In this paper we study the asymptotic behavior of solutions to the (p,q)-equation −Δpu−Δqu=f(x,u)inΩ,u=0on∂Ω,as p→1+, where N≥2, 1<p<q<1∗≔N/(N−1) and f is a Carathéodory 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∇u|∇u|−Δqu=f(x,u)inΩ,u=0on∂Ω,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.
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