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 and 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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[9]

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

[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. 

[11]

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

[12]

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

[13]

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

[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. 

[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.

[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.

[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.

[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. 

[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.

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[9]

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

[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. 

[11]

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

[12]

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

[13]

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

[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. 

[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.

[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.

[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.

[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. 

[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.

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