Positive Solutions for a Nonhomogeneous Kirchhoff Equation with the Asymptotical Nonlinearity in R3

Abstract

We study the following nonhomogeneous Kirchhoff equation: $-(a+b{\int }_{R^{\mathrm{3}}}\mathrm{‍}|\nabla u{|}^{\mathrm{2}}dx)\mathrm{\Delta }u+u=k(x)f(u)+h(x),x\in R^{\mathrm{3}},u\in H^{\mathrm{1}}(R^{\mathrm{3}}),u>\mathrm{0},x\in R^{\mathrm{3}}$, where $f$ is asymptotically linear with respect to $t$ at infinity. Under appropriate assumptions on $k,f$, and $h$, existence of two positive solutions is proved by using the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory.