We consider an anisotropic Dirichlet problem which is driven by the ▫$(p(z),q(z))$▫-Laplacian (that is, the sum of a ▫$p(z)$▫-Laplacian and a ▫$q(z)$▫-Laplacian). The reaction (source) term, is a ...Carathéodory function which asymptotically as ▫$x \pm\infty$▫ can be resonant with respect to the principal eigenvalue of ▫$(-\Delta_{p(z)}, W^{1,p(z)}_0(\Omega))$▫. 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.
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