Existence of positive solutions for the fourth-order elliptic boundary value problems
Abstract This paper is concerned with the existence of a positive solution of the nonlinear fourth-order elliptic boundary value problem { Δ 2 u = f ( x , u , Δ u ) , x ∈ Ω , u = Δ u = 0 , x ∈ ∂ Ω , $$ \left \{ \textstyle\begin{array}{l} {\Delta}^{2} u = f(x,\,u,\,\Delta u),\qquad x\in \Omega , \\ u...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2025-04-01
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| Series: | Boundary Value Problems |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s13661-025-02047-1 |
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| Summary: | Abstract This paper is concerned with the existence of a positive solution of the nonlinear fourth-order elliptic boundary value problem { Δ 2 u = f ( x , u , Δ u ) , x ∈ Ω , u = Δ u = 0 , x ∈ ∂ Ω , $$ \left \{ \textstyle\begin{array}{l} {\Delta}^{2} u = f(x,\,u,\,\Delta u),\qquad x\in \Omega , \\ u=\Delta u=0, \qquad x\in \partial \Omega , \end{array}\displaystyle \right . $$ where Ω is a bounded smooth domain in R N $\mathbb{R}^{N}$ , f : Ω ‾ × R + × R − → R + $f: \overline{\Omega}\times \mathbb{R}^{+}\times \mathbb{R}^{-}\to \mathbb{R}^{+}$ is a continuous function. Under two inequality conditions of f ( x , ξ , η ) $f(x,\,\xi ,\,\eta )$ when | ( ξ , η ) | $|(\xi ,\,\eta )|$ is small and large, an existence result of positive solutions is obtained. The inequality conditions is related to the principal eigenvalue λ 1 $\lambda _{1}$ of the Laplace operator −Δ with the boundary condition u | ∂ Ω = 0 $u|_{\partial \Omega}=0$ . The discussion is based on the fixed-point index theory in cones. |
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| ISSN: | 1687-2770 |