Constant sign and nodal solutions for resonant double phase problems

Abstract

We consider a double phase Dirichlet problem with a reaction which asymptotically as \(x \rightarrow \pm \infty\) can be resonant with respect to the principle eigenvalue \(\hat{\lambda}_{1}>0\) of the Dirichlet weighted \(p\)-Laplacian. Using variational tools, together with truncation and comparison techniques and critical groups, we show that the problem has at least three bounded solutions which are ordered and we provide sign information for all of them (positive, negative and nodal).