The asymptotic behavior of constant sign and nodal solutions of (p,q)-Laplacian problems as p goes to 1

In this paper we study the asymptotic behavior of solutions to the (p,q)-equation -Delta(p)u-Delta(q)u=f(x,u )in Omega, u=0 on partial derivative Omega, as p -> 1+, where N >= 2, & colone;1<p<q<1 & lowast;& colone;N/(N-1) and f is a Carath & eacute;odory function that grows superlinearly and subcritically. Based on a Nehari manifold treatment, we are able to prove that the (1,q)-Laplace problem given by -div(del u|del u|)-Delta(q)u=f(x,u) in Omega, u=0 on partial derivative Omega, has at least two constant sign solutions and one sign-changing solution, whereby the sign-changing solution has least energy among all sign-changing solutions. Furthermore, the solutions belong to the usual Sobolev space W-0(1,q)(Omega) which is in contrast with the case of 1-Laplacian problems, where the solutions just belong to the space BV(Omega) of all functions of bounded variation. As far as we know this is the first work dealing with (1,q)-Laplace problems even in the direction of constant sign solutions.