| With the development of elastic mechanics,the study of variational problems with nonstandard growth conditions is an important topic.p(x)-growth conditions can be regarded as an important case of nonstandard growth conditions.The study of p(x)-Laplacian equation arises from nonlinear elasticity theory,image restoration,and many results have been obtained.This paper investigates the existence of nontrivial solutions of a class of p(x)-Laplacian equations without any growth and Ambrosetti-Rabinowitz conditions,based on the theoretical knowledge of variable exponent Lebesgue space Lp(x)(Ω)and variable exponent Sobolev space W1,p(x)(Ω).The approaches we used are variational methods,the critical point methods and analytical techniques.First,employing the cutoff function approach,we show that auxiliary problem has at least one nontrivial solution through the mountain pass theorem.Furthermore,we obtain nontrivial solutions for original problems usi ng De Giorgi iteration.The results presented here extend some recent contributions obtained for problems driven by the p(x)-Laplacian. |
