We consider an anisotropic Dirichlet problem which is driven by the $\left(p\right(z),q(z\left)\right)$-Laplacian (that is, the sum of a $p\left(z\right)$-Laplacian and a $q\left(z\right)$-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 $(-{\Delta }_{p\left(z\right)},W_0^{1,p\left(z\right)}\left(\Omega \right))$. 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.

中文翻译：

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共振各向异性（p，q）-方程

我们考虑各向异性Dirichlet问题，它是由 $（p（ž），q（ž））$-Laplacian（即， $p（ž）$-拉普拉斯和 $q（ž）$-Laplacian），即反应（来源）项，是Carathéodory函数，渐近表示为 $X\pm \infty$ 可以相对于主特征值共振 $（-{\Delta }_{p（ž）}，{\mathrm{w\; ^}}_0^{\mathrm{1个}，p（ž）}（\Omega ））$。首先使用截断技术和变异微积分的直接方法，我们产生了两个恒定符号的光滑解。实际上，我们表明存在最小的正解和最大的负解。然后，通过结合变分工具，适当的截断技术和临界群理论，我们证明了存在于两个极端之间的节点（符号变化）解决方案的存在。