Poincaré–Sobolev equations in the hyperbolic space

Abstract

We study the a priori estimates, existence/nonexistence of radial sign changing solution, and the Palais–Smale characterisation of the problem $${-\Delta_{{\mathbb B}^{N}}u - \lambda u = |u|^{p-1}u, u\in H^1({\mathbb B}^{N})}$$ in the hyperbolic space $${{\mathbb B}^{N}}$$ where $${1 < p\leq\frac{N+2}{N-2}}$$ . We will also prove the existence of sign changing solution to the Hardy–Sobolev–Mazya equation and the critical Grushin problem.