Normalized Solutions For A Class Of Schr(?)dinger Type Equations Involving The P-Laplacian

The purpose of this thesis is to study the existence of normalized solutions for a class of Schr(?)dinger type equations involving the p-Laplacian(p≥2)via the variational methods and elliptic equations theory.In general.the existence of the normalized solution for a Schrodinger equation can be transformed to the existence of the critical point of the functional corresponding to the equation on a constraint set.Our work includes the following two parts:In Chapter 3.we consider a Schrodinger type equation involving the p-Laplacian on a bounded domain of the N dimensional space.According to the relationship between p and N.we study the following two cases:the first case is that p<N and the nonlinear term is a power one.In this case.by constructing an appropriate constraint set.we prove that the functional corresponding to the equation has a local minimum point on the set via the Sobolev embedding theorem on bounded domains.thus we obtain a normalized solution for the equation:The second case is that p is greater than or equal to N and the nonlinear term satisfies a critical Trudinger-Moser exponential growth condition.In this case.by using a generalized Trudinger-Moser inequality and the eigenvalue theory for the p-Laplacian.we show that the corresponding functional also has a local minimum point on the preceding constraint set.so that a normalized solution for the equation.In Chapter 4.we consider a Schrodinger type equations involving the p-Laplacia.n on the whole N dimensional space.We will study the following three cases:the first case is that p<N and the nonlinear term is a general form which satisfies a subcritical Sobolev growth condition.In this case.by constructing a special minimization sequence on a Pohozaev type constraint set.using the principle of concentration-compactness and the estimation of the ground state energy.we obtain a existence result of the normalized solutions for the equation;The second case is that p=N and the nonlinear term satisfies a subcritical Trudinger-Moser exponential growth condition.In this case,similar to the idea of proof in the first case,we construct a Pohozaev type constraint set,and combine the Trudinger-Moser inequality on unbounded domains to prove that a special minimization sequence for the functional on the constraint set is sequential compact,so the limit function is a critical point for the functional on the constraint set,thereby obtaining the existence result of the normalized solutions for the equation;The third case is that p<N and the nonlinear term is a mixed power one which contains the critical Sobolev exponent.In this case,our main idea is to use subcritical Sobolev exponents to approximate the critical Sobolev exponent.Combining the Sobolev compact embedding theorem in radial symmetric spaces and careful estimations for the supremum of the corresponding functional,we obtain a existence result of the normalized solutions for the equation.