Concentration of solutions of a semilinear PDE with slow spatial dependence (Q1004331)

The paper deals with existence of solutions \(u\in H^2(\mathbb R^n)\) for small positive \(\varepsilon\) of the semilinear equation \[ -\varepsilon^2 \nabla\cdot \big(P(x)\nabla u\big) +F(V(x),u)=0,\quad x\in \mathbb R^n, \] that concentrate at a single point \(b\in \mathbb R^n\) as \(\varepsilon\to0,\) and where \(V(x)\) is a multidimensional potential with values in \(\mathbb R^r.\)