Small Solutions Of P(x)-Laplacian Equations And An Open Problem

We mainly investigate the existence of weak solutions to p(x)-Laplacian equations in this paper.We have managed to solve an open problem proposed by Professor X.L.Fan during the 5-th ISAAC held at Catania University in Italy and extended the result of M. MihÇŽilescu,V.RÇŽdulescu.In order to get access to our results,we have used truncation technique and critical point theory.By investigating the truncated problem,we could prove that the original problem admits a sequence of small solutions.What's more,the regularity theory on p(x)-Laplacian equations and the symmetric mountain pass lemma have played essential roles in our proof.In addition,we give some useful corollaries which are all new and interesting results.