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Semilinear nonlocal elliptic equations with critical and supercritical exponents
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 |
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.
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. Alves, M. Delgado, M. 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'Aglio, V. De Cicco, G. 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 Ó, E. 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 Ó, E. 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 Figueiredo, M. 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. Silva, M. 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. Alves, M. Delgado, M. 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'Aglio, V. De Cicco, G. 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 Ó, E. 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 Ó, E. 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 Figueiredo, M. 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. Silva, M. 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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