November  2016, 15(6): 2489-2507. doi: 10.3934/cpaa.2016046

Positive solutions for Robin problems with general potential and logistic reaction

1. 

Department of Mathematics, Missouri State University, Spring eld, MO 65804

2. 

Department of Mathematics, National Technical University, Zografou Campus, Athens 15780

Received  February 2016 Revised  August 2016 Published  September 2016

We consider a semilinear Robin problem driven by the negative Laplacian plus an indefinite and unbounded potential and a superdiffusive lotistic-type reaction. We prove bifurcation results describing the dependence of the set of positive solutions on the parameter of the problem. We also establish the existence of extreme positive solutions and determine their properties.
Citation: Shouchuan Hu, Nikolaos S. Papageorgiou. Positive solutions for Robin problems with general potential and logistic reaction. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2489-2507. doi: 10.3934/cpaa.2016046
References:
[1]

Memoirs, AMS, vol. 196, no. 905, 2008. doi: 10.1090/memo/0915.  Google Scholar

[2]

Comm. Pure. Appl. Anal., 13 (2014), 1075-1086. doi: 10.3934/cpaa.2014.13.1075.  Google Scholar

[3]

Ann. Mat. Pura Appl., 193 (2013), 1-21. doi: 10.1007/s10231-012-0263-0.  Google Scholar

[4]

Discrete Contin. Dynam. Systems, 33 (2013), 123-140.  Google Scholar

[5]

Discrete Contin. Dynam. Systems, 35 (2015), 99-116. doi: 10.3934/dcds.2015.35.99.  Google Scholar

[6]

Acta Math. Sinica, 22 (2006), 665-670. doi: 10.1007/s10114-005-0696-0.  Google Scholar

[7]

Discrete Contin. Dynam. Systems, 24 (2009), 405-440. doi: 10.3934/dcds.2009.24.405.  Google Scholar

[8]

Comm. Pure. Appl. Anal., 9 (2010), 1507-1527. doi: 10.3934/cpaa.2010.9.1507.  Google Scholar

[9]

Chapman & Hall/CRC, Boca Raton, 2006.  Google Scholar

[10]

Discrete Contin. Dynam. Systems, 34 (2014), 2037-2060.  Google Scholar

[11]

Math. Biosci., 33 (1977), 35-49.  Google Scholar

[12]

Kluwer Academic Publishers, Dordrecht, the Netherlands, 1997. doi: 10.1007/978-1-4615-6359-4.  Google Scholar

[13]

Discrete Contin. Dynam. Systems, 35 (2015), 4859-4887. doi: 10.3934/dcds.2015.35.4859.  Google Scholar

[14]

Funkcialaj Ekvacioj, 55 (2012), 1-15. doi: 10.1619/fesi.55.1.  Google Scholar

[15]

Annali Math. Pura Appl., 192 (2013), 297-315. doi: 10.1007/s10231-011-0224-z.  Google Scholar

[16]

Discrete Contin. Dynam. Systems, 35 (2015), 3087-3102. doi: 10.3934/dcds.2015.35.3087.  Google Scholar

[17]

Israel J. Math., 201 (2014), 761-796. doi: 10.1007/s11856-014-1050-y.  Google Scholar

[18]

Contemp. Math., 595 (2013), 293-315. doi: 10.1090/conm/595/11801.  Google Scholar

[19]

Revista Mat. Complutense, 19 (2016), 91-126. doi: 10.1007/s13163-015-0181-y.  Google Scholar

[20]

J. Diff. Equas., 256 (2014), 2449-2479. doi: 10.1016/j.jde.2014.01.010.  Google Scholar

[21]

Trans. Amer. Math. Soc., 367 (2015), 8723-8756. doi: 10.1090/S0002-9947-2014-06518-5.  Google Scholar

[22]

Discrete Contin. Dynam. Systems, 35 (2015), 5003-5036 doi: 10.3934/dcds.2015.35.5003.  Google Scholar

[23]

Forum Math., doi: 101515/forum-2-12-0042. doi: 10.1515/forum-2012-0042.  Google Scholar

[24]

Discrete Contin. Dynam. Systems, 34 (2014), 2657-2667. doi: 10.3934/dcds.2014.34.2657.  Google Scholar

[25]

Discrete Contin. Dynam. Systems, 34 (2014), 761-787.  Google Scholar

[26]

Proc. Amer. Math. Soc., 129 (2001), 433-441. doi: 10.1090/S0002-9939-00-05723-3.  Google Scholar

[27]

J. Diff. Equas., 173 (2001), 138-144. doi: 10.1006/jdeq.2000.3914.  Google Scholar

[28]

J. Diff. Equas., 93 (1991), 283-310. doi: 10.1016/0022-0396(91)90014-Z.  Google Scholar

[29]

Discrete Contin. Dynam. Systems, 34 (2014), 4947-4966. doi: 10.3934/dcds.2014.34.4947.  Google Scholar

show all references

References:
[1]

Memoirs, AMS, vol. 196, no. 905, 2008. doi: 10.1090/memo/0915.  Google Scholar

[2]

Comm. Pure. Appl. Anal., 13 (2014), 1075-1086. doi: 10.3934/cpaa.2014.13.1075.  Google Scholar

[3]

Ann. Mat. Pura Appl., 193 (2013), 1-21. doi: 10.1007/s10231-012-0263-0.  Google Scholar

[4]

Discrete Contin. Dynam. Systems, 33 (2013), 123-140.  Google Scholar

[5]

Discrete Contin. Dynam. Systems, 35 (2015), 99-116. doi: 10.3934/dcds.2015.35.99.  Google Scholar

[6]

Acta Math. Sinica, 22 (2006), 665-670. doi: 10.1007/s10114-005-0696-0.  Google Scholar

[7]

Discrete Contin. Dynam. Systems, 24 (2009), 405-440. doi: 10.3934/dcds.2009.24.405.  Google Scholar

[8]

Comm. Pure. Appl. Anal., 9 (2010), 1507-1527. doi: 10.3934/cpaa.2010.9.1507.  Google Scholar

[9]

Chapman & Hall/CRC, Boca Raton, 2006.  Google Scholar

[10]

Discrete Contin. Dynam. Systems, 34 (2014), 2037-2060.  Google Scholar

[11]

Math. Biosci., 33 (1977), 35-49.  Google Scholar

[12]

Kluwer Academic Publishers, Dordrecht, the Netherlands, 1997. doi: 10.1007/978-1-4615-6359-4.  Google Scholar

[13]

Discrete Contin. Dynam. Systems, 35 (2015), 4859-4887. doi: 10.3934/dcds.2015.35.4859.  Google Scholar

[14]

Funkcialaj Ekvacioj, 55 (2012), 1-15. doi: 10.1619/fesi.55.1.  Google Scholar

[15]

Annali Math. Pura Appl., 192 (2013), 297-315. doi: 10.1007/s10231-011-0224-z.  Google Scholar

[16]

Discrete Contin. Dynam. Systems, 35 (2015), 3087-3102. doi: 10.3934/dcds.2015.35.3087.  Google Scholar

[17]

Israel J. Math., 201 (2014), 761-796. doi: 10.1007/s11856-014-1050-y.  Google Scholar

[18]

Contemp. Math., 595 (2013), 293-315. doi: 10.1090/conm/595/11801.  Google Scholar

[19]

Revista Mat. Complutense, 19 (2016), 91-126. doi: 10.1007/s13163-015-0181-y.  Google Scholar

[20]

J. Diff. Equas., 256 (2014), 2449-2479. doi: 10.1016/j.jde.2014.01.010.  Google Scholar

[21]

Trans. Amer. Math. Soc., 367 (2015), 8723-8756. doi: 10.1090/S0002-9947-2014-06518-5.  Google Scholar

[22]

Discrete Contin. Dynam. Systems, 35 (2015), 5003-5036 doi: 10.3934/dcds.2015.35.5003.  Google Scholar

[23]

Forum Math., doi: 101515/forum-2-12-0042. doi: 10.1515/forum-2012-0042.  Google Scholar

[24]

Discrete Contin. Dynam. Systems, 34 (2014), 2657-2667. doi: 10.3934/dcds.2014.34.2657.  Google Scholar

[25]

Discrete Contin. Dynam. Systems, 34 (2014), 761-787.  Google Scholar

[26]

Proc. Amer. Math. Soc., 129 (2001), 433-441. doi: 10.1090/S0002-9939-00-05723-3.  Google Scholar

[27]

J. Diff. Equas., 173 (2001), 138-144. doi: 10.1006/jdeq.2000.3914.  Google Scholar

[28]

J. Diff. Equas., 93 (1991), 283-310. doi: 10.1016/0022-0396(91)90014-Z.  Google Scholar

[29]

Discrete Contin. Dynam. Systems, 34 (2014), 4947-4966. doi: 10.3934/dcds.2014.34.4947.  Google Scholar

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