Abstract
AbstractWe are interested in the existence of multiple weak solutions for the Neumann elliptic problem involving the anisotropic-Laplacian operator, on a bounded domain with smooth boundary. We work on the anisotropic variable exponent Sobolev space, and by using a consequence of the local minimum theorem due to Bonanno, we establish existence of at least one weak solution under algebraic conditions on the nonlinear term. Also, we discuss existence of at least two weak solutions for the problem, under algebraic conditions including the classical Ambrosetti–Rabinowitz condition on the nonlinear term. Furthermore, by employing a three critical point theorem due to Bonanno and Marano, we guarantee the existence of at least three weak solutions for the problem in a special case.
Highlights
Our study is conducted in the framework of the anisotropic variable exponent Lebesgue–Sobolev space
We are interested in the existence of multiple weak solutions for the Neumann elliptic problem involving the anisotropic p(x)-Laplacian operator, on a bounded domain with smooth boundary
We work on the anisotropic variable exponent Sobolev space, and by using a consequence of the local minimum theorem due to Bonanno, we establish existence of at least one weak solution under algebraic conditions on the nonlinear term
Summary
Martin Bohner*, Giuseppe Caristi, Fariba Gharehgazlouei, and Shapour Heidarkhani Existence and Multiplicity of Weak Solutions for a Neumann Elliptic Problem with p(x)-Laplacian https://doi.org/10.1515/msds-2020-0108 Received May 6, 2020; accepted June 28, 2020
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