On a Kirchhoff Singular p ( x ) $p(x)$ -Biharmonic Problem with Navier Boundary Conditions
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On a Kirchhoff Singular p(x)-Biharmonic Problem with Navier Boundary Conditions Khaled Kefi1,2 · Kamel Saoudi3,4 · Mohammed Mosa Al-Shomrani5
Received: 11 October 2019 / Accepted: 27 July 2020 © Springer Nature B.V. 2020
Abstract The purpose of the present paper is to study the existence of solutions for the following nonhomogeneous singular Kirchhoff problem involving the p(x)-biharmonic operator: ⎧ ⎨ M(t) 2 u + a(x)|u|p(x)−2 u = g(x)u−γ (x) ∓ λf (x, u), in , p(x) ⎩
u = u = 0,
on ∂,
where ⊂ RN , (N ≥ 3) be a bounded domain with C 2 boundary, λ is a positive parameter, γ : −→ (0, 1) be a continuous function, p ∈ C() with 1 < p − := inf p(x) ≤ p + := x∈
p ∗ (x) N Np(x) , g ∈ L p∗ (x)+γ (x)−1 (). We assume that M(t) is sup p(x) < , as usual, p ∗ (x) = 2 N − 2p(x) x∈ a continuous function with 1 (|u|p(x) + a(x)|u|p(x) )dx, t := p(x)
B K. Kefi
[email protected] K. Saoudi [email protected] M.M. Al-Shomrani [email protected]
1
Faculty of Computer Science and Information Technology, Northern Border University, Rafha, Saudi Arabia
2
Mathematics Department, Faculty of Sciences Tunis El Manar, 1060 Tunis, Tunisia
3
College of Sciences at Dammam, University of Imam Abdulrahman Bin Faisal, 31441 Dammam, Saudi Arabia
4
Basic and Applied Scientific Research Center, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, 31441, Dammam, Saudi Arabia
5
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
K. Kefi et al.
and assumed to verify assertions (M1)-(M3) in Sect. 3, moreover f (x, u) are assumed to satisfy assumptions (f1)-(f6). In the proofs of our results we use variational techniques and monotonicity arguments combined with the theory of the generalized Lebesgue Sobolev spaces. Keywords Kirchhoff problem · Navier boundary condition · Singular problem · p(x)-Biharmonic operator · Variational methods · Existence results · Generalized Lebesgue Sobolev spaces Mathematics Subject Classification 35J20 · 35J60 · 35G30 · 35J35
1 Introduction In this paper, we investigate the following singular Kirchhoff problem involving the p(x)biharmonic operator: ⎧ ⎨ M(t) 2 u + a(x)|u|p(x)−2 u = g(x)u−γ (x) ∓ λf (x, u), in , p(x) (P∓λ ) ⎩ u = u = 0, on ∂, where be a smooth bounded domain in RN (N ≥ 3) with C 2 boundary, λ is a positive parameter, γ ∈ C() and satisfied the following conditions: 0 < γ − = inf γ (x) ≤ γ + = supγ (x) < 1. x∈
x∈
p ∈ C() with 1 < p − := inf p(x) ≤ p + := sup p(x) < x∈
x∈
p ∗ (x) p ∗ (x)+γ (x)−1
N Np(x) , as usual, p ∗ (x) = , 2 N − 2p(x)
() and almost every where positive in . g∈L In the sequel and throughout the manuscript, X will denote the Sobolev space 1,p(x) (). W 2,p(x) () ∩ W0 The operator 2p(x) u := (|u|p(x)−2 u) is called the p(x)-biharmonic operator of fourth order where p is a continuous non-constant function. This differential operator is a natural generalization of the p-biharmonic operator 2p u := (|u|p−2 u), where p > 1 is a real constant. However, the
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