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Nonsymmetric elliptic operators with Wentzell boundary conditions in general domains
Positive solutions for Robin problems with general potential and logistic reaction
1. | Department of Mathematics, Missouri State University, Springeld, MO 65804 |
2. | Department of Mathematics, National Technical University, Zografou Campus, Athens 15780 |
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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