Andrzej Szulkin: A concentration phenomenon in a semilinear elliptic equation

We consider the problem

-∆u + V(x)u = Q(x)|u|^{p-2}u,     x ∈ Ω, u ∈ H^1_0(Ω),

where Ω ⊂ ℝ^N is a domain containing the origin, 2 < p < 2^*:=2N/(N-2), V is bounded, nonegative and the spectrum σ(-∆+V) ⊂ (0,∞). Further, we assume that Q is bounded, positive on a small ball centered at the origin and negative outside a slightly larger ball. We show, starting from Maxwell's equations, that this problem models the propagation of light in a waveguide (an optical cable). Then we show that there exist positive solutions and that they concentrate at the origin as the size of the ball tends to 0. We also consider the same problem with Q positive on two spots of small size and show that ground state solutions concentrate at one of these spots.