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High perturbations of quasilinear problems with double criticality
Mathematische Zeitschrift ( IF 1.0 ) Pub Date : 2021-04-23 , DOI: 10.1007/s00209-021-02757-z
Claudianor O. Alves , Prashanta Garain , Vicenţiu D. Rădulescu

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

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{\Phi }u=f(x,u) &{}\quad \text {in } \Omega ,\\ u=0 &{}\quad \text {on }\partial \Omega , \end{array} \right. \end{aligned}$$(P)

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.



中文翻译:

具有双重临界的拟线性问题的高扰动

本文关注以下类拟线性问题解的定性分析

$$ \ begin {aligned} \ left \ {\ begin {array} {ll}-\ Delta _ {\ Phi} u = f(x,u)&{} \ quad \ text {in} \ Omega,\\ u = 0&{} \ quad \ text {on} \ partial \ Omega,\ end {array} \ right。\ end {aligned} $$(P)

其中\(\ Delta _ {\ Phi} u = \ mathrm {div} \ ,, \\ varphi(x,| \ nabla u |)\ nabla u)\)\(\ Phi(x,t)= \ int _ {0} ^ {| t |} \ varphi(x,s)s \,ds \)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\(\欧米茄_N,\欧米茄_p \)\({\划线{\欧米茄}} _ N \ cap {\ overline {\ Omega}} _ p = \ emptyset \)。本文的特点是\(-\ Delta _ {\ Phi} u \)的行为类似于\(-\ Delta _N u \)\(\ Omega _N \)\(-\ Delta _p u \)\(\ Omega _p \),并表示\(f:\ Omega \ times {\ mathbb {R}} \ rightarrow {\ mathbb {R}} \)就像\(e ^ {\ alpha | t | ^ {\ frac {N} {N- 1}}} \)\(\ Omega _N \)上,并在\(\ Omega _p \)上作为\(| t | ^ {p ^ {*}-2} t \)\(\ Omega _p \)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。

更新日期:2021-04-23
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