Two solutions for a fourth order nonlocal problem with indefinite potentials

• Published in: Manuscripta mathematica Vol. 160; no. 1-2; pp. 199 - 215
• Main Authors: Figueiredo, Giovany M., Furtado, Marcelo F., da Silva, João Pablo P.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2019; Springer Nature B.V
• Subjects: Algebraic Geometry; Boundary conditions; Calculus of Variations and Optimal Control; Optimization; Geometry; Lie Groups; Mathematics; Mathematics and Statistics; Number Theory; Topological Groups; Variational methods; Primary 35J60; Secondary 35J20
• ISSN: 0025-2611; 1432-1785
• DOI: 10.1007/s00229-018-1057-5

We study the nonlocal equation Δ 2 u - m ∫ Ω | ∇ u | 2 d x Δ u = λ a ( x ) | u | q - 2 u + b ( x ) | u | p - 2 u , in Ω , subject to the boundary condition u = Δ u = 0 on ∂ Ω . For m continuous and positive we obtain a nonnegative solution if 1 < q < 2 < p ≤ 2 N / ( N - 4 ) and λ > 0 small. If the affine case m ( t ) = α + β t , we obtain a second solution if 4 < p < 2 N / ( N - 4 ) and N ∈ { 5 , 6 , 7 } . In the proofs we apply variational methods.