On a Schrödinger–Kirchhoff Type Equation Involving the Fractional p-Laplacian without the Ambrosetti–Rabinowitz Condition
DOI:
https://doi.org/10.15407/mag20.01.041Ключові слова:
p-оператор Лапласа, дробовий простiр Соболєва, рiвняння типу Шредiнгера–Кiрхгофа, умова Амброзеттi–Рабiновiца, варiацiйнi методиАнотація
У цiй статтi ми розглядаємо iснування та множиннiсть багатьох слабких розв’язкiв для наступного дробового рiвняння типу Шредiнгера–Кiрхгофа: \begin{align*} \left(a+b\displaystyle\iint_{\mathbb{R}^{2N}}\displaystyle\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\mathrm{d}x\mathrm{d}y\right)^{p-1} & \times (-\Delta)^s_pu+\lambda V(x)|u|^{p-2}u \\ & =f(x,u)+h(x)\ \mathrm{ in } \ \mathbb{R}^N,\end{align*} де $N > sp$, $a,b > 0$ — константи, $\lambda$ — параметр, $(-\Delta)^s_p$ — дробовий $p-$оператор Лапласа з $0 < s < 1 < p < \infty$, нелiнiйнiсть $f(x,u)$ i потенцiальна функцiя $V (x)$ задовольняють деякi прийнятнi припущення. За таких умов одержано деякi новi результати для достатньо великих $\lambda >
0$ шляхом застосування варiацiйних методiв.
Mathematical Subject Classification 2020: 35A15, 35J60, 35R11
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