Nodal solutions for singularly perturbed equations with critical exponential growth (Q872013)

The authors consider the problem \[ -\varepsilon^2\Delta_x u+ V(z)u= f(u)\quad\text{in }\Omega,\tag{1} \] \[ u= 0\quad\text{on }\partial\Omega,\tag{2} \] where \(\varepsilon> 0\) and \(\Omega\) is a domain in \(\mathbb{R}^2\), not necessarily bounded, with empty or smooth boundary. Under appropriate conditions on \(V\) and \(f\) the existence and concentration behaviour of nodal solutions for (1)--(2) are established.