Semilinear elliptic problems in ℝ^N: the interplay between the potential and the nonlinear term

2022/12/26 by Elves Alves de Barros e Silva, Silva, Elves Alves de Barros e, Sérgio H. Monari Soares +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2212.13300

openalex publication_date 2022/12/26 · openalex created_date 2023/01/06 · openalex updated_date 2026/07/28

Abstract

It is considered a semilinear elliptic partial differential equation in ℝ^N with a potential that may vanish at infinity and a nonlinear term with subcritical growth. A positive solution is proved to exist depending on the interplay between the decay of the potential at infinity and the behavior of the nonlinear term at the origin. The proof is based on a penalization argument, variational methods, and L^∞ estimates. Those estimates allow dealing with settings where the nonlinear source may have supercritical, critical, or subcritical behavior near the origin. Results that provide the existence of multiple and infinitely many solutions when the nonlinear term is odd are also established.

Related

• A new approach to the study of elliptic semilinear equations
• Nontrivial solutions for semilinear elliptic systems via Orlicz-Sobolev theory
• Ground state solutions for a semilinear elliptic problem with critical-subcritical growth
• A semilinear elliptic equation with competing powers and a radial potential
• Multiple solutions to a semilinear elliptic equation with a sharp change of sign in the nonlinearity