Fourth-order nonlocal type elliptic problems with indefinite nonlinearities

Edcarlos D. Silva (), J. C. Albuquerque () and T. R. Cavalcante ()
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Edcarlos D. Silva: Universidade Federal de Goiás
J. C. Albuquerque: Universidade Federal de Pernambuco
T. R. Cavalcante: Universidade Federal de Tocantins

Partial Differential Equations and Applications, 2021, vol. 2, issue 2, 1-22

Abstract: Abstract In this work we establish the existence of at least one weak solution and one ground state solution for the following class of fourth-order nonlocal elliptic problems $$ \left\{ \begin{array}{ll} \Delta ^{2} u - g\left( \int _{\Omega } |\nabla u|^{2} \, {\text {d}}x\right) \Delta u = \mu a(x)|u|^{q-2}u + b(x)|u|^{p-2}u&{}\quad {\text {in}}\,\, \Omega , \\ u = \Delta u = 0 &{}\quad {\text {on }} \,\, \partial \Omega , \end{array} \right. $$ Δ 2 u - g ∫ Ω | ∇ u | 2 d x Δ u = μ a ( x ) | u | q - 2 u + b ( x ) | u | p - 2 u in Ω , u = Δ u = 0 on ∂ Ω , where $$ N \ge 5$$ N ≥ 5 , $${\Omega \subset {\mathbb {R}}^{N}}$$ Ω ⊂ R N is a smooth bounded domain, $${\Delta ^{2}} = \Delta \circ \Delta $$ Δ 2 = Δ ∘ Δ is the biharmonic operator, $$\mu > 0$$ μ > 0 , $$ 1

Keywords: Fourth-order elliptic problems; Indefinite elliptic problems; Nehari method; Concave-convex nonlinearities; 35G15; 35G20; 35G25 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s42985-021-00078-8

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