We consider a nonlinear Dirichlet problem driven by a variable exponent $p$-Laplacian plus an indefinite potential term. The reaction has the competing effects of a parametric concave (sublinear) term and a convex (superlinear) perturbation (the anisotropic concave–convex problem). We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the positive parameter $\lambda$ varies. Also, we prove the existence of minimal positive solutions.

中文翻译：

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具有无限势和竞争非线性的各向异性方程

我们考虑由变量指数驱动的非线性Dirichlet问题 $p$-拉普拉斯语加上不确定的潜在条件。该反应具有参量凹（亚线性）项和凸（超线性）微扰（各向异性凹-凸问题）的竞争效应。我们证明了一个分叉型定理，它将正解集的变化描述为正参数$\lambda$不同。此外，我们证明了最小正解的存在。