Existence of solutions for an elliptic equation with indefinite weight (Q878526)

The authors deal with the existence of solutions of the following elliptic equation: \[ -\Delta_x u= V(x)u+ f(x,u),\quad u\in H^1(\mathbb{R}^N),\tag{1} \] where \(V(x)\) is a function with possibly changing sign, \(f\) is a continuous function on \(\mathbb{R}^N\times \mathbb{R}\). Under appropriate assumptions on \(f(x,u)\), the existence of \(m-n\) pairs of nontrivial solutions (\(m> n\), \(m\) and \(n\) are integers) of (1) is proved. There are two main difficulties in considering (1): a) loss of compactness; b) since \(V(x)\) may change sign, it leads to difficulty in verifying the Palais-Smale condition.