SOLUTIONS WITH MULTIPLE ALTERNATE SIGN PEAKS ALONG A BOUNDARY GEODESIC TO A SEMILINEAR DIRICHLET PROBLEM

We study the existence of sign-changing multiple interior spike solutions for the following Dirichlet problem epsilon 2 Delta v - v + f(v) = 0 in Omega, v = 0 on partial derivative Omega, where Omega is a smooth and bounded domain of R-N, epsilon is a small positive parameter, f is a superlinear, subcritical and odd nonlinearity. In particular we prove that if Omega has a plane of symmetry and its intersection with the plane is a two- dimensional strictly convex domain, then, provided that k is even and sufficiently large, a k- peak solution exists with alternate sign peaks aligned along a closed curve near a geodesic of partial derivative Omega.