J. Nonl. Evol. Equ. Appl. 2020 (7), pp. 117-148, published on September 30, 2020:

Structural stability of p(x)-Laplace problems

K. Kansie

LAboratoire de Mathématiques et Informatique (LA.M.I), UFR. Sciences et Techniques, Université Nazi Boni, 01 BP 1091 Bobo 01, Bobo-Dioulasso, Burkina Faso

S. Ouaro

LAboratoire de Mathématiques et Informatique (LA.M.I), UFR. Sciences Exactes et Appliquées, Université Ouaga I Pr Joseph KI ZERBO, 03 BP 7021 Ouaga 03, Ouagadougou, Burkina Faso

Received on May 20, 2018
Accepted on January 30, 2020

Communicated by Khalil Ezzinbi

Abstract.  In this paper, we study the structural stability (i.e., the continuous dependence on coefficients) of solutions of the nonlinear homogeneous Neumann boundary value problems involving the p(x)-Laplace of the form
(Pbn)   { b(un) − div an(x, ∇un) = fn in Ω,
an(x, ∇un).η = 0 on ∂Ω,
where Ω is an open bounded domain of RN (N ≥ 3) with smooth boundary ∂Ω and η is the outer unit normal to ∂Ω. Here, b : R → R is a continuous, onto and non-decreasing function such that b(0) = 0; (an(x, ξ))n∈Nis a family of applications which verify the classical Leray-Lions hypotheses but with a variable summability exponent pn(x) such that 1 < p ≤ pn(.) ≤ p+ < +∞ and (fn(x, ξ))n∈N ⊂ L1(Ω).
Keywords:Generalized Lebesgue-Sobolev spaces, Leray-Lions operator, Weak solution, Renormalized solution, Thermorheological fluids, Continuous dependence, Young measures..
2010 Mathematics Subject Classification:   n/a.

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