Weighted Elliptic Equations in Dimension N with Subcritical and Critical Double Exponential Nonlinearities
DOI:
https://doi.org/10.15407/mag19.03.527Ключові слова:
нерiвнiсть Трудiнґера–Мозера, нелiнiйнiсть подвiй- ного експоненцiального зростання, критичнi експоненти, рiвень компактностiАнотація
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.
Посилання
A. Adimurthi, Existence results for the semilinear Dirichlet problem with critical growth for the $n$-Laplacian, Houst. J. Math. 7 (1991), 285--298.
A. Adimurthi and Y. Yang, An interpolation of Hardy inequality and Trudinger Moser inequality in $R^{N}$ and its applications, Int. Math. Res. Not. 13 (2010) 2394--2426. https://doi.org/10.1093/imrn/rnp194
A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical points theory and applications, J. Funct. Anal. 14 (1973), 349--381. https://doi.org/10.1016/0022-1236(73)90051-7
R. Aris, The Mathematical Theory of Diffusion and Reaction in Permeable Catalyst, 1, 2, Clarendon Press Oxford, 1975.
G. Astrita and G. Marrucci, Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill, New York, 1974.
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, (2010). https://doi.org/10.1007/978-0-387-70914-7
M. Calanchi and B. Ruf, Trudinger-Moser type inequalities with logarithmic weights in dimension $N$, Nonlinear Anal. 121 (2015), 403--411. https://doi.org/10.1016/j.na.2015.02.001
M. Calanchi, B. Ruf, and F. Sani, Elliptic equations in dimension $2$ with double exponential nonlinearities, NoDEA Nonlinear Differential Equations Appl. 24 (2017), No. 3, 29. https://doi.org/10.1007/s00030-017-0453-y
M. Calanchi, E. Massa, and B. Ruf, Weighted Trudinger-Moser inequalities and associated Liouville type equations, Proc. Amer. Math. Soc. 146 (2018), 5243--5256. https://doi.org/10.1090/proc/14189
G. Cerami, An existence criterion for the critical points on unbounded manifolds, (Italian) Istit. Lombardo Accad. Sci. Lett. Rend. 112 (1978), No. 2, 332--336.
D.G. Costa and C.A. Magalhäes, Variational elliptic problems which are nonquadratic at infinity, Nonlinear Anal. 23 (1994), 1401--1412. https://doi.org/10.1016/0362-546X(94)90135-X
S. Deng, Existence of solutions for some weighted mean field equations in dimension $N$, Appl. Math. Lett. 100 (2020), 106010. https://doi.org/10.1016/j.aml.2019.106010
S. Deng, T. Hu, and C.Tang, $N$-laplacian problems with critical double exponential nonlinearities, Discrete Contin. Dyn. Syst. 41 (2021), 987--1003. https://doi.org/10.3934/dcds.2020306
D.G. de Figueiredo, O.H. Miyagaki, and B. Ruf, Elliptic equations in $R^{2}$ with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations 3 (1995), 139--153. https://doi.org/10.1007/BF01205003
F. Fang and S.B. Liu, Nontrivial solutions of superlinear $p$-Laplacian equations, J. Math. Anal. Appl. 351 (2009), 138--146. https://doi.org/10.1016/j.jmaa.2008.09.064
L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on $R^N$, Proc. Roy. Soc. Edinburgh. 129 (1999), 787--809. https://doi.org/10.1017/S0308210500013147
A. Kufner, Weighted Sobolev spaces, John Wiley and Sons Ltd., 1985.
N. Lam and G. Lu, Existence and multiplicity of solutions to equations of $N$-Laplacian type with critical exponential growth in $R^{N}$, J. Funct. Anal. 262 (2012), No. 3, 1132--1165. https://doi.org/10.1016/j.jfa.2011.10.012
P.L. Lions, The Concentration-Compactness Principle in the Calculus of Variations, Part 1, Revista Iberoamericana 11 (1985), 185--201. https://doi.org/10.4171/RMI/6
O. Miyagaki and M. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Differential Equations 245 (2008), 3628--3638. https://doi.org/10.1016/j.jde.2008.02.035
P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence 12 (1990), 629--639. https://doi.org/10.1109/34.56205
M. Schechter, The Use of Cerami Sequences in Critical Point Theory, Abstr. Appl. Anal. 2007 (2007), 58948. https://doi.org/10.1155/2007/58948
M. Schechter and W. Zou, Superlinear problems, Pacific J. Math. 214 (2004), 145--160. https://doi.org/10.2140/pjm.2004.214.145
R.E. Volker, Nonlinear f low in porus media by finite elements, J. Hydraulics Division 95 (1969), 2093--2114. https://doi.org/10.1061/JYCEAJ.0002207
M. Willem and W. Zou, On a Schrödinger equation with periodic potential and spectrum point zero, Indiana Univ. Math. J. 52 (2003), 109--132. https://doi.org/10.1512/iumj.2003.52.2273
Y.Yang, Existence of positive solutions to quasi-linear elliptic equations with exponential growth in the whole Euclidean space, J. Funct. Anal. 262 (2012), No. 4, 1679--1704. https://doi.org/10.1016/j.jfa.2011.11.018
C. Zhang, Concentration-Compactness principle for Trudinger-Moser inequalities with logarithmic weights and their applications, Nonlinear Anal. 197 (2020), 1--22. https://doi.org/10.1016/j.na.2020.111845
W.M. Zou, Variant fountain theorems and their applications, Manuscripta Math. 104 (2001), 343--358. https://doi.org/10.1007/s002290170032
