Multiplicity of solutions for a quasilinear elliptic equation with (p, q)-Laplacian and critical exponent on RN

The multiplicity of solutions fora (p,q)-Laplacian equation involving critical exponent -Delta(p)u - Delta(q)u = lambda V(x)vertical bar u vertical bar(k-2)u + K(x)vertical bar u vertical bar(p*-2)u, X is an element of R-N is considered. By variational methods and the concentration-compactness principle, we prove that the problem possesses infinitely many weak solutions with negative energy for lambda is an element of (0, lambda*). Moreover, the existence of infinitely many solutions with positive energy is also given for all lambda > 0 under suitable conditions.