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

Positive solutions for Robin problems with general potential and logistic reaction

Shouchuan Hu 1, and  Nikolaos S. Papageorgiou 2,

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.
Keywords: Robin problem., Indefinite and unbounded potential, semilinear equation, positive solution, superdiffusive reaction, bifurcation-type theorem.
Mathematics Subject Classification: 35J20, 35J60, 58E0.
Citation: Shouchuan Hu, Nikolaos S. Papageorgiou. Positive solutions for Robin problems with general potential and logistic reaction. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2489-2507. doi: 10.3934/cpaa.2016046
References:
[1]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Degree Theory for Operators of Monotone Type and Nonlinear Elliptic Equations with Inequality Constraints, Memoirs, AMS, vol. 196, no. 905, 2008. doi: 10.1090/memo/0915.

[2]

G. Barletta, R. Livrea and N. S. Papageorgiou, A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian, Comm. Pure. Appl. Anal., 13 (2014), 1075-1086. doi: 10.3934/cpaa.2014.13.1075.

[3]

T, Cardinali, N. S. Papageorgiou and P. Rubbioni, Bifurcation phenomena for nonlinear superdiffusive Neumann equations of logistic type, Ann. Mat. Pura Appl., 193 (2013), 1-21. doi: 10.1007/s10231-012-0263-0.

[4]

A. Castro, J. Cossio and C. Vélez, Existence and qualitative properties of solutions for nonlinear Dirichlet problems, Discrete Contin. Dynam. Systems, 33 (2013), 123-140.

[5]

G. Dai and R. Ma, Unilateral global bifurcation for p-Laplacian with non-p-1-linearization nonlinearity, Discrete Contin. Dynam. Systems, 35 (2015), 99-116. doi: 10.3934/dcds.2015.35.99.

[6]

W. Dong and J. T. Chen, Existence and multiplicity results for a degenerate elliptic equation, Acta Math. Sinica, 22 (2006), 665-670. doi: 10.1007/s10114-005-0696-0.

[7]

M. Filippakis, A. Kristaly and N. S. Papageorgiou, Existence of five nonzero solutions with exact sign for a $p$-Laplacian equation, Discrete Contin. Dynam. Systems, 24 (2009), 405-440. doi: 10.3934/dcds.2009.24.405.

[8]

M. Filippakis, D. O'Regan and N. S. Papageorgiou, Positive solutions and bifurcation phenomena for nonlinear elliptic equations of logistic type: the superdiffusive case, Comm. Pure. Appl. Anal., 9 (2010), 1507-1527. doi: 10.3934/cpaa.2010.9.1507.

[9]

L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Chapman & Hall/CRC, Boca Raton, 2006.

[10]

L. Gasinski and N. S. Papageorgiou, Dirichlet $(p,q)$-equations at resonance, Discrete Contin. Dynam. Systems, 34 (2014), 2037-2060.

[11]

M. F. Gurtin and R. C. MacComy, On the diffusion of biological population, Math. Biosci., 33 (1977), 35-49.

[12]

Shouchuan Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis Volume I: Theory, Kluwer Academic Publishers, Dordrecht, the Netherlands, 1997. doi: 10.1007/978-1-4615-6359-4.

[13]

K. M. Hui and S. Kim, Existence of Neumann and singular solutions of the fast diffusion equation, Discrete Contin. Dynam. Systems, 35 (2015), 4859-4887. doi: 10.3934/dcds.2015.35.4859.

[14]

S. Kyritsi and N. S. Papageorgiou, A bifurcation-type result for nonlinear Neumann eigenvalue problems, Funkcialaj Ekvacioj, 55 (2012), 1-15. doi: 10.1619/fesi.55.1.

[15]

S. Kyritsi and N. S. Papageorgiou, Multiple solutions for superlinear Dirichlet problems with an indefinite potential, Annali Math. Pura Appl., 192 (2013), 297-315. doi: 10.1007/s10231-011-0224-z.

[16]

S. A. Marano and S. J. N. Mosconi, Multiple solutions to elliptic inclusions via critical point theory on closed convex sets, Discrete Contin. Dynam. Systems, 35 (2015), 3087-3102. doi: 10.3934/dcds.2015.35.3087.

[17]

N. S. Papageorgiou and F. Papalini, Seven solutions for superlinear Dirichlet problems with sign information for sublinear equations with indefinite and unbounded potential and no symmetries, Israel J. Math., 201 (2014), 761-796. doi: 10.1007/s11856-014-1050-y.

[18]

N. S. Papageorgiou and V. D. Radulescu, Semilinear Neumann problems with indefinite and unbounded potential and crossing nonlinearity, Contemp. Math., 595 (2013), 293-315. doi: 10.1090/conm/595/11801.

[19]

N. S. Papageorgiou and V. D. Radulescu, Robin problems with indefinite unbounded potential and reaction of arbitrary growth, Revista Mat. Complutense, 19 (2016), 91-126. doi: 10.1007/s13163-015-0181-y.

[20]

N. S. Papageorgiou and V. D. Radulescu, Multiple solutions with precise sign information for parametric Robin problems, J. Diff. Equas., 256 (2014), 2449-2479. doi: 10.1016/j.jde.2014.01.010.

[21]

N. S. Papageorgiou and V. D. Radulescu, Multiplicity of solutions for resonant Neumann problems with an indefinite and unbounded potential, Trans. Amer. Math. Soc., 367 (2015), 8723-8756. doi: 10.1090/S0002-9947-2014-06518-5.

[22]

N. S. Papageorgiou and V. D. Radulescu, Bifurcation of positive solutions for nonlinear nonhomogeneous Robin and Neumann problems with competing nonlinearities, Discrete Contin. Dynam. Systems, 35 (2015), 5003-5036 doi: 10.3934/dcds.2015.35.5003.

[23]

N. S. Papageorgiou and G. Smyrlis, On a class of parametric Neumann problems with indefinite and unbounded potential, Forum Math., doi: 101515/forum-2-12-0042. doi: 10.1515/forum-2012-0042.

[24]

P. Poláčik, On the multiplicity of nonnegative solutions with a nontrivial nodal set for elliptic equations on symmetric domains, Discrete Contin. Dynam. Systems, 34 (2014), 2657-2667. doi: 10.3934/dcds.2014.34.2657.

[25]

P. Sacks and M. Warma, Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1$-data, Discrete Contin. Dynam. Systems, 34 (2014), 761-787.

[26]

S. Takeuchi, Positive solutions of a degenerate elliptic equation with a logistic reaction, Proc. Amer. Math. Soc., 129 (2001), 433-441. doi: 10.1090/S0002-9939-00-05723-3.

[27]

S. Takeuchi, Multiplicity result for a degenerate elliptic equation with a logistic reaction, J. Diff. Equas., 173 (2001), 138-144. doi: 10.1006/jdeq.2000.3914.

[28]

X. Wang, Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Diff. Equas., 93 (1991), 283-310. doi: 10.1016/0022-0396(91)90014-Z.

[29]

X. Yu, Liouville type theorem for nonlinear elliptic equation with general nonlinearity, Discrete Contin. Dynam. Systems, 34 (2014), 4947-4966. doi: 10.3934/dcds.2014.34.4947.

show all references

References:
[1]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Degree Theory for Operators of Monotone Type and Nonlinear Elliptic Equations with Inequality Constraints, Memoirs, AMS, vol. 196, no. 905, 2008. doi: 10.1090/memo/0915.

[2]

G. Barletta, R. Livrea and N. S. Papageorgiou, A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian, Comm. Pure. Appl. Anal., 13 (2014), 1075-1086. doi: 10.3934/cpaa.2014.13.1075.

[3]

T, Cardinali, N. S. Papageorgiou and P. Rubbioni, Bifurcation phenomena for nonlinear superdiffusive Neumann equations of logistic type, Ann. Mat. Pura Appl., 193 (2013), 1-21. doi: 10.1007/s10231-012-0263-0.

[4]

A. Castro, J. Cossio and C. Vélez, Existence and qualitative properties of solutions for nonlinear Dirichlet problems, Discrete Contin. Dynam. Systems, 33 (2013), 123-140.

[5]

G. Dai and R. Ma, Unilateral global bifurcation for p-Laplacian with non-p-1-linearization nonlinearity, Discrete Contin. Dynam. Systems, 35 (2015), 99-116. doi: 10.3934/dcds.2015.35.99.

[6]

W. Dong and J. T. Chen, Existence and multiplicity results for a degenerate elliptic equation, Acta Math. Sinica, 22 (2006), 665-670. doi: 10.1007/s10114-005-0696-0.

[7]

M. Filippakis, A. Kristaly and N. S. Papageorgiou, Existence of five nonzero solutions with exact sign for a $p$-Laplacian equation, Discrete Contin. Dynam. Systems, 24 (2009), 405-440. doi: 10.3934/dcds.2009.24.405.

[8]

M. Filippakis, D. O'Regan and N. S. Papageorgiou, Positive solutions and bifurcation phenomena for nonlinear elliptic equations of logistic type: the superdiffusive case, Comm. Pure. Appl. Anal., 9 (2010), 1507-1527. doi: 10.3934/cpaa.2010.9.1507.

[9]

L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Chapman & Hall/CRC, Boca Raton, 2006.

[10]

L. Gasinski and N. S. Papageorgiou, Dirichlet $(p,q)$-equations at resonance, Discrete Contin. Dynam. Systems, 34 (2014), 2037-2060.

[11]

M. F. Gurtin and R. C. MacComy, On the diffusion of biological population, Math. Biosci., 33 (1977), 35-49.

[12]

Shouchuan Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis Volume I: Theory, Kluwer Academic Publishers, Dordrecht, the Netherlands, 1997. doi: 10.1007/978-1-4615-6359-4.

[13]

K. M. Hui and S. Kim, Existence of Neumann and singular solutions of the fast diffusion equation, Discrete Contin. Dynam. Systems, 35 (2015), 4859-4887. doi: 10.3934/dcds.2015.35.4859.

[14]

S. Kyritsi and N. S. Papageorgiou, A bifurcation-type result for nonlinear Neumann eigenvalue problems, Funkcialaj Ekvacioj, 55 (2012), 1-15. doi: 10.1619/fesi.55.1.

[15]

S. Kyritsi and N. S. Papageorgiou, Multiple solutions for superlinear Dirichlet problems with an indefinite potential, Annali Math. Pura Appl., 192 (2013), 297-315. doi: 10.1007/s10231-011-0224-z.

[16]

S. A. Marano and S. J. N. Mosconi, Multiple solutions to elliptic inclusions via critical point theory on closed convex sets, Discrete Contin. Dynam. Systems, 35 (2015), 3087-3102. doi: 10.3934/dcds.2015.35.3087.

[17]

N. S. Papageorgiou and F. Papalini, Seven solutions for superlinear Dirichlet problems with sign information for sublinear equations with indefinite and unbounded potential and no symmetries, Israel J. Math., 201 (2014), 761-796. doi: 10.1007/s11856-014-1050-y.

[18]

N. S. Papageorgiou and V. D. Radulescu, Semilinear Neumann problems with indefinite and unbounded potential and crossing nonlinearity, Contemp. Math., 595 (2013), 293-315. doi: 10.1090/conm/595/11801.

[19]

N. S. Papageorgiou and V. D. Radulescu, Robin problems with indefinite unbounded potential and reaction of arbitrary growth, Revista Mat. Complutense, 19 (2016), 91-126. doi: 10.1007/s13163-015-0181-y.

[20]

N. S. Papageorgiou and V. D. Radulescu, Multiple solutions with precise sign information for parametric Robin problems, J. Diff. Equas., 256 (2014), 2449-2479. doi: 10.1016/j.jde.2014.01.010.

[21]

N. S. Papageorgiou and V. D. Radulescu, Multiplicity of solutions for resonant Neumann problems with an indefinite and unbounded potential, Trans. Amer. Math. Soc., 367 (2015), 8723-8756. doi: 10.1090/S0002-9947-2014-06518-5.

[22]

N. S. Papageorgiou and V. D. Radulescu, Bifurcation of positive solutions for nonlinear nonhomogeneous Robin and Neumann problems with competing nonlinearities, Discrete Contin. Dynam. Systems, 35 (2015), 5003-5036 doi: 10.3934/dcds.2015.35.5003.

[23]

N. S. Papageorgiou and G. Smyrlis, On a class of parametric Neumann problems with indefinite and unbounded potential, Forum Math., doi: 101515/forum-2-12-0042. doi: 10.1515/forum-2012-0042.

[24]

P. Poláčik, On the multiplicity of nonnegative solutions with a nontrivial nodal set for elliptic equations on symmetric domains, Discrete Contin. Dynam. Systems, 34 (2014), 2657-2667. doi: 10.3934/dcds.2014.34.2657.

[25]

P. Sacks and M. Warma, Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1$-data, Discrete Contin. Dynam. Systems, 34 (2014), 761-787.

[26]

S. Takeuchi, Positive solutions of a degenerate elliptic equation with a logistic reaction, Proc. Amer. Math. Soc., 129 (2001), 433-441. doi: 10.1090/S0002-9939-00-05723-3.

[27]

S. Takeuchi, Multiplicity result for a degenerate elliptic equation with a logistic reaction, J. Diff. Equas., 173 (2001), 138-144. doi: 10.1006/jdeq.2000.3914.

[28]

X. Wang, Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Diff. Equas., 93 (1991), 283-310. doi: 10.1016/0022-0396(91)90014-Z.

[29]

X. Yu, Liouville type theorem for nonlinear elliptic equation with general nonlinearity, Discrete Contin. Dynam. Systems, 34 (2014), 4947-4966. doi: 10.3934/dcds.2014.34.4947.

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