Concentrating Solutions For A Planar Elliptic Problem With Large Nonlinear Exponent And Robin Boundary Condition

Let Omega subset of R-2 be a bounded domain with smooth boundary and b(x) > 0 a smooth function defined on partial derivative Omega. We study the following Robin boundary value problem:{Delta u + u(p) = 0 in Omega, u > 0 in Omega, partial derivative u/partial derivative v + lambda b(x)u = 0 on partial derivative Omega,where v denotes the exterior unit vector normal to partial derivative Omega, 0 < lambda < + infinity and p > 1 is a large exponent. We construct solutions of this problem which exhibit concentration as p -> + infinity and simultaneously as lambda -> + infinity at points that get close to the boundary, and show that in general the set of solutions of this problem exhibits a richer structure than the problem with Dirichlet boundary condition.