PROBLEM ELLIPTIC ANISOTROPIC NONLINEAR IN RN WITH VARIABLE EXPONENT AND LOCALLY INTEGRABLE DATA
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Date
2025-10-30
Journal Title
Journal ISSN
Volume Title
Publisher
University of Mohamed Boudiaf M'Sila
Abstract
This work is devoted to establishing the existence of weak solutions for a certain class
of nonlinear anisotropic elliptic equations, where the involved exponents vary with po sition and the coercivity condition may degenerate. The equations under consideration
take the following general form
B(u) + H(x, u) = f, x ∈ R
N , N ≥ 2
where f is locally integrable on R
N and the operator
B(u) = −
N
X
i=1
Di(ei(x, u)bi(x, u, Du))
is properly defined between W0
1,p(.)
(Ω), (Ω or R
N )and its dual. Suppose that bi
: R
N ×R×
R
N −→ R, are a Carathéodory functions.
The functions ei
: R
N × R −→ R are Carathéodory functions and satisfying the following
condition
η
(1 + |u|)
γi(x)
≤ ei(x, u) ≤ µ,
where η, µ are strictly positeve real numbers and γi(x) ≥ 0, i = 1, ..., N are continuous
functions on R
N . And H : R
N × R −→ R be a Carathéodory functions. The differential
operetor B is not coercive if u is large.
The core strategy of the proof involves deriving local estimates for a sequence of appro priately constructed approximate problems, followed by a limiting process. The findings
presented here extend known results from the constant exponent framework and also
build upon certain results discussed in [12].
Description
Keywords
Anisotropic equations, Variable exponents, Nonlinear elliptic problem, Weak solutions, Locally integrable data