We consider the log-perturbed Brézis–Nirenberg problem on the hyperbolic space and study the existence and non-existence of solutions. We show that for positive values of the parameter, there exists an H^1-solution, while for negative values of the parameter, no positive solution exists within a reasonably general class. Since the logarithmic term in the perturbation changes sign, Pohozaev-type identities do not yield any non-existence results. The main contribution of this article is the derivation of an “almost” precise lower asymptotic decay estimate for positive solutions when the parameter is negative, which ultimately leads to the non-existence result.

We study positive solutions to sublinear elliptic problems on the exterior of a ball in R2. For a class of positone problems, we establish the existence of multiple positive solutions for a range of the parameter and uniqueness of positive solutions for either sufficiently large or small values of the parameter. Additionally, we obtain an existence result for a semipositone problem. Our results extend the study of similar problems on exterior domains in Rn, n > 2.

We prove the compactness of the solution operator for a class of singular semilinear elliptic problems on the exterior of a ball in Rn, n ≥ 3. Compactness of solution operators for similar problems in Rn, n ≥ 2 is also established. Further, using these compactness results and employing Schauder-Tychonoff fixed point theorem, we prove the existence of a positive solution to classes of semipositone problems with asymptotically linear reaction terms.

We study positive solutions to superlinear elliptic semipositone problems on the exterior of a ball in Rn. Recently, several authors have studied positive radial solutions to this problem assuming that the weight function is radial. We allow non-radial weights and study the existence and non-existence of positive solutions. We prove the existence of a positive solution for small values of the parameter using variational methods. A non-existence result is established for large values of parameter.