Local and global behavior of solutions to 2D-elliptic equation with exponentially-dominated nonlinearities

Article type: Research Article

Authors: Suzuki, Takashi^{; *}

Affiliations: Center for Mathematical Modeling and Data Science, Unit of Mathematical Science, Osaka University, Japan

Correspondence: [*] Corresponding author. E-mail: suzuki@sigmath.es.osaka-u.ac.jp.

Abstract: We study the family of blowup solutions to semilinear elliptic equations in two-space dimensions with exponentially-dominated nonnegative nonlinearities. Such a family admits an exclusion of the boundary blowup, finiteness of blowup points, and pattern formation. Then, Hamiltonian control of the location of blowup points, residual vanishing, and mass quantization arise under the estimate from below of the nonlinearity. Finally, if the principal growth rate of nonlinearity is exactly exponential and the residual part has a gap relative to this term, there is a locally uniform estimate of the solution which ensures its asymptotic non-degeneracy.

Keywords: Semilinear elliptic equation, blowup analysis, point vortex Hamiltonian, recursive hierarchy, Onsager’s theory

DOI: 10.3233/ASY-211713

Journal: Asymptotic Analysis, vol. 128, no. 4, pp. 465-494, 2022