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Multiple nodal solutions of planar Stein-Weiss equations

Eudes M. Barboza, Eduardo De S. Böer, Olímpio H. Miyagaki, Claudia R. Santana

TL;DR

This work addresses the existence of multiple nodal solutions to planar Stein-Weiss problems with nonlinearities exhibiting subcritical or critical exponential growth in the Trudinger-Moser sense. The authors develop a gluing construction combined with a Nehari-type manifold approach to build radial ground states that change sign exactly $k$ times for every $k\in\mathbb N$, by partitioning $\mathbb R^2$ into $k+1$ symmetric regions and solving a coupled, multi-component variational problem. They establish both subcritical (Section 3) and critical (Section 4) exponential regimes, using a reduced functional on a product of radial spaces and a Brouwer degree argument to assemble the global solution from localized pieces. The results extend nodal-solution techniques to nonlocal Stein-Weiss terms with exponential nonlinearities, providing a robust variational framework for radial nodal states in planar nonlocal elliptic problems with potential applications to related convolution-type equations.

Abstract

In this paper, our goal is to investigate the existence of multiple nodal solutions to a class of planar Stein-Weiss problems involving a nonlinearity $f$ with subcritical or critical growth in the sense of Trudinger-Moser. To achieve this, we combine a gluing approach with the Nehari manifold argument. We demonstrate that for any positive integer $k\in \mathbb{N}$, the problem studied has at least one radially symmetrical ground state solution that changes sign exactly $k$-times.

Multiple nodal solutions of planar Stein-Weiss equations

TL;DR

This work addresses the existence of multiple nodal solutions to planar Stein-Weiss problems with nonlinearities exhibiting subcritical or critical exponential growth in the Trudinger-Moser sense. The authors develop a gluing construction combined with a Nehari-type manifold approach to build radial ground states that change sign exactly times for every , by partitioning into symmetric regions and solving a coupled, multi-component variational problem. They establish both subcritical (Section 3) and critical (Section 4) exponential regimes, using a reduced functional on a product of radial spaces and a Brouwer degree argument to assemble the global solution from localized pieces. The results extend nodal-solution techniques to nonlocal Stein-Weiss terms with exponential nonlinearities, providing a robust variational framework for radial nodal states in planar nonlocal elliptic problems with potential applications to related convolution-type equations.

Abstract

In this paper, our goal is to investigate the existence of multiple nodal solutions to a class of planar Stein-Weiss problems involving a nonlinearity with subcritical or critical growth in the sense of Trudinger-Moser. To achieve this, we combine a gluing approach with the Nehari manifold argument. We demonstrate that for any positive integer , the problem studied has at least one radially symmetrical ground state solution that changes sign exactly -times.

Paper Structure

This paper contains 10 sections, 14 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Hypotheses and main results
  3. Outline
  4. Preliminaries
  5. Subcritical case
  6. Critical points of the functional $E$ in a Nehari type manifold
  7. Minimizing sequence of $E\mid_{\mathcal{N}_k}$
  8. Proof of main Theorem \ref{['teo1']}
  9. Critical case
  10. Proof of main Theorem \ref{['teo2']}

Key Result

Lemma 1

$($Weighted Hardy-Littlewood-Sobolev inequality$)$ Let $1 < r,s < +\infty$, $0 < \mu < N$, $\gamma + \beta \geq 0$, $0 < \gamma +\beta +\mu \leq N$, $g \in L^r(\mathbb{R}^N)$ and $h\in L^s(\mathbb{R}^N)$. Then, there exists a sharp constant $C(r,s, N,\alpha,\beta,\mu)$ such that where and In addition, for all $h \in L^s(\mathbb{R}^N)$, we have where $t$ verifies

Theorems & Definitions (27)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Lemma 3
  • proof
  • Remark 3
  • Lemma 4
  • ...and 17 more