Superlinear Problems For The N-Laplacian With Critical Exponential Growth

As one of the hot topics in the field of science,the elliptic equation has gradually got people's attention.The complex phenomena in the field of science has been researched carefully,so people have more and more new ideas and new methods.Since 1934,Friedrich has studied the solutions to elliptic equation,the research of elliptic equation obtained fruitful results.After several decades of development,it has become an important method to study elliptic equations by using fixed point theory,variational method and critical point theory.In recent years,many scholars have studied the elliptic equations by means of relatively mature nonlinear analysis methods,and obtained many good results.In this paper,we deal with the existence of positive solutions to the following elliptic equation by using the mountain theorem,variational method and the critical point theory.where:N?2,??R,.We shall assume that V?x?satisfies the following conditions:?V?₁V?x?is bounded from below;?V₂?V?x?^-1belongs to.And we assume that f satisfies the following conditions:?1?critical exponential growth and superlinear growth;?2?monotonicity and eigenvalue;?3?infinitesimal of higher order.Based on the above assumptions and Mountain theorem,it is proved that under suitable conditions for all ??0,the problem has at least one nontrivial solution.It is worth mentioning that we study the problem without the AR condition,namely,there is a constantm ??2 such thatIt is well known that the AR condition is very strong and not easy to satisfy such as .Therefore,the problem we study in this paper is more general and meaningful.