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Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay
$ N- $Laplacian problems with critical double exponential nonlinearities
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China |
$ \left\{ {\begin{array}{*{20}{l}}{ - {\rm{div}}(\omega (x)|\nabla u(x){|^{N - 2}}\nabla u(x)) = f(x,u),\;\;\quad }&{\;\;\;\;\;{\rm{in}}\;B;}\\{u = 0,\;\;\quad }&{\;\;\;\;\;{\rm{on}}\;\partial B,}\end{array}} \right.{\rm{ }}\;\;\;\;\;\;\;\left( 1 \right)$ |
$ B $ |
$ \mathbb{R}^N $ |
$ N\geq 2 $ |
$ \omega(x) $ |
$ f(x,u) $ |
$ B\times\mathbb{R} $ |
$ \exp\{e^{\alpha |u|^{\frac{N}{N-1}}}\} $ |
$ |u|\to\infty $ |
$ \alpha>0 $ |
References:
[1] |
A. Ambrosetti and P. H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[2] |
L. Boccardo and F. Murat,
Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal., 19 (1992), 581-597.
doi: 10.1016/0362-546X(92)90023-8. |
[3] |
M. Calanchi, Some weighted inequalities of Trudinger-Moser type, in: Progress in Nonlinear Differential Equations and Appl., Birkhäuser, 85 (2014), 163–174. |
[4] |
M. Calanchi, E. Massa and B. Ruf,
Weighted Trudinger-Moser inequalities and associated Liouville type equations, Proc. Amer. Math. Soc., 146 (2018), 5243-5256.
doi: 10.1090/proc/14189. |
[5] |
M. Calanchi and B. Ruf,
On Trudinger-Moser type inequalities with logarithmic weights, J. Differ. Equ., 258 (2015), 1967-1989.
doi: 10.1016/j.jde.2014.11.019. |
[6] |
M. Calanchi and B. Ruf,
Trudinger-Moser type inequalities with logarithmic weights in dimension $N$, Nonlinear Anal., 121 (2015), 403-411.
doi: 10.1016/j.na.2015.02.001. |
[7] |
M. Calanchi, B. Ruf and F. Sani, Elliptic equations in dimension 2 with double exponential nonlinearities, NoDEA Nonlinear Differential Equations Appl., 24 (2017), 29, 18 pp.
doi: 10.1007/s00030-017-0453-y. |
[8] |
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.
doi: 10.1007/BF01205003. |
[9] |
S. Deng, Existence of solutions for some weighted mean field equations in dimension $N$, Appl. Math. Lett., 100 (2020), 106010, 7 pp.
doi: 10.1016/j.aml.2019.106010. |
[10] |
J. M. B. do Ó,
Semilinear Dirichlet problems for the $N$-Laplacian in $\bf R^N$ with nonlinearities in the critical growth range, Differential Integral Equations, 9 (1996), 967-979.
|
[11] |
Y. Jabri, The Mountain Pass Theorem, Variant, Generalizations and some Applications, Encyclopedia of Mathematics and its Applications, 95, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511546655.![]() ![]() ![]() |
[12] | |
[13] |
J. Moser,
A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092.
doi: 10.1512/iumj.1971.20.20101. |
[14] |
V. H. Nguyen,
Remarks on the Moser-Trudinger type inequality with logarithmic weights in dimension $N$, Proc. Amer. Math. Soc., 147 (2019), 5183-5193.
doi: 10.1090/proc/14566. |
[15] |
P. Roy,
On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions, Discrete Contin. Dyn. Syst., 39 (2019), 5207-5222.
doi: 10.3934/dcds.2019212. |
[16] |
N. S. Trudinger,
On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.
doi: 10.1512/iumj.1968.17.17028. |
[17] |
M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996.
doi: 10.1007/978-1-4612-4146-1. |
show all references
References:
[1] |
A. Ambrosetti and P. H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[2] |
L. Boccardo and F. Murat,
Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal., 19 (1992), 581-597.
doi: 10.1016/0362-546X(92)90023-8. |
[3] |
M. Calanchi, Some weighted inequalities of Trudinger-Moser type, in: Progress in Nonlinear Differential Equations and Appl., Birkhäuser, 85 (2014), 163–174. |
[4] |
M. Calanchi, E. Massa and B. Ruf,
Weighted Trudinger-Moser inequalities and associated Liouville type equations, Proc. Amer. Math. Soc., 146 (2018), 5243-5256.
doi: 10.1090/proc/14189. |
[5] |
M. Calanchi and B. Ruf,
On Trudinger-Moser type inequalities with logarithmic weights, J. Differ. Equ., 258 (2015), 1967-1989.
doi: 10.1016/j.jde.2014.11.019. |
[6] |
M. Calanchi and B. Ruf,
Trudinger-Moser type inequalities with logarithmic weights in dimension $N$, Nonlinear Anal., 121 (2015), 403-411.
doi: 10.1016/j.na.2015.02.001. |
[7] |
M. Calanchi, B. Ruf and F. Sani, Elliptic equations in dimension 2 with double exponential nonlinearities, NoDEA Nonlinear Differential Equations Appl., 24 (2017), 29, 18 pp.
doi: 10.1007/s00030-017-0453-y. |
[8] |
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.
doi: 10.1007/BF01205003. |
[9] |
S. Deng, Existence of solutions for some weighted mean field equations in dimension $N$, Appl. Math. Lett., 100 (2020), 106010, 7 pp.
doi: 10.1016/j.aml.2019.106010. |
[10] |
J. M. B. do Ó,
Semilinear Dirichlet problems for the $N$-Laplacian in $\bf R^N$ with nonlinearities in the critical growth range, Differential Integral Equations, 9 (1996), 967-979.
|
[11] |
Y. Jabri, The Mountain Pass Theorem, Variant, Generalizations and some Applications, Encyclopedia of Mathematics and its Applications, 95, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511546655.![]() ![]() ![]() |
[12] | |
[13] |
J. Moser,
A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092.
doi: 10.1512/iumj.1971.20.20101. |
[14] |
V. H. Nguyen,
Remarks on the Moser-Trudinger type inequality with logarithmic weights in dimension $N$, Proc. Amer. Math. Soc., 147 (2019), 5183-5193.
doi: 10.1090/proc/14566. |
[15] |
P. Roy,
On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions, Discrete Contin. Dyn. Syst., 39 (2019), 5207-5222.
doi: 10.3934/dcds.2019212. |
[16] |
N. S. Trudinger,
On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.
doi: 10.1512/iumj.1968.17.17028. |
[17] |
M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996.
doi: 10.1007/978-1-4612-4146-1. |
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