We consider a parametric Dirichlet problem driven by the sum of a p-Laplacian (\(p>2\)) and a Laplacian (a (p, 2)-equation). The reaction consists of an asymmetric \((p-1)\)-linear term which is resonant as \(x \rightarrow - \infty \), plus a concave term. However, in this case the concave term enters with a negative sign. Using variational tools together with suitable truncation techniques and Morse theory (critical groups), we show that when the parameter is small the problem has at least three nontrivial smooth solutions.

中文翻译：

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（p，2）-具有交叉非线性和凹项的方程

我们考虑由p -Laplacian（\（p> 2 \））和Laplacian（a（p，2）方程）之和驱动的参数Dirichlet问题。反应由一个非对称\（（p-1）\）-线性项（谐振为\（x \ rightarrow-\ infty \））和一个凹项组成。但是，在这种情况下，凹项以负号输入。使用变分工具以及合适的截断技术和莫尔斯理论（临界组），我们表明，当参数较小时，问题至少具有三个非平凡的光滑解。