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