Variational Approach for a Robin Problem Involving Non Standard Growth Conditions

Khaled Kefi and Mohammed Mosa Al-Shomrani
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Khaled Kefi: Faculty of Computer Sciences and Information Technology, Northern Border University, Arar 73222, Saudi Arabia
Mohammed Mosa Al-Shomrani: Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Mathematics, 2022, vol. 10, issue 7, 1-8

Abstract: In this manuscript, we study a Robin problem driven by the p ( x ) -Laplacian with two parameters. − div ( | ∇ w | p ( x ) − 2 ∇ w ) = λ V ( x ) | w | q ( x ) − 2 w , x ∈ Q , | ∇ w | p ( x ) − 2 ∂ w ∂ n + θ ( x ) | w | p ( x ) − 2 = β V 1 ( x ) | w | r ( x ) − 2 w . x ∈ ∂ Q . Here, Q is a regular bounded domain in R N , λ , β > 0 , p , q are continuous functions on Q ¯ , ∂ w ∂ n is the outer unit normal derivative on ∂ Q , θ ∈ L ∞ ( ∂ Q ) , such that e s s inf x ∈ ∂ Q θ ( x ) > 0 , V is an indefinite function in L s ( x ) ( Q ) and V 1 is a non-negative one in L s 1 ( x ) ( ∂ Q ) . Using variational tools, we show the existence of a non-trivial solution.

Keywords: Robin problem; Sobolev spaces with variable exponents; weak solution (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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