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 $-{\Delta }_{p}u-{\Delta }_{q}u=f(x,u)\text{in}\Omega ,u=0\text{on}\partial \Omega ,$as $p\to 1^{+}$, where $N\ge 2$, $1<p<q<1^{\ast }≔N/(N-1)$ 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 $-div\left(\frac{\nabla u}{|\nabla u|}\right)-{\Delta }_{q}u=f(x,u)\text{in}\Omega ,u=0\text{on}\partial \Omega ,$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 $W_0^{1,q}\left(\Omega \right)$ which is in contrast with the case of 1-Laplacian problems, where the solutions just belong to the space $BV\left(\Omega \right)$ 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.