Abstract
PurposeIn the present paper, the authors will discuss the solvability of a class of nonlinear anisotropic elliptic problems (P), with the presence of a lower-order term and a non-polynomial growth which does not satisfy any sign condition which is described by an N-uplet of N-functions satisfying the Δ2-condition, within the fulfilling of anisotropic Sobolev-Orlicz space. In addition, the resulting analysis requires the development of some new aspects of the theory in this field. The source term is merely integrable.Design/methodology/approachAn approximation procedure and some priori estimates are used to solve the problem.FindingsThe authors prove the existence of entropy solutions to unilateral problem in the framework of anisotropic Sobolev-Orlicz space with bounded domain. The resulting analysis requires the development of some new aspects of the theory in this field.Originality/valueTo the best of the authors’ knowledge, this is the first paper that investigates the existence of entropy solutions to unilateral problem in the framework of anisotropic Sobolev-Orlicz space with bounded domain.
Highlights
Let Ω be a bounded domain of RN ðN ≥ 2Þ
We consider the unilateral elliptical operators whose nonlinearity is given by a vector of N-functions like
The full terms of this licence may be seen at http:// creativecommons.org/licences/by/4.0/legalcode
Summary
Let Ω be a bounded domain of RN ðN ≥ 2Þ. The aim behind this paper is the study of boundary value problems for a class of nonlinear anisotropic elliptic equations. In the Orlicz space, Benkirane and Bennouna in [11] demonstrated the existence of entropy solutions to the following nonlinear elliptic problem:. No previous research has investigated the existence of entropy solutions to unilateral problem ðPÞ with the second term as an operator with growth described by an nuplet of N-functions satisfying the Δ2Àcondition, within the fulfilling of anisotropic SobolevOrlicz space with bounded domain, the function biðx; u; ∇uÞ does not satisfy any sign condition and the source f is merely integrable. The rest of this paper is organized as follows: In Section 2, we give some definitions and fundamental properties of anisotropic Sobolev-Orlicz spaces. The N-function M(t) satisfies the Δ2Àcondition as long as there exist positive numbers c > 1 and t0 ≥ 0 such as for t ≥ t0 we have t bðtÞ ≤ c M ðtÞ:. The Following function: M ðzÞ 1⁄4 j z jb ð j lnjzj j þ 1 Þ; with b > 1 check the Δ2-condition and (18)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
