Abstract

We study the existence of weak solutions for the following class of problems {αΔ2u+βΔu=μu+γh(x,u)in Ω,B(u)=0on ∂Ω,\left\{ \begin{array}{lcl} \alpha \Delta^{2}u +\beta \Delta u = \mu u +\gamma h(x,u)&\text{in}\ \Omega,\\[1mm] B(u) = 0 &\text{on}\ \partial\Omega, \end{array} \right. where Ω⊂RN\Omega\subset\mathbb{R}^N is a bounded smooth domain, N≥1N\geq 1, α≥0\alpha\geq0, −∞<β<αλ1-\infty <\beta<\alpha\lambda_1, λ1\lambda_1 is the first eigenvalue of (−Δ,H01(Ω))(-\Delta, H^1_0(\Omega)), μ∈(0,μˉ)\mu\in(0,\bar{\mu}), μˉ<μ2\bar{\mu}<\mu_2, μ2\mu_2 is the second eigenvalue of the problem (αΔ2u+βΔu,H01(Ω)∩H2(Ω)),(\alpha\Delta^2u+\beta\Delta u,H^1_0(\Omega)\cap H^2(\Omega)), γ≠0\gamma\neq0 is a real parameter and h:Ω‾×R→Rh:\overline{\Omega}\times\mathbb{R} \to \mathbb{R} is a Carathéodory function verifying some conditions, the boundary condition B(u)=0B(u)=0 on ∂Ω\partial \Omega means that u=Δu=0u=\Delta u=0 on ∂Ω\partial\Omega when α>0\alpha>0 and u=0u=0 on ∂Ω\partial \Omega when α=0\alpha=0. In this article, we revisit the arguments of Landesman-Lazer, both in the global and local aspects.

Author information

Contact details are reproduced from the original publication and may be historical.

Giovany M. Figueiredo

Dep. de Matemática, Universidade de Brasília, Brasília, Brazil

giovany@unb.br

Segundo M. A. Salirrosas

Dep. de Matemática, Universidade de Brasília, Brasília, Brazil

semaarsa@gmail.com

Keywords

  • Biharmonic operator
  • Landesman-Lazer condition

Mathematics Subject Classification

35J60; 35J60, 35J10, 35J20

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Suggested citation

G. M. Figueiredo, S. M. A. Salirrosas, L. Soriano. “Spectral Study about Biharmonic Operator with Landesman-Lazer Condition.” Journal of Convex Analysis 33 (2026), No. 1&2, 131–144. https://doi.org/10.68381/jca33008

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