Bound states to critical quasilinear Schrödinger equations (Q421134)

The authors prove the existence of a positive bound-state solution for singularly perturbed equations \[ -\varepsilon^2\Delta u +V(x) u -\varepsilon^2[\Delta (u^2)]u= |u|^{2(2^*)-2}u + g(u) \] in \(\mathbb{R}^N\). Here, \(V\) is a Hölder continuous function bounded below by a positive constant, \(2^* = \frac{2N}{N-2}\) is the critical Sobolev exponent and \(g\) is a function of sub-critical growth. Each of the solutions found has the property that it has a unique maximum point which tents to a local minimum of \(V\) as \(\varepsilon\) tends to zero.