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 −We will also prove the existence of sign changing solution to the Hardy-Sobolev-Mazya equation and the critical Grushin problem.

Elliptic problems on the ball endowed with Funk-type metrics

“…Thus, we consider the space of radially symmetric functions in H 2 1 (B n (1)), i.e., H r (B n (1)) = {u ∈ H 2 1 (B n (1)) : u(x) = u(|x|)}. By using a Strauss-type inequality, Bhakta and Sandeep [2] proved that the embedding H r (B n (1)) ֒→ L p (B n (1)) is compact for every p ∈ (2, 2 * ). [Note that in [2] the Poincaré ball model is used which is conformally equivalent to the Klein ball model.]…”

“…By using a Strauss-type inequality, Bhakta and Sandeep [2] proved that the embedding H r (B n (1)) ֒→ L p (B n (1)) is compact for every p ∈ (2, 2 * ). [Note that in [2] the Poincaré ball model is used which is conformally equivalent to the Klein ball model.] Moreover, for every u ∈ H r (B n (1)), the Strauss-estimate shows that u(x) → 0 as |x| → 1.…”

Infinitely many solutions for Hardy-Hénon type elliptic system in hyperbolic space

“…This result is contrasted with the result in Euclidean space due to [12]. Afterward, Bhakta and Sandeep [4] had investigated the priori estimates, existence/nonexistence of radial sign changing solutions of problem (1.7). In [5], the classification of radial solutions of problem (1.7) was done by Bonforte etc.…”

Existence results for semilinear problems in the two-dimensional hyperbolic space involving critical growth

“…So the next step is to characterize all sign changing solutions. Existence of sign changing solutions has been investigated in ( [10], [11]). Furthermore, extension to general manifolds are also discussed AMS subject classifications : 35J20, 35J60 , 35J61, 35B33, 58J05 keywords: Semilinear elliptic problem, Hyperbolic space, Critical growth, Moser-Trudinger inequality. in [9].…”