Work

Bifurcation diagrams of semilinear elliptic equations for supercritical nonlinearities in two dimensions

Year: 2025

Type: article

Abstract: We consider the Gelfand problem with general supercritical nonlinearities in the two-dimensional unit ball. In this paper, we prove the non-existence of an unstable solution for any positive small parameter $$\lambda $$ λ . The result implies that once the bifurcation curve emanates from the starting point, then the curve never approaches $$\lambda =0$$ λ = 0 . As a result, we obtain the existence of a radial singular solution. In addition, we prove the uniformly boundedness of finite Morse index solutions. As a result, we prove that the bifurcation curve has infinitely many turning points. We remark that these properties are well-known in N dimensions with $$3\le N \le 9$$ 3 ≤ N ≤ 9 and less known in two dimensions. Our results clarify that the bifurcation structure is solely determined by the supercriticality of the nonlinearities if $$2\le N\le 9$$ 2 ≤ N ≤ 9 .

Source: Nonlinear Differential Equations and Applications NoDEA

DOI: 10.1007/s00030-025-01043-9

Author: Kenta Kumagai

Institution: Institute of Science Tokyo

Language: English

Topic: Nonlinear Partial Differential Equations

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,190

Funder: Japan Society for the Promotion of Science