Asymptotics and Symmetries of Ground-state and Least Energy Nodal Solutions for Boundary-value Problems With Slowly Growing Superlinearities

We study the problems -Delta u = f(theta)(u) in Omega, u = 0 partial derivative Omega -Delta u + u = f(theta)(u) in Omega, partial derivative(nu)u = 0 on partial derivative Omega, where f(theta) is a slowly superlinearly growing nonlinearity, and Omega is a bounded domain. Namely, we are interested in generalizing the results obtained in [4], where the model nonlinearity f(theta)(u) = |u|(theta-2)u was considered in the case of Dirichlet boundary conditions. We derive the asymptotic behaviour of ground state and least energy nodal solutions when theta -> 2, leading to symmetry results for theta small. Our assumptions permit us to study some typical nonlinearities such as a superlinear perturbation of a small pure power or the sum of small powers and slowly exponentialy growing nonlinearities in dimension 2