Nontrivial Solutions for Asymmetric Kirchhoff Type Problems

We consider a class of particular Kirchhoff type problems with a right-hand side nonlinearity which exhibits an asymmetric growth at +infinity and -infinity in R-N (N = 2,3). Namely, it is 4-linear at -infinity and 4-superlinear at +infinity. However, it need not satisfy the Ambrosetti-Rabinowitz condition on the positive semiaxis. Some existence results for nontrivial solution are established by combining Mountain Pass Theorem and a variant version of Mountain Pass Theorem with Moser-Trudinger inequality.