Multiple Positive Solutions for Semilinear Elliptic Equations in ℝ N Involving Concave-Convex Nonlinearities and Sign-Changing Weight Functions

Abstract

We study the existence and multiplicity of positive solutions for the following semilinear elliptic equation $-\Delta u+u=\lambda a\left(x\right)|u{|}^{q-2}u+b(x\left)\right|u{|}^{p-2}u$in ${ℝ}^N$, $u\in H^1\left({ℝ}^N\right)$, where $\lambda >0$ , $1<q<2<p<2^{*}(2^{*}=2N/(N-2)$ if $N\ge 3$, $2^{*}=\infty$ if $N=\mathrm{1,2}$), $a\left(x\right)$, $b\left(x\right)$ satisfy suitable conditions, and $a\left(x\right)$ may change sign in ${ℝ}^N$.