Existence of infinitely many solutions for nonlinear Poisson-Poisson elliptic systems

Abstract

We study a nonlinear Poisson-Poisson system in a bounded domain Ω⊂ℝ^N, N≥ 3, whose reduced form is the nonlocal elliptic equation −Δ u+λ(−Δ)⁻¹u=|u|^p−2u. We assume that λ∈(−λ₁(Ω)²,∞), where λ₁(Ω) denotes the first Dirichlet eigenvalue of −Δ in Ω, 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 H₀¹(Ω). 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,ϕ_{u_k}) such that |u_k|_λ²→∞. A three-dimensional finite element experiment illustrates two distinct discrete solutions, one of which changes sign.

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BibTeX RIS

BibTeX

@article{ninamollisaca2026existence,
  title = {{Existence of infinitely many solutions for nonlinear Poisson-Poisson elliptic systems}},
  author = {Raul Nina-Mollisaca and Julio Huayta-Centeno and Mauricio Sepulveda-Cortes and Octavio Vera-Villagran},
  journal = {Electronic Journal of Differential Equations},
  year = {2026},
  month = oct,
  url = {http://ejde.math.txstate.edu/Volumes/2026/83/abstr.html},
  note = {CC BY},
}

RIS

TY  - JOUR
TI  - Existence of infinitely many solutions for nonlinear Poisson-Poisson elliptic systems
AU  - Raul Nina-Mollisaca
AU  - Julio Huayta-Centeno
AU  - Mauricio Sepulveda-Cortes
AU  - Octavio Vera-Villagran
PY  - 2026
DA  - 2026/10/23
JO  - Electronic Journal of Differential Equations
UR  - http://ejde.math.txstate.edu/Volumes/2026/83/abstr.html
AB  - We study a nonlinear Poisson-Poisson system in a bounded domain Ω⊂ℝ^N, N≥ 3, whose reduced form is the nonlocal elliptic equation −Δ u+λ(−Δ)^−1u=|u|^p−2u. We assume that λ∈(−λ_1(Ω)^2,∞), where λ_1(Ω) denotes the first Dirichlet eigenvalue of −Δ in Ω, 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(Ω). 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,ϕ_u_k) such that |u_k|_λ^2→∞. A three-dimensional finite element experiment illustrates two distinct discrete solutions, one of which changes sign.
SN  - 1072-6691
ER  -

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