Existence Of Positive Solutions To(2,p)-Laplacian Equation

In this article,we study the existence of solutions to the following(2,p)-Laplacian equation(?),where Δpu = div(1(?)u|p-2(?)u),Ω is a smooth bounded domain in RN with N ≥ 1,f ∈C(Ω× R,R).The existence of the solutions to the(2,p)-Laplacian equation is a theoretical research topic and has a certain value for improving the qualitative theory to the(2,p)-Laplacian equation.The(2,p)-Laplacdan equation,having rich application background and practical significance,is a mathematical model of problems in many fields,such as physics,chemistry and biology science.This article is divided into two chapters,where we study the existence of positive solu-tions to the(2,p)-Laplacian equation with two cases of p∈(2,∞)and p ∈(1,2),respectively.In the first chapter,we consider it with p ∈(2,∞).Some compactness conditions of the energy functional corresponding to the(2,p)-Laplacian equation are obtained by using the Fucik spectrum theory of the p-Laplacian operator.Then the existence of positive solutions to a kind of(2,p)-Laplacian equation is discussed.In the second chapter,when p ∈(1,2),using Ekeland variational principle and Mountain pass theorem,we examine the existenceΩof positive solutions to two kinds of(2,p)-Laplacian equation.