Abstract
We investigate existence and multiplicity of weak solutions for fourth‐order problems involving the Leray–Lions type operators in variable exponent spaces and improve a result of Bonanno and Chinnì (2011). We use variational methods and apply a multiplicity theorem of Bonanno and Marano (2010).
Highlights
The objective of this work is to study the existence of solutions for the following problems involving the Leray-Lions type operators in variable exponent spaces
Where Ω is a bounded domain in RN≥2 with smooth boundary ∂Ω, λ > 0 is a parameter, f is a Caratheodory function, p satisfies infx∈Ω p(x)
We point out that the extension from the p-biharmonic problem to the p(x)-biharmonic problem is nontrivial since the p(x)-biharmonic problem possesses a more complicated structure, for example it is nonhomogeneous and it usually does not have the so-called first eigenvalue
Summary
Where Ω is a bounded domain in RN≥2 with smooth boundary ∂Ω, λ > 0 is a parameter, f is a Caratheodory function, p. Is the Leray-Lions operator, where a is a Caratheodory function satisfying some suitable supplementary conditions Investigations of this type of operators has been going on in various fields, e.g. in electrorheological fluids (see Ruzicka [1]), elasticity theory (see Zhikov [2]), stationary thermorheological viscous flows of non-Newtonian fluids (see Antontsev and Rodrigues [3]), image processing (see Chen, Levine and Rao [4]), and mathematical description of the processes filtration of barotropic gas through a porous medium Proved the existence of three solutions for a problem without small perturbations of the nonlinear term, whenever p(x) > N. proved the existence of three solutions for a problem without small perturbations of the nonlinear term, whenever p(x) > N
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