31 research outputs found

### Existence Results For Semilinear Problems in the Two Dimensional Hyperbolic Space Involving Critical Growth

We consider semilinear elliptic problems on two-dimensional hyperbolic space
involving critical growth. We first establish the Palais-Smale(P-S) condition
and using (P-S) condition we obtain existence of solutions. In addition, we
also explore existence of infinitely many sign changing solutions as well.Comment: Some new results adde

### Improved higher order Poincar\'e inequalities on the hyperbolic space via Hardy-type remainder terms

The paper deals about Hardy-type inequalities associated with the following
higher order Poincar\'e inequality:
$\left( \frac{N-1}{2} \right)^{2(k -l)} := \inf_{ u \in C_{c}^{\infty}
\setminus \{0\}} \frac{\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{k} u|^2
\ dv_{\mathbb{H}^{N}}}{\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{l} u|^2
\ dv_{\mathbb{H}^{N}} }\,,$ where $0 \leq l < k$ are integers and $\mathbb{H}^{N}$ denotes the
hyperbolic space. More precisely, we improve the Poincar\'e inequality
associated with the above ratio by showing the existence of $k$ Hardy-type
remainder terms. Furthermore, when $k = 2$ and $l = 1$ the existence of further
remainder terms are provided and the sharpness of some constants is also
discussed. As an application, we derive improved Rellich type inequalities on
upper half space of the Euclidean space with non-standard remainder terms.Comment: 17 page

### Sharp Poincar\'e-Hardy and Poincar\'e-Rellich inequalities on the hyperbolic space

We study Hardy-type inequalities associated to the quadratic form of the
shifted Laplacian $-\Delta_{\mathbb H^N}-(N-1)^2/4$ on the hyperbolic space
${\mathbb H}^N$, $(N-1)^2/4$ being, as it is well-known, the bottom of the
$L^2$-spectrum of $-\Delta_{\mathbb H^N}$. We find the optimal constant in the
resulting Poincar\'e-Hardy inequality, which includes a further remainder term
which makes it sharp also locally. A related inequality under suitable
curvature assumption on more general manifolds is also shown. Similarly, we
prove Rellich-type inequalities associated with the shifted Laplacian, in which
at least one of the constant involved is again sharp.Comment: Final version. To appear in JF

### Nondegeneracy of positive solutions of semilinear elliptic problems in the hyperbolic space

In this article, we will study the nondegeneracy properties of positive finite energy solutions of the equation -Δu - λu = |u|p-1u in the hyperbolic space. We will show that the degeneracy occurs only in an N-dimensional subspace. We will prove that the positive solutions are nondegenerate in the case of geodesic balls