Acta Applicandae Mathematicae ( IF 1.2 ) Pub Date : 2020-07-30 , DOI: 10.1007/s10440-020-00352-8 Khaled Kefi , Kamel Saoudi , Mohammed Mosa Al-Shomrani
The purpose of the present paper is to study the existence of solutions for the following nonhomogeneous singular Kirchhoff problem involving the \(p(x)\)-biharmonic operator:
$$\begin{aligned} \left \{ \textstyle\begin{array}{l} M(t)\Big(\Delta ^{2}_{p(x)}u+a(x)|u|^{p(x)-2}u\Big) =g(x)u^{-\gamma (x)} \mp \lambda f(x,u),\quad \mbox{in }\Omega , \\ \Delta u=u=0, \quad \mbox{on }\partial \Omega , \end{array}\displaystyle \right . \end{aligned}$$where \(\Omega \subset {\mathbb{R}}^{N}\), \((N\geq 3)\) be a bounded domain with \(C^{2}\) boundary, \(\lambda \) is a positive parameter, \(\gamma : \overline{\Omega }\longrightarrow (0,1)\) be a continuous function, \(p\in C(\overline{\Omega })\) with \(\displaystyle 1< p^{-}:=\inf _{x\in \Omega }p(x)\leq p^{+}:=\sup _{x \in \Omega }p(x)<\frac{N}{2}\), as usual, \(p^{*}(x)=\displaystyle \frac{N p(x)}{N-2p(x)}\), \(g \in L^{\frac{p^{*}(x)}{p^{*}(x)+\gamma (x)-1}}(\Omega )\). We assume that \(M(t)\) is a continuous function with
$$ t:=\int _{\Omega }\frac{1}{p(x)}(|\Delta u|^{p(x)}+a(x)|u|^{p(x)})dx, $$and assumed to verify assertions (M1)-(M3) in Sect. 3, moreover \(f(x,u)\) are assumed to satisfy assumptions (f1)-(f6). In the proofs of our results we use variational techniques and monotonicity arguments combined with the theory of the generalized Lebesgue Sobolev spaces.
中文翻译：
关于Kirchhoff奇异p（x）$ p（x）$-具有Navier边界条件的双调和问题
本文的目的是研究以下涉及\（p（x）\）-双调和算子的非齐奇奇异Kirchhoff问题的解的存在性：
$$ \ begin {aligned} \ left \ {\ textstyle \ begin {array} {l} M（t）\ Big（\ Delta ^ {2} _ {p（x）} u + a（x）| u | ^ {p（x）-2} u \ Big）= g（x）u ^ {-\ gamma（x）} \ mp \ lambda f（x，u），\ quad \ mbox {in} \ Omega，\ \ \ Delta u = u = 0，\ quad \ mbox {on} \ partial \ Omega，\ end {array} \ displaystyle \ right。\ end {aligned} $$其中\（\ Omega \ subset {\ mathbb {R}} ^ {N} \），\（（N \ geq 3）\）是边界为\（C ^ {2} \）的有界域，\（\ lambda \）是一个正参数，\（\ gamma：\ overline {\ Omega} \ longrightarrow（0,1）\）是一个连续函数，\（p \ in C（\ overline {\ Omega}）\\具有\（\ displaystyle 1 <p ^ {-}：= \ inf _ {x \ in \ Omega} p（x）\ leq p ^ {+}：= \ sup _ {x \ in \ Omega} p（x） <\ frac {N} {2} \），像往常一样，\（p ^ {*}（x）= \ displaystyle \ frac {N p（x）} {N-2p（x）} \），\（ g \ in L ^ {\ frac {p ^ {*}（x）} {p ^ {*}（x）+ \ gamma（x）-1}}（\ Omega} \）。我们假设\（M（t）\）是具有
$$ t：= \ int _ {\ Omega} \ frac {1} {p（x）}（| \ Delta u | ^ {p（x）} + a（x）| u | ^ {p（x） }）dx，$$并假设要验证Sect中的断言（M1） - （M3）。此外，如图3所示，假定\（f（x，u）\）满足假设（f1） - （f6）。在我们的结果证明中，我们使用了变分技术和单调论证，并结合了广义Lebesgue Sobolev空间的理论。
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