Resonant Anisotropic (p,q)-Equations - Leszek Gasiński; Nikolaos S. Papageorgiou (Mathematics, 2020)
We consider an anisotropic Dirichlet problem which is driven by the <semantics>(p(z),q(z))</semantics>-Laplacian (that is, the sum of a <semantics>p(z)</semantics>-Laplacian and a <semantics>q(z)</semantics>-Laplacian), The reaction (source) term, is a Carathéodory function which asymptotically as <semantics>x±∞</semantics> can be resonant with respect to the principal eigenvalue of <semantics>(−Δp(z),<msubsup>W01,p(z)</msubsup>(Ω))</semantics>. First using truncation techniques and the direct method of the calculus of variations, we produce two smooth solutions of constant sign. In fact we show that there exist a smallest positive solution and a biggest negative solution. Then by combining variational tools, with suitable truncation techniques and the theory of critical groups, we show the existence of a nodal (sign changing) solution, located between the two extremal ones.