MULTI-PEAK SOLUTIONS FOR A PLANAR ROBIN NONLINEAR ELLIPTIC PROBLEM WITH LARGE EXPONENT

We consider the elliptic equation Delta u+u(p) = 0 in a bounded smooth domain Omega in R-2 subject to the Robin boundary condition partial derivative u/partial derivative v + lambda b(x)u = 0. Here v denotes the unit outward normal vector on partial derivative Omega, b(x) is a smooth positive function defined on partial derivative Omega, 0 < lambda < +infinity, and p is a large exponent. For any fixed lambda large we find topological conditions on Omega which ensure the existence of a positive solution with exactly m peaks separated by a uniform positive distance from the boundary and each from other as p -> +infinity and lambda -> +infinity. In particular, for a nonsimply connected domain such solution exists for any m >= 1.