Positive solutions of semipositone elliptic problems with critical Trudinger-Moser nonlinearities
Keywords
Semipositone $N$-Laplacian problems, critical Trudinger-Moser nonlinearities, positive solutions, uniform $C^{1, \alpha}$ a priori estimatesAbstract
We prove the existence of a positive solution to a semipositone $N$-Laplacian problem with a critical Trudinger-Moser nonlinearity. The proof is based on obtaining uniform $C^{1,\alpha}$ a priori estimates via a compactness argument. Our result is new even in the semilinear case $N = 2$, and our arguments can easily be adapted to obtain positive solutions of more general semipositone problems with critical Trudinger-Moser nonlinearities.References
Adimurthi,Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the n-Laplacian, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1990), no. 3, 393–413.
D.G. de Figueiredo, J.M. do Ó and B. Ruf, Elliptic equations and systems with critical Trudinger–Moser nonlinearities, Discrete Contin. Dyn. Syst. 30 (2011), no. 2, 455–476.
D.G. de Figueiredo, O.H. Miyagaki and B. Ruf, Elliptic equations in R2 with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations 3 (1995), no. 2, 139–153.
D.G. de Figueiredo, O.H. Miyagaki and B. Ruf, Corrigendum: “Elliptic equations in R2 with nonlinearities in the critical growth range”, Calc. Var. Partial Differential Equations 4 (1996), no. 2, p. 203.
J.M.B. do Ó, Semilinear Dirichlet problems for the N -Laplacian in RN with nonlinearities in the critical growth range, Differential Integral Equations 9 (1996), no. 5, 967–979.
M. Guedda and L. Véron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal. 13 (1989), no. 8, 879–902.
G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), no. 11, 1203–1219.
P.-L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 (1982), no. 4, 441–467.
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case I, Rev. Mat. Iberoam. 1 (1985), no. 1, 145–201.
J. Moser A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970/1971), 1077–1092.
N.S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–483.
J.L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), no. 3, 191–202.
Y. Yang and K. Perera, N -Laplacian problems with critical Trudinger–Moser nonlinearities, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 16 (2016), no. 4, 1123–1138.
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