MULTIPLE SOLUTIONS FOR A HAMILTONIAN ELLIPTIC SYSTEM WITH SIGN-CHANGING PERTURBATION

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doi:10.1007/s10473-025-0218-3

Abstract: In this paper, we study the elliptic system
$\begin{equation*} \left\{ \begin{array}{ll} -\Delta u +V(x) u=|v|^{p-2}v-\lambda_2|v|^{s_2-2}v, \\ -\Delta v + V(x)v=|u|^{p-2}u-\lambda_1|u|^{s_1-2}u, \\ u,v \in H^1(\mathbb{R}^N) \end{array} \right.\end{equation*}$
with strongly indefinite structure and sign-changing nonlinearity. We overcome the absence of the upper semi-continuity assumption which is crucial in traditional variational methods for strongly indefinite problems. By some new tools and techniques we proved the existence of infinitely many geometrically distinct solutions if parameters $\lambda_1, \lambda_2>0$ small enough. To the best of our knowledge, our result seems to be the first result about infinitely many solutions for Hamiltonian system involving sign-changing nonlinearity.

Key words: variational method, strongly indefinite, elliptic system, multiple solutions, sign-changing nonlinearity

CLC Number:

35J20

Peng Chen, Longjiang Gu, Yan Wu. MULTIPLE SOLUTIONS FOR A HAMILTONIAN ELLIPTIC SYSTEM WITH SIGN-CHANGING PERTURBATION[J].Acta mathematica scientia,Series B, 2025, 45(2): 602-614.

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