In this paper we study the existence, multiplicity and concentration behaviour of ground states for a class of quasilinear Schrödinger equations with critical growth. By using a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem. We relate the number of positive solutions with the topology of the set where the potential attains its minimum value. The proofs are based on the Ljusternik-Schnirelmann theory and variational methods.

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We study the existence of a class of nonlinear elliptic equation with Neumann boundary condition, and obtain infinitely many nodal solutions. The study of such a problem is based on the variational methods and critical point theory. We prove the conclusion by using the symmetric mountain-pass theorem under the Cerami condition.

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By means of critical point and index theories, we obtain the existence and multiplicity of sign-changing solutions for some elliptic problems with strong resonance at infinity, under weaker conditions. 2000 Mathematics Subject Classification: 35J65; 58E05.