Least energy solutions to the Dirichlet problem for the equation -Δu = f (x, u)

Let Omega be a bounded smooth domain in R-N. We prove a general existence result of least energy solutions and least energy nodal ones for the problem {-Delta u = f(x, u) in Omega u = 0 on partial derivative Omega (P) where f is a Caratheodory function. Our result includes some previous results related to special cases of f. Finally, we propose some open questions concerning the global minima of the restriction on the Nehari manifold of the energy functional associated with (P) when the nonlinearity is of the type f(x, u) = lambda vertical bar u vertical bar(s-2)u - mu vertical bar u vertical bar(r-2)u, with s, r is an element of(1, 2) and lambda, mu > 0