Existence and multiplicity of positive solutions for indefinite semilinear elliptic problems in \(\mathbb R^N\) (Q2877580)

The authors are concerned with a class of indefinite semilinear elliptic problems \(-\Delta u+u=\left| u\right| ^{p-2}u+f(x)\left| u\right| ^{q-2}u\) in \(\mathbb{R}^{N},\) \(0\leq u\in H^{1}(\mathbb{R}^{N}), \) where \(2<q\leq p<2^{\ast }\) (\(2^{\ast }=2N/N-2\) if \(N\geq 3,\) and \(2^{\ast }=\infty \) if \(N=1,2)\) and \(f\) is a continuous function in \(\mathbb{R}^{N}.\) The main purpose of the paper is to use the shape of the graph of the function \(f\) to prove the existence and multiplicity of positive solutions of the previous problem. To this end, they consider the problem in which \(f\) has the form \(f_{\lambda }(x)=\lambda f_{+}(x)-\lambda f_{-}(x),\) where the nonnegative function \(f_{+}\) and \(f_{-}\) satisfy certain conditions. The authors prove the existence of a positive solution for all \(\lambda \in \mathbb R\) and then they establish the existence of two positive solutions for \(\lambda \) sufficiently small, by a variational method.