On the p ( X )-Kirchhoff-Type Equation Involving the p ( X )-Biharmonic Operator via the Genus Theory
- PDF / 185,388 Bytes
- 12 Pages / 594 x 792 pts Page_size
- 59 Downloads / 216 Views
ON THE p(x)-KIRCHHOFF-TYPE EQUATION INVOLVING THE p(x)-BIHARMONIC OPERATOR VIA THE GENUS THEORY S. Taarabti,1,2 Z. El Allali,3 and K. Ben Haddouch4
UDC 517.9
The paper deals with the existence and multiplicity of nontrivial weak solutions for the p(x)-Kirchhofftype problem 1 0 Z 1 |∆u|p(x) dxA∆2p(x) u = f (x, u) in ⌦, −M@ p(x) ⌦
u = ∆u = 0
on
@⌦.
By using the variational approach and the Krasnosel’skii genus theory, we prove the existence and multiplicity of solutions for the p(x)-Kirchhoff-type equation.
1. Introduction In the present paper, we consider the following problem: 0 Z @ −M
⌦
1
1 |∆u|p(x) dxA∆2p(x) u = f (x, u) p(x)
in
⌦, (1.1)
u = ∆u = 0 on @⌦, where ⌦ is a bounded domain in RN , N ≥ 2, with smooth boundary @⌦, � � ∆2p(x) u = ∆ |∆u|p(x)−2 ∆u
is the p(x)-biharmonic operator, and p is a continuous function on ⌦ with 1 < p(x) < N. We assume that M (t) and f (x, t) satisfy the following assumptions: (M 1 ) M : R+ ! R+ is a continuous function and satisfies the (polynomial growth) condition m1 tβ−1 M (t) m2 t↵−1
1
for all t > 0, real numbers m1 and m2 such that 0 < m1 m2 , and ↵ ≥ β > 1;
National School of Applied Sciences, University Ibnou Zohr, Agadir, Morocco; e-mail: [email protected]. Corresponding author. 3 Multidisciplinary Faculty of Nador, Mohammed I University, Oujda, Morocco; e-mail: [email protected]. 4 National School of Applied Sciences, Sidi Mohammed Ben Abdellah University, Fes, Morocco; e-mail: [email protected]. 2
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 6, pp. 842–851, June, 2020. Ukrainian DOI: 10.37863/umzh.v72i6.6019. Original article submitted April 28, 2017; revision submitted November 7, 2017. 978
0041-5995/20/7206–0978
c 2020
Springer Science+Business Media, LLC
O N THE p(x)-K IRCHHOFF -T YPE E QUATION I NVOLVING THE p(x)-B IHARMONIC O PERATOR VIA THE G ENUS T HEORY 979
(f 1 ) f : ⌦ ⇥ R ! R is a continuous function such that d1 |t|s(x)−1 f (x, t) d2 |t|q(x)−1 for all t ≥ 0 and all x 2 ⌦, where d1 and d2 are positive constants, and s, q 2 C(⌦) are such that 1 < s(x) < q(x) < p⇤ (x)
Data Loading...
