Existence of solution for an indefinite weight quasilinear problem with variable exponent
- 1 December 2013
- journal article
- research article
- Published by Taylor & Francis Ltd in Complex Variables and Elliptic Equations
- Vol. 58 (12), 1655-1666
- https://doi.org/10.1080/17476933.2012.702421
Abstract
We study the nonlinear boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in ℝ N with smooth boundary; λ, μ are positive real numbers; p 1, p 2, q, α are a continuous functions on ; V 1 and V 2 are weight functions in the generalized Lebesgues spaces and respectively, such that V 1 > 0 in an open set Ω0 ⊂ Ω and V 2 ≥ 0 on Ω. We prove, under appropriate conditions that for any μ > 0, there exists a λ* large enough, such that for any λ ≥ λ*, the above nonhomogeneous quasilinear problem has a non-trivial positive weak solution. Moreover, under supplementary conditions on these functions, we establish that for any μ, λ > 0, the problem has a non-trivial solution. The proof relies on some variational method.Keywords
This publication has 21 references indexed in Scilit:
- Multiplicity of solutions for -polyharmonic elliptic Kirchhoff equationsNonlinear Analysis, 2011
- 𝑝(𝑥)-Laplacian with indefinite weightProceedings of the American Mathematical Society, 2011
- A multiplicity of solutions for a nonlinear degenerate problem involving ap(x)-Laplace-type operatorComplex Variables and Elliptic Equations, 2010
- Mountain pass and Ekeland's principle for eigenvalue problem with variable exponent†Complex Variables and Elliptic Equations, 2009
- Uniqueness of solutions of the parabolic variational inequality with variable exponent of nonlinearityNonlinear Analysis, 2009
- Existence and Non-Existence Results for Quasilinear Elliptic Exterior Problems with Nonlinear Boundary ConditionsCommunications in Partial Differential Equations, 2008
- A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluidsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2006
- Existence and multiplicity of solutions for p(x)p(x)-Laplacian equations in RNRN☆Nonlinear Analysis, 2004
- Existence of solutions for p(x)-Laplacian Dirichlet problemNonlinear Analysis, 2003
- Electrorheological FluidsScience, 1992