In this paper, we deal with the existence of positive solutions for semipositone (p,N)-Laplacian problems with critical Trudinger–Moser nonlinearities in a bounded domain:

$$\begin{aligned} \left\{ \begin{array}{clll} -\varDelta _p u-\varDelta _N u=\lambda u^{N-1}e^{\beta u^{N'}} - \mu , &{} \text {in}\,\varOmega ;\\ u>0, &{} \text {in}\,\varOmega ;\\ u=0,&{} \text {on}\,\partial \varOmega . \end{array} \right. \end{aligned}$$

We obtain the positive solutions by combining variational methods with regularity arguments. And the main novelty here is to obtain a uniform \(\mathcal {C}^{1,\alpha }\) priori estimate of the weak solution. Our arguments can be also adapted to seek positive solutions of more general semipositone problems.

中文翻译：

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具有临界Trudinger-Moser非线性的半正（p，N）-Laplacian问题的正解

在本文中，我们处理有限域中具有临界Trudinger-Moser非线性的半正（p，N）-Laplacian问题正解的存在：

$$ \ begin {aligned} \ left \ {\ begin {array} {clll}-\ varDelta _p u- \ varDelta _N u = \ lambda u ^ {N-1} e ^ {\ beta u ^ {N'} }-\ mu，＆{} \ text {in} \，\ varOmega; \\ u> 0，＆{} \ text {in} \，\ varOmega; \\ u = 0，＆{} \ text {on } \，\ partial \ varOmega。\ end {array} \ right。\ end {aligned} $$

我们通过将变分方法与正则参数相结合来获得正解。并且这里的主要新颖之处是获得弱解的统一\（\ mathcal {C} ^ {1，\ alpha} \）先验估计。我们的论点也可以用于寻求更一般的半正问题的正解。