Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Semilinear elliptic problems in $\mathbb{R}^N$: the interplay between the potential and the nonlinear term

Elves Alves de Barros e Silva, Sergio H. Monari Soares

Abstract

It is considered a semilinear elliptic partial differential equation in $\mathbb{R}^N$ 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 $L^\infty$ 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.

Semilinear elliptic problems in $\mathbb{R}^N$: the interplay between the potential and the nonlinear term

Abstract

It is considered a semilinear elliptic partial differential equation in 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 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.
Paper Structure (5 sections, 15 theorems, 92 equations)

This paper contains 5 sections, 15 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Estimates
  4. Existence of a positive solution
  5. Existence of multiple solutions

Key Result

Theorem 1.1

Suppose $V$ satisfies $({V_1})$-$({V_2})$ and $f$ satisfies $({f_1})$-$({f_3})$. Then there is $\Lambda^* >0$ such that (eq1) possesses a positive solution provided $\Lambda(R) \geq \Lambda^*$.

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Proposition 2.1: mountain pass geometry
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Corollary 2.4
  • ...and 16 more