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## On a class of semipositone problems with singular Trudinger-Moser nonlinearities

 Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA

* Corresponding author: Kanishka Perera

Received  December 2019 Revised  April 2020 Published  November 2020

We prove the existence of positive solutions for a class of semipositone problems with singular Trudinger-Moser nonlinearities. The proof is based on compactness and regularity arguments.

Citation: Shiqiu Fu, Kanishka Perera. On a class of semipositone problems with singular Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020452
##### References:
 [1] Ad imurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications, NoDEA Nonlinear Differential Equations Appl., 13 (2007), 585-603.  doi: 10.1007/s00030-006-4025-9.  Google Scholar [2] I. Ali, A. Castro and R. Shivaji, Uniqueness and stability of nonnegative solutions for semipositone problems in a ball, Proc. Amer. Math. Soc., 117 (1993), 775-782.  doi: 10.1090/S0002-9939-1993-1116249-5.  Google Scholar [3] A. Ambrosetti, D. Arcoya and B. Buffoni, Positive solutions for some semi-positone problems via bifurcation theory, Differential Integral Equations, 7 (1994), 655-663.   Google Scholar [4] A. Castro, D. G. de Figueredo and E. Lopera, Existence of positive solutions for a semipositone $p$-Laplacian problem, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 475-482.  doi: 10.1017/S0308210515000657.  Google Scholar [5] A. Castro and R. Shivaji, Nonnegative solutions for a class of nonpositone problems, Proc. Roy. Soc. Edinburgh Sect. A, 108 (1988), 291-302.  doi: 10.1017/S0308210500014670.  Google Scholar [6] M. Chhetri, P. Drábek and R. Shivaji, Existence of positive solutions for a class of $p$-Laplacian superlinear semipositone problems, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 925-936.  doi: 10.1017/S0308210515000220.  Google Scholar [7] D. G. Costa, H. Ramos Quoirin and H. Tehrani, A variational approach to superlinear semipositone elliptic problems, Proc. Amer. Math. Soc., 145 (2017), 2661-2675.  doi: 10.1090/proc/13426.  Google Scholar

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##### References:
 [1] Ad imurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications, NoDEA Nonlinear Differential Equations Appl., 13 (2007), 585-603.  doi: 10.1007/s00030-006-4025-9.  Google Scholar [2] I. Ali, A. Castro and R. Shivaji, Uniqueness and stability of nonnegative solutions for semipositone problems in a ball, Proc. Amer. Math. Soc., 117 (1993), 775-782.  doi: 10.1090/S0002-9939-1993-1116249-5.  Google Scholar [3] A. Ambrosetti, D. Arcoya and B. Buffoni, Positive solutions for some semi-positone problems via bifurcation theory, Differential Integral Equations, 7 (1994), 655-663.   Google Scholar [4] A. Castro, D. G. de Figueredo and E. Lopera, Existence of positive solutions for a semipositone $p$-Laplacian problem, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 475-482.  doi: 10.1017/S0308210515000657.  Google Scholar [5] A. Castro and R. Shivaji, Nonnegative solutions for a class of nonpositone problems, Proc. Roy. Soc. Edinburgh Sect. A, 108 (1988), 291-302.  doi: 10.1017/S0308210500014670.  Google Scholar [6] M. Chhetri, P. Drábek and R. Shivaji, Existence of positive solutions for a class of $p$-Laplacian superlinear semipositone problems, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 925-936.  doi: 10.1017/S0308210515000220.  Google Scholar [7] D. G. Costa, H. Ramos Quoirin and H. Tehrani, A variational approach to superlinear semipositone elliptic problems, Proc. Amer. Math. Soc., 145 (2017), 2661-2675.  doi: 10.1090/proc/13426.  Google Scholar
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