In this article, we study the existence of nontrivial weak solutions for the following boundary value problem:

$$ - {\Delta _\gamma }u = f(x,u)\;{\rm{in}}\;\Omega ,\;\;\;\;\;u = 0\;{\rm{on}}\;\partial \Omega ,$$

where Ω is a bounded domain with smooth boundary in \({\mathbb{R}^N},\;\Omega \cap \left\{ {{x_j} = 0} \right\} \ne \emptyset \) for some j, Δ_γ is a subelliptic linear operator of the type

$${\Delta _\gamma }: = \sum\limits_{j = 1}^N {{\partial _{{x_j}}}(\gamma _j^2{\partial _{{x_j}}}),\;\;\;\;\;{\partial _{{x_j}}}: = {\partial \over {\partial {x_j}}}} ,\;\;\;\;\;N \ge 2,$$

where γ(x) = (γ₁(x), γ₂(x), …, γ_N(x)) satisfies certain homogeneity conditions and degenerates at the coordinate hyperplanes and the nonlinearity f(x, ξ) is of subcritical growth and does not satisfy the Ambrosetti-Rabinowitz (AR) condition.

中文翻译：

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半线性Δγ-微分方程边值问题的非平凡解

在本文中，我们研究以下边值问题的非平凡弱解的存在：

$$-{\ Delta _ \ gamma} u = f（x，u）\; {\ rm {in}} \; \ Omega，\; \; \; \; \; u = 0 \; {\ rm {on}} \; \ partial \ Omega，$$

其中Ω是具有光滑边界的有界区域\（{\ mathbb {R} ^ N}，\; \欧米茄\帽\左\ {{{x_j} = 0} \右\} \ NE \ emptyset \）为一些Ĵ，Δ _γ是所述类型的近椭圆形线性算

$$ {\ Delta _ \ gamma}：= \ sum \ limits_ {j = 1} ^ N {{\ partial _ {{x_j}}}（\ gamma _j ^ 2 {\ partial _ {{x_j}}}） ，\; \; \; \; \; \; {\ partial _ {{{x_j}}}}：= {\ partial \ over {\ partial {x_j}}}}}，\; \; \; \; \; N \ ge 2，$$

其中γ（X）=（γ ₁（X），γ ₂（X），...，γ _Ñ（X））满足一定的均匀性条件和退化在坐标超平面和非线性˚F（X，ξ）是亚临界生长并且不满足Ambrosetti-Rabinowitz（AR）的条件。