×

Some existence results for elliptic systems with exponential nonlinearities and convection terms in dimension two. (English) Zbl 1512.35250

Summary: In this paper, we establish the existence of solutions to a class of elliptic systems. The nonlinearities include exponential growth terms and convection terms. The exponential growth term means it could be critical growth at \(\infty \). The Trudinger-Moser inequality is used to deal with it. The convection term means it involves the gradient of unknown function. The strong convergence of sequences is employed to overcome the difficulties caused by convection terms. The variational methods are invalid and the Galerkin method and an approximation scheme are applied to obtain four different solutions. Our results supplements those from [Anderson L. A. de Araujo and M. Montenegro, J. Differ. Equations 264, No. 3, 2270–2286 (2018; Zbl 1388.35074)].

MSC:

35J57 Boundary value problems for second-order elliptic systems
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J61 Semilinear elliptic equations
47H11 Degree theory for nonlinear operators

Citations:

Zbl 1388.35074
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] C.O. Alves and G.M. Figueiredo, Existence of positive solution for a planar Schrodinger-Poisson system with exponential growth, J. Math. Phys. 60 (2019), no. 1, 011503, 13. · Zbl 1410.35194
[2] H. Amann and M.G. Crandall, On some existence theorems for semilinear elliptic equations, Indiana Univ. Math. J. 27 (1978), no. 5, 779-790. · Zbl 0391.35030
[3] A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994), no. 2, 519-543. · Zbl 0805.35028
[4] H. Brezis and R.E.L. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations 2 (1977), no. 6, 601-614. · Zbl 0358.35032
[5] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. · Zbl 1220.46002
[6] H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal. 10 (1986), no. 1, 55-64. · Zbl 0593.35045
[7] D.M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in \(\mathbb R^2\), Comm. Partial Differential Equations 17 (1992) no. 3-4, 407-435. · Zbl 0763.35034
[8] D. Cassani and C. Tarsi, Existence of solitary waves for supercritical Schrodinger systems in dimension two, Calc. Var. Partial Differential Equations 54 (2015), no. 2, 1673-1704. · Zbl 1327.35099
[9] A.L.A. de Araujo and L.F.O. Faria, Positive solutions of quasilinear elliptic equations with exponential nonlinearity combined with convection term, J. Differential Equations 267 (2019), no. 8, 4589-4608. · Zbl 1421.35168
[10] A.L.A. de Araujo and M. Montenegro, Existence of solution for a general class of elliptic equations with exponential growth, Ann. Mat. Pura Appl. (4) 195 (2016), no. 5, 1737-1748. · Zbl 1454.35187
[11] 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), no. 3, 2270-2286. · Zbl 1388.35074
[12] D.G. de Figueiredo, O.H. Miyagaki and B. Ruf, Elliptic equations in \(\mathbb R^2\) with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations 3 (1995), no. 2, 139-153. · Zbl 0820.35060
[13] D.G. de Figueiredo, O.H. Miyagaki and B. Ruf, Corrigendum: “Elliptic equations in \(\mathbb R^2\) with nonlinearities in the critical growth range”, Calc. Var. Partial Differential Equations 4 (1996), no. 2, 203. · Zbl 0847.35048
[14] L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, second edition, 2010. · Zbl 1194.35001
[15] F. Faraci, D. Motreanu and D. Puglisi, Positive solutions of quasilinear elliptic equations with dependence on the gradient, Calc. Var. Partial Differential Equations 54 (2015), no. 1, 525-538. · Zbl 1326.35159
[16] L.F.O. Faria, O.H. Miyagaki and D. Motreanu, Comparison and positive solutions for problems with the \((p, q)\)-Laplacian and a convection term, Proc. Edinb. Math. Soc. (2) 57 (2014), no. 3, 687-698. · Zbl 1315.35114
[17] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, reprint of the 1998 edition. · Zbl 1042.35002
[18] D. Guo, Nonlinear Functional Analysis, Modern Mathematical Foundation, Higher Education Press, Beijing, third edition, 2015.
[19] G. Jiang, Y. Liu and Z. Liu, Transition between nonlinear and linear eigenvalue problems, J. Differential Equations 269 (2020), no. 12, 10919-10936. · Zbl 1450.35206
[20] S. Kesavan, Topics in Functional Analysis and Applications, John Wiley and Sons, IncNew York, 1989. · Zbl 0666.46001
[21] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970/1971), 1077-1092. · Zbl 0203.43701
[22] N.S. Papageorgiou, V.D. Radulescu and D.D. Repovs, Nonlinear analysis-theory and methods, Springer Monographs in Mathematics, Springer, Cham, 2019. · Zbl 1414.46003
[23] N.S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473-483. · Zbl 0163.36402
[24] M. Willem, Functional Analysis, Fundamentals and Applications, Cornerstones, Birkhauser-Springer, New York, 2013. · Zbl 1284.46001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.