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Note on the multiplicity of solutions for nonlinear scalar field equations with a critical inverse-square potential

Bartosz Bieganowski, Daniel Strzelecki

TL;DR

This work addresses the multiplicity of solutions to a nonlinear scalar field equation with the critical inverse-square Hardy potential at the threshold of coercivity. By formulating the problem in the enlarged energy space $X^1(\mathbb{R}^N)$ and leveraging the oddness of the nonlinearity $g$, the authors apply Ikoma's abstract critical point theorem within symmetry-restricted subspaces to generate infinitely many radial and nonradial standing-wave solutions. The main contributions include proving the existence of an unbounded sequence of solutions with $P(u_j)=0$ and establishing regularity, extending multiplicity results to a singular, non-coercive, non-translation-invariant setting. The results have implications for standing waves of nonlinear Schrödinger equations with critical inverse-square potentials and illustrate how symmetry and variational methods overcome the lack of coercivity.

Abstract

We are interested in the multiplicity of solutions to the following scalar field equation $$ -Δu - \frac{(N-2)^2}{4|x|^2} u = g(u), \quad \mbox{in } \mathbb{R}^N \setminus \{0\}. $$ We establish the existence of infinitely many radial and non-radial solutions.

Note on the multiplicity of solutions for nonlinear scalar field equations with a critical inverse-square potential

TL;DR

This work addresses the multiplicity of solutions to a nonlinear scalar field equation with the critical inverse-square Hardy potential at the threshold of coercivity. By formulating the problem in the enlarged energy space and leveraging the oddness of the nonlinearity , the authors apply Ikoma's abstract critical point theorem within symmetry-restricted subspaces to generate infinitely many radial and nonradial standing-wave solutions. The main contributions include proving the existence of an unbounded sequence of solutions with and establishing regularity, extending multiplicity results to a singular, non-coercive, non-translation-invariant setting. The results have implications for standing waves of nonlinear Schrödinger equations with critical inverse-square potentials and illustrate how symmetry and variational methods overcome the lack of coercivity.

Abstract

We are interested in the multiplicity of solutions to the following scalar field equation We establish the existence of infinitely many radial and non-radial solutions.
Paper Structure (6 sections, 2 theorems, 24 equations)

This paper contains 6 sections, 2 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Functional and variational setting
  3. The X1 space
  4. Symmetry classes.
  5. Variational setting
  6. Main result

Key Result

Theorem 2.2

Suppose that $J_{\mid X} \in {\mathcal{C}}^1(X, \mathbb{R})$ is an even functional, i.e., $J_{\mid X}(-u) = J_{\mid X}(u)$ for all $u \in X$. Assume further that: Then there exists a sequence $(u_j)_{j=1}^\infty \subset X$ such that $(J_{\mid X})'(u_j) = 0$, $P(u_j)=0$, and $J(u_j) = c_j \to \infty$ as $j \to \infty$.

Theorems & Definitions (4)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 3.1
  • proof