Triple Solutions for p(x)-Triharmonic Problems Using Ricceri’s Critical Point Theorem
Анотація
Ми досліджуємо нелінійну еліптичну задачу шостого порядку, керовану тригармонічним оператором зі змінним показником:
$$
\left\{
\begin{aligned}
& -\Delta^3_{p(x)} u + |u|^{p(x)-2}u = \lambda f(x,u) + \mu g(x,u), && x \in \Omega, \\
& u = \Delta u = \Delta^2 u = 0,& & x \in \partial\Omega,
\end{aligned}
\right.
$$
де $\Omega \subset \mathbb{R}^{N}$ ($N > 3$) є гладкою обмеженою областю, $p \in C(\overline{\Omega})$ задовольняє умову $p^{-} > \max\left\{3, \tfrac{N}{3} \right\}$, а $f$ та $g$ є функціями Каратеодорі. Застосовуючи теорему Річчері про три критичні точки та використовуючи вкладення простору $W^{3,p(\cdot)}(\Omega) \cap W^{1,p(\cdot)}_0(\Omega)$ у простори Лебега, ми доводимо існування щонайменше трьох обмежених слабких розв'язків для відкритої множини параметрів $(\lambda, \mu)$. Це розширює теорію кратності на тригармонічний випадок зі змінним показником та крайовими умовами Нав'є.
Mathematical Subject Classification 2020: 35D30, 35J55, 35J65, 35A15
Ключові слова:
тригармонічний оператор зі змінним показником, нелінійне еліптичне рівняння шостого порядку, теорема Річчері, слабкі розв'язки, кратність, крайові умови Нав'єПосилання
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