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Existence of solutions for a nonhomogeneous Dirichlet problem involving ๐‘(๐‘ฅ)-Laplacian operator and indefinite weight

Authors
Type
Published Article
Journal
Boundary Value Problems
Publisher
Springer (Biomed Central Ltd.)
Publication Date
Oct 26, 2019
Volume
2019
Issue
1
Identifiers
DOI: 10.1186/s13661-019-1276-z
Source
Springer Nature
Keywords
License
Green

Abstract

We obtain multiplicity and uniqueness results in the weak sense for the following nonhomogeneous quasilinear equation involving the ๐‘(๐‘ฅ)-Laplacian operator with Dirichlet boundary condition: โˆ’ฮ”๐‘(๐‘ฅ)๐‘ข+๐‘‰(๐‘ฅ)|๐‘ข|๐‘ž(๐‘ฅ)โˆ’2๐‘ข=๐‘“(๐‘ฅ,๐‘ข)in ๐›บ,๐‘ข=0 on โˆ‚๐›บ, where ฮฉ is a smooth bounded domain in โ„๐‘, V is a given function with an indefinite sign in a suitable variable exponent Lebesgue space, ๐‘“(๐‘ฅ,๐‘ก) is a Carathรฉodory function satisfying some growth conditions. Depending on the assumptions, the solutions set may consist of a bounded infinite sequence of solutions or a unique one. Our technique is based on a symmetric version of the mountain pass theorem.

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