Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 86, pp. 1-21.

Title: Existence and multiplicity of solutions to triharmonic problems

Authors: Qifan Wei (North China Electric Power Univ., Beijing,China)
         Xuemei Zhang (North China Electric Power Univ. Beijing, China)

Abstract:
The authors consider the triharmonic equation
$$
 (-\Delta)^3u+c_1\Delta^2 u+c_2\Delta u=h(x)|u|^{p-2} u+g(x,u)
$$
in $\Omega$, where $p\in(1,2)$, subject to Navier boundary conditions.
Based on the least action principle, the Ekeland's variational principle and a
variant version of mountain pass lemma, we analyze the existence and multiplicity
of nontrivial solutions to the above problem. In addition, we obtain the first
eigenvalue of triharmonic operator and consider its structure.
The conclusions are illustrated with several examples.

Submitted June 23, 2025. Published August 18, 2025.
Math Subject Classifications: 35J30, 35J40.
Key Words: Triharmonic equation; Ekeland's variational principle; mountain pass lemma;
  existence and multiplicity of solutions