dimensions. We also obtain an estimate for the gradient of the smaller principal curvature in 2 dimensions.

(0.1)

\begin{equation}\label{eq:0.1} \left\{ \begin{array}{ll} \displaystyle -\Delta_{\mathbb{H}^{N}}u=|v|^{p-1}v x, \\ \displaystyle -\Delta_{\mathbb{H}^{N}}v=|u|^{q-1}u, \\ \end{array} \right. \end{equation}

Let Ω be a bounded domain with smooth boundary in ℝ2, q∈[1,2) and x1, x2,. . .,xm ∈ Ω. In this paper we are concerned with the following type of problem:

\[ -\Delta u-\lambda|\nabla u|^q = \rho^{2}e^{u}, \]

In this note, we introduce an approximation of harmonic maps via a sequence of exponentially harmonic maps. We then reestablish the existence theorem of harmonic maps due to Eells and Sampson.

We study a semilinear elliptic problem on a compact Riemannian manifold with boundary, subject to an inhomogeneous Neumann boundary condition. Under various hypotheses on the nonlinear terms, depending on their behaviour in the origin and infinity, we prove multiplicity of solutions by using variational arguments.

In this paper, we first investigate the Dirichlet problem for coupled vortex equations. Secondly, we give existence results for solutions of the coupled vortex equations on a class of complete noncompact Kähler manifolds which include simply-connected strictly negative curved manifolds, Hermitian symmetric spaces of noncompact type and strictly pseudo-convex domains equipped with the Bergmann metric.

We derive a weighted ${{L}^{2}}$-estimate of the Witten spinor in a complete Riemannian spin manifold $({{M}^{n}},\,g)$ of non-negative scalar curvature which is asymptotically Schwarzschild. The interior geometry of $M$ enters this estimate only via the lowest eigenvalue of the square of the Dirac operator on a conformal compactification of $M$.