Global multiplicity of positive solutions for a sublinear elliptic equation in RN

We establish global multiplicity of positive solutions (existence and nonexistence theory) for the following problem$\{\begin{array}{c}\Delta u+\lambda h\left(x\right)f\left(u\right)=0\text{in}{R}^N,\\ u>0\text{in}{R}^N,\\ u\in D^{1,2}\left(R^N\right),\end{array}$ where $N\ge 3$, $\lambda >0$ is a parameter, $0\le h\in L^{\infty }\left(R^N\right)$ and f is a sublinear nonlinearity at ∞. In order to obtain our results we use a combination of the sub- super solution method and variational techniques. For instance, we need to implement a relevant result of type $D^{1,2}\left(R^N\right)$ versus X local minimizer for some appropriate space X.