Electron. J. Differential Equations, Vol. 2026 (2026), No. 83, pp. 1-17.
Existence of infinitely many solutions for nonlinear Poisson-Poisson elliptic systems
Raul Nina-Mollisaca, Julio Huayta-Centeno, Mauricio Sepulveda-Cortes, Octavio Vera-Villagran
Abstract:
We study a nonlinear Poisson-Poisson system in a bounded domain
\(\Omega\subset\mathbb{R}^N\), \(N\geq 3\), whose reduced form is the
nonlocal elliptic equation
\(-\Delta u+\lambda(-\Delta)^{-1}u=|u|^{p-2}u\).
We assume that \(\lambda\in(-\lambda_1(\Omega)^2,\infty)\), where
\(\lambda_1(\Omega)\) denotes the first Dirichlet eigenvalue of
\(-\Delta\) in \(\Omega\), and that the exponent is subcritical,
\(2< p< 2^*=2N/(N-2)\).
We first show that, throughout this coercive range, the quadratic part
of the associated energy functional defines a norm equivalent to the
standard norm of \(H_0^1(\Omega)\).
We then establish the Palais-Smale condition on the Nehari manifold
and use the Krasnosel'skii genus to prove the existence of infinitely
many pairs of nontrivial weak solutions.
More precisely, there exists a sequence of solutions
\((u_k,\phi_{u_k})\) such that \(\|u_k\|_\lambda^2\to\infty\).
A three-dimensional finite element experiment illustrates two distinct
discrete solutions, one of which changes sign.
Submitted August 4, 2026. Published October 22, 2026.
Math Subject Classifications: 35J47, 35J60, 35A15, 58E05, 65N30.
Key Words: Nonlinear elliptic system; inverse Laplacian; nonlocal elliptic equation;
Nehari manifold; Krasnosel'skii genus; infinitely many solutions; finite element method.
DOI: 10.58997/ejde.2026.83
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Raul Nina Mollisaca
Departamento de Matemática y Estadística
Universidad Nacional Jorge Basadre Grohmann
Tacna, Perú
email: raninamo@unjbg.edu.pe
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Julio Huayta-Centeno
Departamento de Educación de Matemática y Computación
Universidad Nacional Jorge Basadre Grohmann
Tacna, Perú
email: jhuaytac@unjbg.edu.pe
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Mauricio Sepúlveda-Cortés
Departamento de Ingeniería Matemática and CI2MA
Universidad de Concepción
Concepción, Chile
email: maursepu@udec.cl
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Octavio Vera-Villagrán
Departamento de Matemática
Universidad de Tarapacá
Av. 18 de Septiembre 2222, Arica, Chile
email: opverav@academicos.uta.cl
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