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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 & 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, (2008).  doi: 10.1090/memo/0915.  Google Scholar

[2]

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

[3]

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

[4]

A. Castro, J. Cossio and C. Vélez, Existence and qualitative properties of solutions for nonlinear Dirichlet problems,, \emph{Discrete Contin. Dynam. Systems}, 33 (2013), 123.   Google Scholar

[5]

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

[6]

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

[7]

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

[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,, \emph{Comm. Pure. Appl. Anal.}, 9 (2010), 1507.  doi: 10.3934/cpaa.2010.9.1507.  Google Scholar

[9]

L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis,, Chapman & Hall/CRC, (2006).   Google Scholar

[10]

L. Gasinski and N. S. Papageorgiou, Dirichlet $(p,q)$-equations at resonance,, \emph{Discrete Contin. Dynam. Systems}, 34 (2014), 2037.   Google Scholar

[11]

M. F. Gurtin and R. C. MacComy, On the diffusion of biological population,, \emph{Math. Biosci.}, 33 (1977), 35.   Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[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,, \emph{Israel J. Math.}, 201 (2014), 761.  doi: 10.1007/s11856-014-1050-y.  Google Scholar

[18]

N. S. Papageorgiou and V. D. Radulescu, Semilinear Neumann problems with indefinite and unbounded potential and crossing nonlinearity,, \emph{Contemp. Math.}, 595 (2013), 293.  doi: 10.1090/conm/595/11801.  Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

[23]

N. S. Papageorgiou and G. Smyrlis, On a class of parametric Neumann problems with indefinite and unbounded potential,, \emph{Forum Math.}, (1015), 2.  doi: 10.1515/forum-2012-0042.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

S. Takeuchi, Multiplicity result for a degenerate elliptic equation with a logistic reaction,, \emph{J. Diff. Equas.}, 173 (2001), 138.  doi: 10.1006/jdeq.2000.3914.  Google Scholar

[28]

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

[29]

X. Yu, Liouville type theorem for nonlinear elliptic equation with general nonlinearity,, \emph{Discrete Contin. Dynam. Systems}, 34 (2014), 4947.  doi: 10.3934/dcds.2014.34.4947.  Google Scholar

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, (2008).  doi: 10.1090/memo/0915.  Google Scholar

[2]

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

[3]

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

[4]

A. Castro, J. Cossio and C. Vélez, Existence and qualitative properties of solutions for nonlinear Dirichlet problems,, \emph{Discrete Contin. Dynam. Systems}, 33 (2013), 123.   Google Scholar

[5]

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

[6]

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

[7]

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

[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,, \emph{Comm. Pure. Appl. Anal.}, 9 (2010), 1507.  doi: 10.3934/cpaa.2010.9.1507.  Google Scholar

[9]

L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis,, Chapman & Hall/CRC, (2006).   Google Scholar

[10]

L. Gasinski and N. S. Papageorgiou, Dirichlet $(p,q)$-equations at resonance,, \emph{Discrete Contin. Dynam. Systems}, 34 (2014), 2037.   Google Scholar

[11]

M. F. Gurtin and R. C. MacComy, On the diffusion of biological population,, \emph{Math. Biosci.}, 33 (1977), 35.   Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[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,, \emph{Israel J. Math.}, 201 (2014), 761.  doi: 10.1007/s11856-014-1050-y.  Google Scholar

[18]

N. S. Papageorgiou and V. D. Radulescu, Semilinear Neumann problems with indefinite and unbounded potential and crossing nonlinearity,, \emph{Contemp. Math.}, 595 (2013), 293.  doi: 10.1090/conm/595/11801.  Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

[23]

N. S. Papageorgiou and G. Smyrlis, On a class of parametric Neumann problems with indefinite and unbounded potential,, \emph{Forum Math.}, (1015), 2.  doi: 10.1515/forum-2012-0042.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

S. Takeuchi, Multiplicity result for a degenerate elliptic equation with a logistic reaction,, \emph{J. Diff. Equas.}, 173 (2001), 138.  doi: 10.1006/jdeq.2000.3914.  Google Scholar

[28]

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

[29]

X. Yu, Liouville type theorem for nonlinear elliptic equation with general nonlinearity,, \emph{Discrete Contin. Dynam. Systems}, 34 (2014), 4947.  doi: 10.3934/dcds.2014.34.4947.  Google Scholar

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