Supercritical Semi-Linear Elliptic Problems Using Variational Principles

The thesis investigates the use of variational methods to study elliptic partial differential equations (PDEs) with supercritical nonlinearities. By focusing on convex subsets of a Banach space, the research overcomes compactness issues typically encountered with nonlinearities that exceed the Sobolev embedding exponent. This enables the effective use of standard variational techniques, leading to existence results for solutions. The work covers two supercritical elliptic problems with different boundary conditions. First, it establishes the existence of a nontrivial solution for a Neumann problem in the unit ball, where the nonlinearity f(u) has a continuous primitive. Then, the dissertation explores a Dirichlet boundary problem in a bounded annular domain, with a supercritical nonlinearity satisfying specific regularity conditions. In both cases, the study demonstrates the existence of solutions under symmetry and monotonicity assumptions, showing how the variational approach applied to convex subsets of the Sobolev space resolves these complex problems despite supercriticality-related compactness issues.

Name thesis:degree_name

Master of Science (M.Sc.)

Level thesis:degree_level

Master's

Discipline thesis:degree_discipline

Mathematics

Grantor dc:publisher

Carleton University

Year dc:date.issued

2025

Author dc:creator

• Sadeghi Kenarsari, Banafsheh

Statement dc:rights

• Copyright © 2025 the author(s). Theses may be used for non-commercial research, educational, or related academic purposes only. Such uses include personal study, distribution to students, research and scholarship. Theses may only be shared by linking to the Carleton University Institutional Repository and no part may be copied without proper attribution to the author; no part may be used for commercial purposes directly or indirectly via a for-profit platform; no adaptation or derivative works are permitted without consent from the copyright owner.

Language dc:language.iso

en

DOI

10.22215/etd/2025-16587

OAI identifier oai:identifier

oai:carleton.scholaris.ca:20.500.14718/43954