Nontrivial Solutions for Asymmetric Kirchhoff Type Problems

Abstract

We consider a class of particular Kirchhoff type problems with a right-hand side nonlinearity which exhibits an asymmetric growth at $+\infty$ and $-\infty$ in ${ℝ}^N(N=2,3)$. Namely, it is 4-linear at $-\infty$ and 4-superlinear at $+\infty$. 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.