In this paper, we discuss the existence of solutions to a nonlinear problem involving an exponential model of the Born–Infeld nonlinear electromagnetism. We establish an existence theorem by variational methods and mathematical analysis. We also show that the solutions obtained are nonnegative.

1.
M.
Born
and
L.
Infeld
, “
Foundations of the new field theory
,”
Nature
132
,
1004
(
1933
);
M.
Born
and
L.
Infeld
,
Proc. R. Soc. A
144
,
425
451
(
1934
).
2.
C. G.
Callan
, Jr.
and
J. M.
Maldacena
, “
Brane dynamics from the Born-Infeld action
,”
Nucl. Phys. B
513
,
198
212
(
1998
).
3.
E.
Bergshoeff
,
E.
Sezgin
,
C. N.
Pope
, and
P. K.
Townsend
, “
The Born-Infeld action from conformal invariance of the open superstring
,”
Phys. Lett. B
188
,
70
74
(
1987
).
4.
W.
Heisenberg
and
H.
Euler
, “
Consequences of Dirac theory of positive
,”
Z. Phys.
98
,
714
732
(
1936
).
5.
C. V.
Costa
,
D. M.
Gitman
, and
A. E.
Shabad
, “
Finite field-energy of a point charge in QED
,”
Phys. Scr.
90
,
074012
(
2015
).
6.
I.
Dymnikova
, “
Vacuum nonsingular black hole
,”
Gen. Relativ. Gravitation
24
,
235
242
(
1992
).
7.
I.
Dymnikova
, “
Regular electrically charged structures in nonlinear electrodynamics coupled to general relativity
,”
Classical Quantum Gravity
21
,
4417
4429
(
2004
).
8.
D. M.
Gitman
and
A. E.
Shabad
, “
A note on ‘Electron self-energy in logarithmic electrodynamics’ by P. Gaete and J. Helayël-Neto
,”
Eur. Phys. J. C
74
,
3186
(
2014
).
9.
S. I.
Kruglov
, “
A model of nonlinear electrodynamics
,”
Ann. Phys.
353
,
299
306
(
2015
).
10.
S. I.
Kruglov
, “
Nonlinear electromagnetic fields as a source of universe acceleration
,”
Int. J. Mod. Phys. A
31
,
1650058
(
2016
).
11.
S. H.
Hendi
, “
Asymptotic charged BTZ black hole solutions
,”
J. High Energy Phys.
2012
,
065
.
12.
S. H.
Hendi
, “
Asymptotic Reissner–Nordström black holes
,”
Ann. Phys.
333
,
282
289
(
2013
).
13.
R.
Bartnik
and
L.
Simon
, “
Spacelike hypersurfaces with prescribed boundary values and mean curvature
,”
Commun. Math. Phys.
87
,
131
152
(
1982
).
14.
S. I.
Kruglov
, “
Dyonic black holes in framework of Born–Infeld-type electrodynamics
,”
Gen. Relativ. Gravitation
51
,
121
(
2019
).
15.
S. I.
Kruglov
, “
Dyonic and magnetic black holes with nonlinear arcsin-electrodynamics
,”
Ann. Phys.
409
,
167937
(
2019
).
16.
Y.
Yang
, “
Classical solutions in the Born–Infeld theory
,”
Proc. R. Soc. London, Ser. A
456
,
615
640
(
2000
).
17.
Z.
Gao
,
S. B.
Gudnason
, and
Y.
Yang
, “
Integer-squared laws for global vortices in the Born–Infeld wave equations
,”
Ann. Phys.
400
,
303
319
(
2019
).
18.
J.
Byeon
,
N.
Ikoma
,
A.
Malchiodi
, and
L.
Mari
, “
Existence and regularity for prescribed Lorentzian mean curvature hypersurfaces, and the Born-Infeld model
,” arXiv:2112.11283 (
2021
).
19.
D.
Bonheure
,
F.
Colasuonno
, and
J.
Földes
, “
On the Born–Infeld equation for electrostatic fields with a superposition of point charges
,”
Ann. Mat. Pura Appl.
198
,
749
772
(
2019
).
20.
D.
Bonheure
,
P.
d’Avenia
, and
A.
Pomponio
, “
On the electrostatic Born–Infeld equation with extended charges
,”
Commun. Math. Phys.
346
,
877
906
(
2016
).
21.
D.
Bonheure
and
A.
Iacopetti
, “
On the regularity of the minimizer of the electrostatic Born–Infeld energy
,”
Arch. Ration. Mech. Anal.
232
,
697
725
(
2019
).
22.
D.
Bonheure
and
A.
Iacopetti
, “
A sharp gradient estimate and W2,q regularity for the prescribed mean curvature equation in the Lorentz-Minkowski space
,”
Arch. Ration. Mech. Anal.
247
,
87
(
2023
).
23.
A.
Haarala
, “
The electrostatic Born-Infeld equations with integrable charge densities
,” arXiv:2006.08208v2 (
2021
).
24.
Y.
Yang
, “
Electromagnetic asymmetry, relegation of curvature singularities of charged black holes, and cosmological equations of state in view of the Born–Infeld theory
,”
Classical Quantum Gravity
39
,
195007
(
2022
).
25.
A.
Azzollini
, “
On a prescribed mean curvature equation in Lorentz–Minkowski space
,”
J. Math. Pures Appl.
106
,
1122
1140
(
2016
).
26.
D.
Bonheure
and
A.
Iacopetti
, “
Spacelike radial graphs of prescribed mean curvature in the Lorentz–Minkowski space
,”
Anal. PDE
12
,
1805
1842
(
2019
).
27.
M. K.-H.
Kiessling
, “
On the quasi-linear elliptic PDE (u/1|u|2)=4πΣkakδsk in physics and geometry
,”
Commun. Math. Phys.
314
,
509
523
(
2012
);
M. K.-H.
Kiessling
,
Commun. Math. Phys.
364
,
825
833
(
2018
).
28.
C. V.
Coffman
and
W. K.
Ziemer
, “
A prescribed mean curvature problem on domains without radial symmetry
,”
SIAM J. Math. Anal.
22
,
982
990
(
1991
).
29.
F.
Obersnel
and
P.
Omari
, “
On a result of C.V. Coffman and W.K. Ziemer about the prescribed mean curvature equation
,”
Discrete Contin. Dyn. Syst. A
31
,
1138
1147
(
2011
).
30.
I.
Bialynicki-Birula
, “
Nonlinear electrodynamics: Variations on a theme by Born and Infeld
,” in
Quantum Theory of Particles and Fields, Festschrift of J. Lopuszanski
, edited by
B.
Jancewicz
and
J.
Lukierski
(
World Scientific
,
Singapore
,
1983
), pp.
31
42
.
31.
A. A.
Tseytlin
, “
Born-Infeld action, supersymmetry and string theory
,” in
The Many Faces of the Superworld
(
World Scientific
,
Singapore
,
2000
), pp.
417
452
.
32.
J.
Beltrán Jiménez
,
L.
Heisenberg
,
G. J.
Olmo
, and
D.
Rubiera–Garcia
, “
Born–Infeld inspired modifications of gravity
,”
Phys. Rep.
727
,
1
129
(
2018
).
33.
B.
Dai
and
R.
Zhang
, “
Ground state solutions for the nonlinear problem involving exponential form of Born-Infeld-like
,”
Differ. Integr. Equations
(
2022
) (unpublished).
34.
S.
Lorca
and
P.
Ubilla
, “
Partial differential equations involving subcritical, critical and supercritical nonlinearities
,”
Nonlinear Anal.: Theory Methods Appl.
56
,
119
131
(
2004
).
You do not currently have access to this content.