Multiple Positive Solutions for Semilinear Elliptic Equations with Sign-Changing Weight Functions in ℝ N

Abstract

Existence and multiplicity of positive solutions for the following semilinear elliptic equation: $-\Delta u+u=a\left(x\right){\left|u\right|}^{p-2}u+\lambda b\left(x\right){\left|u\right|}^{q-2}u$ in ${ℝ}^N$, $u\in H^1\left({ℝ}^N\right)$, are established, where $\lambda >0,1<q<2<p<2^{*}(2^{*}=2N/(N-2)$ if $N\ge 3,2^{*}=\infty$ if $N=1,2)$, $a$, $b$ satisfy suitable conditions, and $b$ maybe changes sign in ${ℝ}^N$. The study is based on the extraction of the Palais-Smale sequences in the Nehari manifold.