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

Abstract

In this paper we study the asymptotic behavior of solutions to the (p,q)-equation [Formula presented]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 W01,q(Ω) which is in contrast with the case of 1-Laplacian problems, where the solutions just belong to the space BV(Ω) 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.