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Existence of solution for elliptic equations with supercritical Trudinger–Moser growth

Part of: Elliptic equations and systems Infinite-dimensional dissipative dynamical systems Partial differential equations

Published online by Cambridge University Press:  16 February 2021

Luiz F. O. Faria  and
Marcelo Montenegro
Luiz F. O. Faria
Affiliation:
Departamento de Matemática, Universidade Federal de Juiz de Fora, ICE, Campus Universitário, Rua José Lourenço Kelmer, s/n, Juiz de Fora, MG, CEP 36036-900, Brazil ([email protected])
Marcelo Montenegro
Affiliation:
Departamento de Matemática, Universidade Estadual de Campinas, IMECC, Rua Sérgio Buarque de Holanda, 651, Campinas, SP, CEP 13083-859, Brazil ([email protected])
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Abstract

This paper is concerned with the existence of solutions for a class of elliptic equations on the unit ball with zero Dirichlet boundary condition. The nonlinearity is supercritical in the sense of Trudinger–Moser. Using a suitable approximating scheme we obtain the existence of at least one positive solution.

Keywords

Elliptic equationpositive solutionsupercritical exponential growth

MSC classification

Primary: 35J25: Boundary value problems for second-order elliptic equations 35B33: Critical exponents
Secondary: 37L65: Special approximation methods (nonlinear Galerkin, etc.) 35B09: Positive solutions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 2 , April 2022 , pp. 291 - 310
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Copyright
Copyright © The Author(s) 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Adams, D. R.. A sharp inequality of J. Moser for higher order derivatives. Ann. Math. 128 (1988), 385–398.CrossRefGoogle Scholar
Adimurthi, and Druet, O.. Blow-up analysis in dimension 2 and a sharp form of Trudinger–Moser inequality. Commun. Partial Differ. Equ. 29 (2004), 295–322.CrossRefGoogle Scholar
Adimurthi. Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the n–Laplacian. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17 (1990), 393–413.Google Scholar
Atkinson, F. V. and Peletier, L. A.. Elliptic equations with critical growth. Math. Inst. Univ. Leiden, Rep. 21, 1986.Google Scholar
Bianchi, G., Chabrowski, J. and Szulkin, A.. On symmetric solutions of an elliptic equation with nonlinearity involving critical Sobolev exponent. Nonlinear Anal. TMA. 25 (1995), 4–59.CrossRefGoogle Scholar
Brezis, H. and Oswald, L.. Remarks on sublinear elliptic equations. Nonlinear Anal. TMA. 10 ( 1986), 55–64.CrossRefGoogle Scholar
Calanchi, M. and Ruf, B.. On Trudinger–Moser type inequalities with logarithmic weights. J. Differ. Equ. 258 (2015), 1967–1989.CrossRefGoogle Scholar
de Araujo, A. L. A. and Faria, L. F. O.. Positive solutions of quasilinear elliptic equations with exponential nonlinearity combined with convection term. J. Differ. Equ. 267 (2019), 4589–4608.CrossRefGoogle Scholar
de Araujo, A. L. A., Faria, L. F. O. and Melo, J. M.. Positive solutions of nonlinear elliptic equations involving supercritial Sobolev exponents without Ambrosetti and Rabinowitz conditions. Calc. Var. Partial Differ. Equ., accepted for publication (2020).CrossRefGoogle Scholar
de Araujo, A. L. A. and Montenegro, M.. Existence of solution for a general class of elliptic equations with exponential growth. Ann. Mat. Pura Appl. 195 (2016), 1737–1748.CrossRefGoogle Scholar
de Araujo, A. L. A. and Montenegro, M.. Existence of solution for a nonvariational elliptic system with exponential growth in dimension two. J. Differ. Equ. 264 (2018), 2270–2286.CrossRefGoogle Scholar
Carrião, P. C., Gonçalves, J. V. and Miyagaki, O. H.. Existence and nonexistence in a class of equations with supercritical growth. Appl. Anal. 74 (2000), 275–287.CrossRefGoogle Scholar
de Figueiredo, D. G., do Ó, J. M. and Ruf, B.. On an inequality by N. Trudinger and J. Moser and related elliptic equations. Commun. Pure Appl. Math. 55 (2002), 135–152.CrossRefGoogle Scholar
de Figueiredo, D. G., Miyagaki, O. H. and Ruf, B.. Elliptic equations in $\mathbb {R}^2$ with nonlinearities in the critical growth range. Calc. Var. Partial Differ. Equ. 3 (1995), 139–153.CrossRefGoogle Scholar
de Freitas, L. R.. Multiplicity of solutions for a class of quasilinear equations with exponential critical growth. Nonlinear Anal. TMA. 95 (2014), 607–624.CrossRefGoogle Scholar
Diening, L., Harjulehto, P., Hasto, P. and Ruž ǐka, M.. Lebesgue and Sobolev spaces with variable exponents. Lecture Notes in Mathematics 2017 (2011.)CrossRefGoogle Scholar
Gilbarg, D. and Trudinger, N. S.. Elliptic Partial Differential Equations of Second Order. Springer-Verlag (2001).CrossRefGoogle Scholar
Hewitt, E. and Stromberg, K.. Real and abstract analysis (Berlin: Springer-Verlag, 1975).Google Scholar
Kajikiya, R.. Comparison theorem and uniqueness of positive solutions for sublinear elliptic equations. Arch. Math. 91 (2008), 427–435.CrossRefGoogle Scholar
Kesavan, S.. Topics in functional analysis and applications (India: John Wiley and Sons, 1989).Google Scholar
Ladyzhenskaya, O. A. and Ural'tseva, N. N.. Linear and quasilinear elliptic equations (New York, London: Academic Press, 1968).Google Scholar
Lam, N. and Lu, G.. Elliptic equations and systems with subcritical and critical exponential growth without the Ambrosetti–Rabinowitz condition. J. Geom. Anal. 24 (2014), 118–143.CrossRefGoogle Scholar
Moser, J.. A sharp form of an inequality by Trudinger. Indiana Univ. Math. J. 20 (1971), 1077–1092.CrossRefGoogle Scholar
Ngô, Q. A. and Nguyen, V. H.. Supercritical Moser–Trudinger inequalities and related elliptic problems. Calc. Var. Partial Differ. Equ. 59 (2020), 69.CrossRefGoogle Scholar
Ngô, Q. A. and Nguyen, V. H.. A supercritical Sobolev type inequality in higher order Sobolev spaces and related higher order elliptic problems. J. Differ. Equ. 268 (2020), 5996–6032.CrossRefGoogle Scholar
Pohožaev, S. I.. The Sobolev embedding in the case pl = n. Proc. of the Technical Scientific Conference on Advances of Scientific Research 1964–1965, Mathematics Section (Moskov. Energet. Inst., Moscow), pp. 158–170 (1965).Google Scholar
Silva, E. A. B. and Soares, S. H. M.. Liouville–Gelfand type problems for the $N$-Laplacian on bounded domains of $\mathbb {R}^n$. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 (1999), 1–30.Google Scholar
Strauss, W. A.. On weak solutions of semi-linear hyperbolic equations. An. Acad. Brasil. Ci. 42 (1970), 645–651.Google Scholar
Trudinger, N. S.. On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17 (1967), 473–483.Google Scholar
Yudovič, V. I.. Some estimates connected with integral operators and with solutions of elliptic equations. Dokl. Akad. Nauk SSSR. 138 (1961), 805–808.Google Scholar