Multiple Solutions to Elliptic Equations on R N with Combined Nonlinearities

Abstract

In this paper, we are concerned with the multiplicity of nontrivial radial solutions for the following elliptic equations $(\mathrm{P}{)}_{\lambda }$: $-\mathrm{\Delta }u+V\left(x\right)u=\lambda Q\left(x\right){\left|u\right|}^{q-\mathrm{2}}u+Q\left(x\right)f\left(u\right),x\in {\mathbb{R}}^N$; $u(x)\to \mathrm{0},$ as $\left|x\right|\to +\mathrm{\infty }$, where , $V$, and $Q$ are radial positive functions, which can be vanishing or coercive at infinity, and $f$ is asymptotically linear at infinity.