Nonlocal elliptic systems of N-Kirchhoff type with exponential growth
DOI:
https://doi.org/10.12775/TMNA.2025.022Keywords
Kirchhoff system, Galerkin approximation, exponential growth, Trudinger-Moser inequalityAbstract
In this work, we are devoted to establishing the existence of positive solutions for a nonlocal elliptic system of $N$-Kirchhoff type on bounded domains in $\mathbb{R}^N$ with $N\geq 2$. The nonlinearity considered in the equation combined a nonlocal term with an exponential term governed by the Trudinger-Moser inequality, which may be subcritical, critical or supercritical. We use the Galerkin approximation together with a variant of the Brouwer Fixed Point Theorem in the product of Sobolev spaces.References
C.O. Alves and D.G. de Figueiredo, Nonvariational Elliptic Systems via Galerkin Methods, Function Spaces, Differential Operators and Nonlinear Analysis: The Hans Triebel Anniversary Volume, 1986.
S. Bernstein, Sur une classe d’equations fonctionnelles aux derives partielles, Izv. Akad. Nauk SSSR, Ser. Mat. 4 (1940), 17–26.
S.M. Boulaaras, R. Guefaifia, B. Cherif and S. Alodhaibi, A new proof of existence of positive weak solutions for sublinear Kirchhoff elliptic systems with multiple parameters, Complexity 2020 (2020), 1–6.
S. Chen, V.D. Rǎdulescu, X. Tang and L. Wen, Planar Kirchhoff equations with critical exponential growth and trapping potential, Math. Z. 302 (2022), 1061–1089.
S. Chen, X.H. Tang and J.Y. Wei, Improved results on planar Kirchhoff-type elliptic problems with critical exponential growth, Z. Angew. Math. Phys. 72 (2021), 1–18.
W. Chen and F. Yu, On a nonhomogeneous Kirchhoff-type elliptic problem with critical exponential in dimension two, Appl. Anal. 101 (2022), 421–436.
A.L.A de Araujo and M. Montenegro, Existence of solution for a nonvariational elliptic system with exponential growth in dimension two, J. Differential Equations 264 (2018), 2270–2286.
D.G. de Figueiredo, J.M. do Ó and B. Ruf, Critical and subcritical elliptic systems in dimension two, Indiana Univ. Math. J. 53 (2005), 1037–1054.
D.G. de Figueiredo, J. M. do Ó and B. Ruf, Non-variational elliptic systems in dimension two: a priori bounds and existence of positive solutions, J. Fixed Point Theory Appl. 4 (2008), 77–96.
S. Deng and X. Tian, Existence and multiplicity of solutions to a Kirchhoff type elliptic system with Trudinger–Moser growth, Results Math. 231 (2022), 1–23.
G.M. Figueiredo and A. Suarez, The sub-supersolution method for Kirchhoff systems: applications, Subject Classification (2000).
G.M. Figueiredo and U.B. Severo, Ground state solution for a Kirchhoff problem with exponential critical growth, Milan J. Math. 84 (2015), 23–39.
S. Fučik, O. John and J. Nečas, On the existence of Schauder bases in Sobolev spaces, Comment. Math. Univ. Carol. 13 (1972), 163–175.
E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer, 1975.
B. Khaldi, Existence of solutions for nonlocal elliptic systems with exponential nonlinearity, Differ. Equ. Appl. 14 (2022), 417–431.
S. Kasavan, Topics in Functional Analysis and Applications, Wiley, New York, 1989.
G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
Y. Leuyacc and S. Soares, On a Hamiltonian system with critical exponential growth, Milan J. Math. 87 (2019), 105–140.
J.L. Lions, On some questions in boundary value problems of mathematical physics, Contemporary Developements in Continuum Mechanics and PDE’s (G.M. de la Penha, L.A. Medeiros, eds.), North-Holland, Amsterdam, 1978.
M. Montenegro, Existence of solutions for some semilinear elliptic systems with singular coefficients, Nonlinear Anal. 2 (2001), 401–415.
T.F. Ma and J.E. M. Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett. 16 (2003), 243–248.
J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970), 1077–1092.
D. Motreanu, V. Motreanu and N.S. Papageorgiou, Multiple constant sign and nodal solutions for Nonlinear Neumann eigenvalue problems, Ann. Sc. Norm. Sup. Pisa Cl. Sci. 5 (2011), 1-27.
S.I. Pohožaev, On a class of quasilinear hyperbolic equations, Math. Sb. 25 (1975), 145–158.
X. Qian and W. Chao, Existence of positive solutions for nonlocal problems with indefinite nonlinearity, Bound. Value Probl. 40 (2020).
J.A.L. Tordecilla, A planar Kirchhoff equation with exponential growth and double nonlocal term, J. Math. Anal. Appl. 537 (2024), 1–8.
J.A.L. Tordecilla, Positive solution for a singular and nonlocal problem of the N Kirchhoff type, Appl. Anal. 103 (2024), no. 18, 3313–3325.
N.S. Trudinger, On imbeddings into Orlicz spaces and applications, J. Math. Mech. 17 (1967), 473–483.
H. Yang and X. Tang, Sign-changing solutions for planer Kirchhoff type problem with critical exponential growth, J. Geom. Anal. 34 (2024), paper no. 178.
L. Zhang, X. Tang and P. Chen, On the planar Kirchhoff-type problem involving supercritical exponential growth, Adv. Nonlinear Anal. 11 (2022), 1412–1446.
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