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Existence and Convergence of Least-Energy Solutions Involving the Logarithmic Schrödinger Operator

Huyuan Chen, Rui Chen, Bobo Hua

TL;DR

The paper addresses existence and convergence for nonlinear nonlocal problems involving the logarithmic Schrödinger operator and the fractional pseudo-relativistic Schrödinger operator. It develops a robust variational framework on nonlocal spaces, leveraging Pitt's inequality, Brezis–Lieb splitting, and Mountain Pass geometry to obtain first existence results for critical and subcritical logarithmic nonlinearities, as well as least-energy solutions for the Brezis–Nirenberg type problem. A key contribution is establishing the convergence of least-energy solutions of the fractional problem to a nontrivial least-energy solution of the limiting logarithmic problem, together with regularity results for sublinear nonlinearities. These results extend the scope of Brezis–Nirenberg type phenomena to a combined nonlocal and logarithmic setting, providing a rigorous bridge between fractional and logarithmic operators and offering tools for further analysis of nonlocal nonlinear equations with logarithmic terms.

Abstract

In this paper, we establish the first existence result for solutions to the critical semilinear equation involving the logarithmic Schrödinger operator with subcritical logarithmic nonlinearities. Additionally, we present the first existence result for least-energy solutions to the Brezis-Nirenberg type problem for the fractional pseudo-relativistic Schrödinger operator with subcritical and critical nonlinearities. Specifically, we demonstrate that the least-energy solutions of the fractional pseudo-relativistic Schrödinger equation converge, up to a subsequence, to a nontrivial least-energy solution of the limiting problem involving the logarithmic Schrödinger operator. Furthermore, we provide regularity result for solutions to the logarithmic Schrödinger equation with sublinear nonlinearities. Our approach relies on uniform positive bounds for elements in the Nehari manifold, least-energy solutions, the Mountain-pass structure, the Palais-Smale condition, variational methods and asymptotic expansion.

Existence and Convergence of Least-Energy Solutions Involving the Logarithmic Schrödinger Operator

TL;DR

The paper addresses existence and convergence for nonlinear nonlocal problems involving the logarithmic Schrödinger operator and the fractional pseudo-relativistic Schrödinger operator. It develops a robust variational framework on nonlocal spaces, leveraging Pitt's inequality, Brezis–Lieb splitting, and Mountain Pass geometry to obtain first existence results for critical and subcritical logarithmic nonlinearities, as well as least-energy solutions for the Brezis–Nirenberg type problem. A key contribution is establishing the convergence of least-energy solutions of the fractional problem to a nontrivial least-energy solution of the limiting logarithmic problem, together with regularity results for sublinear nonlinearities. These results extend the scope of Brezis–Nirenberg type phenomena to a combined nonlocal and logarithmic setting, providing a rigorous bridge between fractional and logarithmic operators and offering tools for further analysis of nonlocal nonlinear equations with logarithmic terms.

Abstract

In this paper, we establish the first existence result for solutions to the critical semilinear equation involving the logarithmic Schrödinger operator with subcritical logarithmic nonlinearities. Additionally, we present the first existence result for least-energy solutions to the Brezis-Nirenberg type problem for the fractional pseudo-relativistic Schrödinger operator with subcritical and critical nonlinearities. Specifically, we demonstrate that the least-energy solutions of the fractional pseudo-relativistic Schrödinger equation converge, up to a subsequence, to a nontrivial least-energy solution of the limiting problem involving the logarithmic Schrödinger operator. Furthermore, we provide regularity result for solutions to the logarithmic Schrödinger equation with sublinear nonlinearities. Our approach relies on uniform positive bounds for elements in the Nehari manifold, least-energy solutions, the Mountain-pass structure, the Palais-Smale condition, variational methods and asymptotic expansion.

Paper Structure

This paper contains 14 sections, 35 theorems, 247 equations.

Table of Contents

  1. Introduction and Main Results
  2. Preliminaries
  3. Definitions and Notations
  4. Fractional Schrödinger Sobolev Inequality and Pitt's Inequality
  5. Convergence Properties
  6. Differentiability of Energy Functional
  7. The Brezis-Lieb Type Lemma
  8. Uniform Bounds for Elements in the Nehari Maniford
  9. Uniform bounds for all Nehari least-energy solutions
  10. Existence of a least-energy solution
  11. The Mountain Pass structure and PS condition
  12. Geometry of the functional J(omega,s)
  13. The proof of main result
  14. Convergence of solutions

Key Result

Theorem 1.1

Let $N \geq 1$, and $\Omega \subset \mathbb{R}^N$ be a bounded open Lipschitz set. Then: (i) For every $k \in \left(0, \frac{4}{N}\right)$ and $\lambda \in \mathbb{R}$, the problem (case1) has a Nehari least-energy solution $u \in \mathcal{H}_0^{\ln}(\Omega) \setminus \{0\}$ and where $\mathcal{T} := \{ \sigma \in C^0([0, 1], \mathcal{H}_0^{\ln}(\Omega)) : \sigma(0) = 0, \sigma(1) \neq 0, J_{\ln}

Theorems & Definitions (36)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 26 more