Non-existence and uniqueness results for supercritical semilinear elliptic equations

TL;DR

This paper establishes non-existence and uniqueness results for supercritical semilinear elliptic equations in star-shaped domains, extending previous work to non-local and additive cases using Rellich-Pohozaev estimates.

Contribution

It extends earlier non-existence and uniqueness results to non-local and additive bifurcation problems in supercritical semilinear elliptic equations.

Findings

Uniqueness holds in certain parameter ranges.

Results apply to both local and non-local equations.

Simplifies proofs of previous results.

Abstract

Non-existence and uniqueness results are proved for several local and non-local supercritical bifurcation problems involving a semilinear elliptic equation depending on a parameter. The domain is star-shaped but no other symmetry assumption is required. Uniqueness holds when the bifurcation parameter is in a certain range. Our approach can be seen, in some cases, as an extension of non-existence results for non-trivial solutions. It is based on Rellich-Pohozaev type estimates. Semilinear elliptic equations naturally arise in many applications, for instance in astrophysics, hydrodynamics or thermodynamics. We simplify the proof of earlier results by K. Schmitt and R. Schaaf in the so-called local multiplicative case, extend them to the case of a non-local dependence on the bifurcation parameter and to the additive case, both in local and non-local settings.