The Concentration Of Solutions For (P,q)-Laplacian Equation

In this paper,the variational method is used to study the concentration of solutions of(p,q)-Laplacian equation.The first chapter describes the background,research significance and present situation of(p,q)-Laplacian equation.The second chapter studies the concentration of solutions for the following double-phase problems with a general nonlinearity:where ε>0 is sufficiently small parameter,N>q>p ≥ 2,the potential function V is a nonnegative and continuous function that has a local minimum,f:R→R is a C1 function and satisfies subcritical growth.Using the variational method,penalization technique and deformation lemma,we prove that the above problem has a positive weak solution uε such that uε has a maximum point xε satisfying(?)dist(xε,M)=0.For such a point xε,vε(x)=uε(εx+xε)converges uniformly as ε→0 to a least energy solution of the limit problem corresponding to the above equation.In addition,we prove that the solution has exponential decay.The third chapter considers the concentration of positive solutions for the following(N,q)Laplacian equation with Trudinger-Moser nonlinearity:where V is a positive continuous function and has a local minimum.ε>0 is a small parameter,2 ≤N<q<+∞,f is a C1 function and satisfies subcritical growth.By using variational method,penalization technique and deformation lemma,we obtain similar results as in the second chapter.In view of the different nonlinear term conditions and p=N,we will encounter some new difficulties,which require more refined processing techniques.