September  2017, 16(5): 1767-1784. doi: 10.3934/cpaa.2017086

On some local-nonlocal elliptic equation involving nonlinear terms with exponential growth

Institut Supérieur des Mathématiques Appliquées et de l'Informatique de Kairouan, Avenue Assad Iben Fourat, 3100 Kairouan, Tunisie

* Corresponding author

Received  October 2016 Revised  March 2017 Published  May 2017

In this paper, we prove the existence of multiple solutions to some intermediate local-nonlocal elliptic equation in the whole two dimensional space. The nonlinearities exhibit an exponential growth at infinity.

Citation: Sami Aouaoui. On some local-nonlocal elliptic equation involving nonlinear terms with exponential growth. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1767-1784. doi: 10.3934/cpaa.2017086
References:
[1]

C. O. Alves, F. J. Corrêa and M. Chipot, On a class of intermediate local-nonlocal elliptic problems Topol. Methods Nonlinear Anal. (2016). doi: 10.12775/TMNA.2016.083.  Google Scholar

[2]

C. O. AlvesM. DelgadoM. A. S. Souto and A. Suarez, Existence of positive solution of a nonlocal logistic population model, Z. Angew. Math. Phys., 66 (2015), 943-953.  doi: 10.1007/s00033-014-0458-x.  Google Scholar

[3]

C. O. Alves and G. M. Figueiredo, On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in ${{\mathbb{R}}^{N}}$, J. Differential Equations, 246 (2009), 1288-1311.  doi: 10.1016/j.jde.2008.08.004.  Google Scholar

[4]

S. Aouaoui, Solutions to quasilinear equations of N-biharmonic type with degenerate coercivity, Electron. J. Differential Equations, 228 (2014), 1-12.   Google Scholar

[5]

S. Aouaoui, A multiplicity result for some integro-differential biharmonic equation in $ \mathbb{R}^4$, J. Part. Diff. Eq., 29 (2016), 102-115.  doi: 10.4208/jpde.v29.n2.2.  Google Scholar

[6]

A. Dall'AglioV. De CiccoG. Giachetti and J. P. Puel, Existence of bounded solutions for nonlinear elliptic equations in unboundd domains, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 431-450.  doi: 10.1007/s00030-004-1070-0.  Google Scholar

[7]

J. M. do ÓE. Medeiros and U. Severo, A nonhomogeneous elliptic problem involving critical growth in dimension two, J. Math. Anal. Appl., 345 (2008), 286-304.  doi: 10.1016/j.jmaa.2008.03.074.  Google Scholar

[8]

J. M. do ÓE. Medeiros and U. Severo, On a quasilinear nonhomogeneous elliptic equation with critical growth in $ \mathbb{R}^N$, J. Differential Equations, 246 (2009), 1363-1386.  doi: 10.1016/j.jde.2008.11.020.  Google Scholar

[9]

I. Ekeland, On the variational principle, J. Math. Anal. App., 47 (1974), 324-353.  doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[10]

D. G. de FigueiredoM. Girardi and M. Matzeu, Semilinear elliptic equations with dependence on the gradient via mountain-pass techniques, Differ. Integral Equ., 17 (2004), 119-126.   Google Scholar

[11]

M. Girardi and M. Matzeu, A compactness result for quasilinear elliptic equations by mountain pass techniques, Rend. Mat. Appl., 29 (2009), 83-95.   Google Scholar

[12]

O. Kavian, Introduction á la théorie des points critiques et applications aux problémes elliptiques Springer-Verlag, Paris, France 1993.  Google Scholar

[13]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.  Google Scholar

[14]

V. Rǎdulescu and P. Pucci, The impact of the mountain pass theory in nonlinear analysis: a mathematical survey, Boll. Unione Mat. Ital. Series Ⅸ, 3 (2010), 543-584.   Google Scholar

[15]

B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $ \mathbb{R}^2$, J. Funct. Anal., 2019 (2005), 340-367.  doi: 10.1016/j.jfa.2004.06.013.  Google Scholar

[16]

R. Servadei, A semilinear elliptic PDE not in divergence form via variational methods, J. Math. Anal. Appl., 383 (2011), 190-199.  doi: 10.1016/j.jmaa.2011.05.017.  Google Scholar

[17]

T. SilvaM. de Souza and J. M. do Ó, Quasilinear nonhomogeneous Schröinger equation with critical exponential growth in $ \mathbb{R}^N$, Topol. Methods Nonlinear Anal., 45 (2015), 615-639.  doi: 10.12775/TMNA.2015.029.  Google Scholar

[18]

E. Tonkes, Solutions to a perturbed critical semilinear equation concerning the $N$ -Laplacian in $ \mathbb{R}^N$, Comment. Math. Univ. Carolin., 40 (1999), 679-699.   Google Scholar

[19]

N. S. Trudinger, On Harnack type inequalities and their applications to quasilinear elliptic equations, Comm. Pure Appl. Math., XX (1967), 721-747.  doi: 10.1002/cpa.3160200406.  Google Scholar

show all references

References:
[1]

C. O. Alves, F. J. Corrêa and M. Chipot, On a class of intermediate local-nonlocal elliptic problems Topol. Methods Nonlinear Anal. (2016). doi: 10.12775/TMNA.2016.083.  Google Scholar

[2]

C. O. AlvesM. DelgadoM. A. S. Souto and A. Suarez, Existence of positive solution of a nonlocal logistic population model, Z. Angew. Math. Phys., 66 (2015), 943-953.  doi: 10.1007/s00033-014-0458-x.  Google Scholar

[3]

C. O. Alves and G. M. Figueiredo, On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in ${{\mathbb{R}}^{N}}$, J. Differential Equations, 246 (2009), 1288-1311.  doi: 10.1016/j.jde.2008.08.004.  Google Scholar

[4]

S. Aouaoui, Solutions to quasilinear equations of N-biharmonic type with degenerate coercivity, Electron. J. Differential Equations, 228 (2014), 1-12.   Google Scholar

[5]

S. Aouaoui, A multiplicity result for some integro-differential biharmonic equation in $ \mathbb{R}^4$, J. Part. Diff. Eq., 29 (2016), 102-115.  doi: 10.4208/jpde.v29.n2.2.  Google Scholar

[6]

A. Dall'AglioV. De CiccoG. Giachetti and J. P. Puel, Existence of bounded solutions for nonlinear elliptic equations in unboundd domains, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 431-450.  doi: 10.1007/s00030-004-1070-0.  Google Scholar

[7]

J. M. do ÓE. Medeiros and U. Severo, A nonhomogeneous elliptic problem involving critical growth in dimension two, J. Math. Anal. Appl., 345 (2008), 286-304.  doi: 10.1016/j.jmaa.2008.03.074.  Google Scholar

[8]

J. M. do ÓE. Medeiros and U. Severo, On a quasilinear nonhomogeneous elliptic equation with critical growth in $ \mathbb{R}^N$, J. Differential Equations, 246 (2009), 1363-1386.  doi: 10.1016/j.jde.2008.11.020.  Google Scholar

[9]

I. Ekeland, On the variational principle, J. Math. Anal. App., 47 (1974), 324-353.  doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[10]

D. G. de FigueiredoM. Girardi and M. Matzeu, Semilinear elliptic equations with dependence on the gradient via mountain-pass techniques, Differ. Integral Equ., 17 (2004), 119-126.   Google Scholar

[11]

M. Girardi and M. Matzeu, A compactness result for quasilinear elliptic equations by mountain pass techniques, Rend. Mat. Appl., 29 (2009), 83-95.   Google Scholar

[12]

O. Kavian, Introduction á la théorie des points critiques et applications aux problémes elliptiques Springer-Verlag, Paris, France 1993.  Google Scholar

[13]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.  Google Scholar

[14]

V. Rǎdulescu and P. Pucci, The impact of the mountain pass theory in nonlinear analysis: a mathematical survey, Boll. Unione Mat. Ital. Series Ⅸ, 3 (2010), 543-584.   Google Scholar

[15]

B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $ \mathbb{R}^2$, J. Funct. Anal., 2019 (2005), 340-367.  doi: 10.1016/j.jfa.2004.06.013.  Google Scholar

[16]

R. Servadei, A semilinear elliptic PDE not in divergence form via variational methods, J. Math. Anal. Appl., 383 (2011), 190-199.  doi: 10.1016/j.jmaa.2011.05.017.  Google Scholar

[17]

T. SilvaM. de Souza and J. M. do Ó, Quasilinear nonhomogeneous Schröinger equation with critical exponential growth in $ \mathbb{R}^N$, Topol. Methods Nonlinear Anal., 45 (2015), 615-639.  doi: 10.12775/TMNA.2015.029.  Google Scholar

[18]

E. Tonkes, Solutions to a perturbed critical semilinear equation concerning the $N$ -Laplacian in $ \mathbb{R}^N$, Comment. Math. Univ. Carolin., 40 (1999), 679-699.   Google Scholar

[19]

N. S. Trudinger, On Harnack type inequalities and their applications to quasilinear elliptic equations, Comm. Pure Appl. Math., XX (1967), 721-747.  doi: 10.1002/cpa.3160200406.  Google Scholar

[1]

Anouar Bahrouni. Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity. Communications on Pure & Applied Analysis, 2017, 16 (1) : 243-252. doi: 10.3934/cpaa.2017011

[2]

Tomasz Cieślak. Trudinger-Moser type inequality for radially symmetric functions in a ring and applications to Keller-Segel in a ring. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2505-2512. doi: 10.3934/dcdsb.2013.18.2505

[3]

Kyril Tintarev. Is the Trudinger-Moser nonlinearity a true critical nonlinearity?. Conference Publications, 2011, 2011 (Special) : 1378-1384. doi: 10.3934/proc.2011.2011.1378

[4]

Xiaobao Zhu. Remarks on singular trudinger-moser type inequalities. Communications on Pure & Applied Analysis, 2020, 19 (1) : 103-112. doi: 10.3934/cpaa.2020006

[5]

Djairo G. De Figueiredo, João Marcos do Ó, Bernhard Ruf. Elliptic equations and systems with critical Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 455-476. doi: 10.3934/dcds.2011.30.455

[6]

Kanishka Perera, Marco Squassina. Bifurcation results for problems with fractional Trudinger-Moser nonlinearity. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 561-576. doi: 10.3934/dcdss.2018031

[7]

Changliang Zhou, Chunqin Zhou. Extremal functions of Moser-Trudinger inequality involving Finsler-Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2309-2328. doi: 10.3934/cpaa.2018110

[8]

Xumin Wang. Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2717-2733. doi: 10.3934/cpaa.2019121

[9]

Prosenjit Roy. On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5207-5222. doi: 10.3934/dcds.2019212

[10]

Guozhen Lu, Yunyan Yang. Sharp constant and extremal function for the improved Moser-Trudinger inequality involving $L^p$ norm in two dimension. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 963-979. doi: 10.3934/dcds.2009.25.963

[11]

Changliang Zhou, Chunqin Zhou. On the anisotropic Moser-Trudinger inequality for unbounded domains in $ \mathbb R^{n} $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 847-881. doi: 10.3934/dcds.2020064

[12]

Nguyen Lam. Equivalence of sharp Trudinger-Moser-Adams Inequalities. Communications on Pure & Applied Analysis, 2017, 16 (3) : 973-998. doi: 10.3934/cpaa.2017047

[13]

Claudianor O. Alves, Giovany M. Figueiredo, Marcelo F. Furtado. Multiplicity of solutions for elliptic systems via local Mountain Pass method. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1745-1758. doi: 10.3934/cpaa.2009.8.1745

[14]

Walter Allegretto, Yanping Lin, Shuqing Ma. On the box method for a non-local parabolic variational inequality. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 71-88. doi: 10.3934/dcdsb.2001.1.71

[15]

Linfeng Mei, Wei Dong, Changhe Guo. Concentration phenomenon in a nonlocal equation modeling phytoplankton growth. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 587-597. doi: 10.3934/dcdsb.2015.20.587

[16]

Cristina Tarsi. Perturbation from symmetry and multiplicity of solutions for elliptic problems with subcritical exponential growth in $\mathbb{R} ^2$. Communications on Pure & Applied Analysis, 2008, 7 (2) : 445-456. doi: 10.3934/cpaa.2008.7.445

[17]

Sami Aouaoui. A multiplicity result for some Kirchhoff-type equations involving exponential growth condition in $\mathbb{R}^2 $. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1351-1370. doi: 10.3934/cpaa.2016.15.1351

[18]

Li Wang, Yang Li, Liwei Zhang. A differential equation method for solving box constrained variational inequality problems. Journal of Industrial & Management Optimization, 2011, 7 (1) : 183-198. doi: 10.3934/jimo.2011.7.183

[19]

Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2001-2029. doi: 10.3934/cpaa.2013.12.2001

[20]

Michael Scheutzow. Exponential growth rate for a singular linear stochastic delay differential equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1683-1696. doi: 10.3934/dcdsb.2013.18.1683

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (23)
  • HTML views (21)
  • Cited by (0)

Other articles
by authors

[Back to Top]