Weak Solutions Of Some Kirchhoff Type Equations In Orlicz-Sobolev Spaces

Kirchhoff equation presented by G. Kirchhoff, a German physicist, in 1883, is an extension of the classical d’Alembert’s wave equation and it provides more accurate descriptions of the process of string vibrations. This equation arises in numerous models such as non-Newtonian fluid mechanic, astrophysics, image pro-cessing, plasma problems, elastic theory and so on. Sobolev spaces is unworkable for nonlinear operators, however, Orlicz-Sobolev (Musielak-Orlicz-Sobolev) spaces pro-vide suitable space framework for such nonlinear problems. So, it is very meaningful to investigate Kirchhoff type equations in Orlicz-Sobolev spaces.In this thesis, we study some Kirchhoff type equations in Orlicz-Sobolev spaces and Musielak-Orlicz-Sobolev spaces. And we give some sufficient conditions for the multiplicity and existence of weak solutions, respectively. The main tool are the theory and methods developed in Orlicz spaces, variation principle and various critical point theory in nonlinear functional analysis.The main results are presented as follows:Firstly, we investigate a Kirchhoff equation with two parameters in RN We prove that this equation has at least three weak solutions in Orlicz-Sobolev spaces by the special structure of Orlicz spaces, critical point theorems given by B. Ricceri and functional analysis. These results generalize the multiplicity of Kirchhoff equations in Sobolev spaces.Secondly, we discuss some Kirchhoff equations in bounded domains with Neu- mann conditions: Such equations are widely investigated in variable exponent spaces. However, lots of experimental results show that a more appropriate framework is necessary. Then, we introduce Musielak-Orlicz-Sobolev spaces. At first, we prove that the functional ρФ still satisfies (S)+ condition by getting rid of the restriction that Ф(x, √t) is convex w.r.t. t∈[0,∞). Then, we establish existence results of this equation:1. When nonlocal term k(t)=1 and Musielak-Orlicz function Ф may not satisfy △2 and ▽2 condition, we prove that there is λ*>0, for every λ∈(0,λ*), this equation has a nontrivial weak solution in Musielak-Orlicz-Sobolev space W1,Ф(Ω) by Ekeland variation principle.2. When nonlocal term is degenerate (k(t) has zero points) and the nonlinear term f satisfies (AR) condition, we prove that there is λ*>0, for every λ∈ (0, λ*), this equation has a nontrivial weak solution in Musielak-Orlicz-Sobolev space by applying Mountain pass theorem and variation techniques.