Existence And Multiplicity Of Weak Solutions For Elliptic Equations Of Fourth Order With Nonstandard Growth Conditions

In this paper, the author firstly shows the interpolation inequalities for derivativesin variable exponent Lebesgue-Sobolev spaces by applying the maximal functionoperator and Sobolev integral representation, As applications, the author proves a compact Sobolev embedding theorem and a new Landau-Komogorov type inequalityfor the second order derivative and discusses the equivalent norms in the space W₀^1,p(x)(Ω)∩W^2,p(x)(Ω). On the base of these conclusions, the author, by the critical point theory, discusses the existence and multiplicity of the weak solutions for the followingelliptic equations of fourth order with nonlinearities have the nonstandard growth conditions of Ambrosetti-Rabinowitz type respectively:andwhereâ–³_p(x)²u :=â–³(|â–³u|^p(x)-2â–³u). At the same time, the existence and multiplicity of the weak solutions for the equations as above without Ambrosetti-Rabinowitz type growth condition are discussed too.