Weighted Elliptic Equations in Dimension N with Subcritical and Critical Double Exponential Nonlinearities
DOI:
https://doi.org/10.15407/mag19.03.527Анотація
У роботi доведено iснування нетривiального розв’язку для такої ва-
гової задачi без умови Амбросеттi–Рабiновiца: $- \mathrm{div} (\sigma(x)|\nabla u|^{N-2} \nabla u) = f(x,u)$ i $u >0$ в $B$, $u=0$ на $\partial B$, де $B$ є одиничною кулею в $\mathbb{R}^N$, $\sigma(x)=\left(\log\left(\frac{e}{|x|}\right)\right)^{N-1}$ є сингулярною логарифмiчною вагою у вкладеннi Трудiнґера–Мозера. Нелiнiйнiсть дає критичне або субкритичне зростання вiдносно нерiвностi Трудiнґера–Мозера. Ми скористалися мiнiмакс технiкою в комбiнацiї з нерiвнiстю Трудiнґера–Мозера, щоб довести iснування розв’язку. Ми запровадили нову умову для зростання та наполягаємо на її важливостi для позбавлення рiвня компактностi.
Mathematical Subject Classification 2020: 46E35, 35J20, 35J33, 35J60.
Ключові слова:
нерiвнiсть Трудiнґера–Мозера, нелiнiйнiсть подвiй- ного експоненцiального зростання, критичнi експоненти, рiвень компактностiПосилання
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