On a class of nonlinear problems involving a p(x)-Laplace type operator



We study the boundary value problem \( - div((\left| {\nabla u} \right|^{p_1 (x) - 2} + \left| {\nabla u} \right|^{p_2 (x) - 2} )\nabla u) = f(x,u) \) in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝ N . Our attention is focused on two cases when \( f(x,u) = \pm ( - \lambda \left| u \right|^{m(x) - 2} u + \left| u \right|^{q(x) - 2} u) \), where m(x) = max{p 1(x), p 2(x)} for any x\( \bar \Omega \) or m(x) < q(x) < N · m(x)/(Nm(x)) for any x\( \bar \Omega \). In the former case we show the existence of infinitely many weak solutions for any λ > 0. In the latter we prove that if λ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a ℤ2-symmetric version for even functionals of the Mountain Pass Theorem and some adequate variational methods.


p(x)-Laplace operator generalized Lebesgue-Sobolev space critical point weak solution electrorheological fluid 


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Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CraiovaCraiovaRomania

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