This paper is concerned with the qualitative analysis of solutions to the following class of quasilinear problems

where \(\Delta _{\Phi }u=\mathrm{div}\,(\varphi (x,|\nabla u|)\nabla u)\) and \(\Phi (x,t)=\int _{0}^{|t|}\varphi (x,s)s\,ds\) is a generalized N-function. We assume that \(\Omega \subset {\mathbb {R}}^N\) is a smooth bounded domain that contains two open regions \(\Omega _N,\Omega _p\) with \({\overline{\Omega }}_N \cap {\overline{\Omega }}_p=\emptyset \). The features of this paper are that \(-\Delta _{\Phi }u\) behaves like \(-\Delta _N u \) on \(\Omega _N\) and \(-\Delta _p u \) on \(\Omega _p\), and that the growth of \(f:\Omega \times {\mathbb {R}} \rightarrow {\mathbb {R}}\) is like that of \(e^{\alpha |t|^{\frac{N}{N-1}}}\) on \(\Omega _N\) and as \(|t|^{p^{*}-2}t\) on \(\Omega _p\) when |t| is large enough. The main result establishes the existence of solutions in a suitable Musielak–Sobolev space in the case of high perturbations with respect to the values of a positive parameter.

本文关注以下类拟线性问题解的定性分析

其中\（\ Delta _ {\ Phi} u = \ mathrm {div} \ ,, \\ varphi（x，| \ nabla u |）\ nabla u）\）和\（\ Phi（x，t）= \ int _ {0} ^ {| t |} \ varphi（x，s）s \，ds \）是广义的N函数。我们假设\（\欧米茄\子集{\ mathbb {R}} ^ N \）是包含两个开口区域的平滑界域\（\欧米茄_N，\欧米茄_p \）与\（{\划线{\欧米茄}} _ N \ cap {\ overline {\ Omega}} _ p = \ emptyset \）。本文的特点是\（-\ Delta _ {\ Phi} u \）的行为类似于\（-\ Delta _N u \）在\（\ Omega _N \）和\（-\ Delta _p u \）上\（\ Omega _p \），并表示\（f：\ Omega \ times {\ mathbb {R}} \ rightarrow {\ mathbb {R}} \）就像\（e ^ {\ alpha | t | ^ {\ frac {N} {N- 1}}} \）在\（\ Omega _N \）上，并在\（\ Omega _p \）上作为\（| t | ^ {p ^ {*}-2} t \）在\（\ Omega _p \）上| t | 足够大。主要结果表明，在对正参数值有高摄动的情况下，在适当的Musielak-Sobolev空间中存在解。