Solutions of perturbed p-Laplacian equations with critical nonlinearity

In this paper, we study a perturbed p-Laplacian equation. Under some given conditions on V(x), we prove that the equation has at least one positive solution provided that epsilon <= E; for any n* is an element of N, it has at least n* pairs solutions if epsilon <= E-n*; and suppose there exists an orthogonal involution T : R-N -> R-N such that V(x), P(x), and K(x) are T-invariant, then it has at least one pair of solutions, which change sign exactly once provided that epsilon <= E, where E and E-n* are sufficiently small positive numbers. Moreover, these solutions u(epsilon) -> 0 in W-1,W- p(R-N) as epsilon -> 0. (C) 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4773228]