Nontrivial solution for asymmetric (p,2)-Laplacian Dirichlet problem

We consider a class of particular (p,2)-Laplacian Dirichlet problems with a right-hand side nonlinearity which exhibits an asymmetric growth at +∞ and −∞. Namely, it is linear at −∞ and superlinear at +∞. However, it need not satisfy the Ambrosetti-Rabinowitz condition on the positive semi-axis. Some existence results for a nontrivial solution are established by the mountain pass theorem and a variant version of the mountain pass theorem in the general case 2<p<N. Similar results are also established by combining the mountain pass theorem and a variant version of the mountain pass theorem with the Moser-Trudinger inequality in the case of p=N.