Multiple existence of sign changing solutions for semilinear elliptic problems on \(\mathbb{R}^N\) (Q5953920)

The goal of this paper is to show the multiple existence of positive solutions and sign changing solutions of the problem NEWLINE\[NEWLINE-\Delta u+u= Q(x)|u|^{p-1}u, \quad u\in H^2(\mathbb{R}^N),NEWLINE\]NEWLINE where \(Q\) is a bounded continuous function, \(N\geq 3\), and \(p\) is a positive number such that \(1<p< {N+ 2\over N-2}\). To this end the author employs a variational approach, that is study critical points of NEWLINE\[NEWLINEI_Q(u)= {1\over 2}\int_{\mathbb{R}^N} \bigl(|\nabla u |^2+|u|^2 \bigr)dx- {1\over p+1} \int_{\mathbb{R}^N} Q(x)|u|^{p+1}dx.NEWLINE\]