November  2014, 13(6): 2749-2766. doi: 10.3934/cpaa.2014.13.2749

Nonlinear Dirichlet problems with a crossing reaction

1. 

College of Mathematics, Shandong Normal University, Jinan, Shandong

2. 

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

Received  January 2014 Revised  June 2014 Published  July 2014

We consider a nonlinear Dirichlet problem driven by the sum of a $p$-Laplacian ($p>2$) and a Laplacian, with a reaction that is ($p-1$)-sublinear and exhibits an asymmetric behavior near $\infty$ and $-\infty$, crossing $\hat{\lambda}_1>0$, the principal eigenvalue of $(-\Delta_p, W^{1,p}_0(\Omega))$ (crossing nonlinearity). Resonance with respect to $\hat{\lambda}_1(p)>0$ can also occur. We prove two multiplicity results. The first for a Caratheodory reaction producing two nontrivial solutions and the second for a reaction $C^1$ in the $x$-variable producing three nontrivial solutions. Our approach is variational and uses also the Morse theory.
Citation: Shouchuan Hu, Nikolaos S. Papageorgiou. Nonlinear Dirichlet problems with a crossing reaction. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2749-2766. doi: 10.3934/cpaa.2014.13.2749
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]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, On $p$-superlinear equations with nonhomogeneous differential operator,, \emph{Nolin. Diff. Equa. Appl.}., (). 

[3]

W. Allegretto and Y. X. Huang, A Picone's identity for the $p$-Laplacian and applications, Nonlin. Anal., 32 (1998), 819-830. doi: 10.1016/S0362-546X(97)00530-0.

[4]

S. Cano-Casanova, Coercivity of elliptic mixed boundary value problems in annulus of $\mathbbR^N$, Discrete and Continuous Dynamical Systems, 32 (2012), 3819-3839. doi: 10.3934/dcds.2012.32.3819.

[5]

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

[6]

L. Cherfils and V. Ilyasov, On the stationary solutions of generalized reaction diffusion equations with $p$ & $q$ Laplacian, Commun. Pure Appl. Anal., 4 (2005), 9-22.

[7]

S. Cingolani and G. Vannella, Critical groups computations on a class of Sobolev Banach spaces via Morse index, Ann. Inst. H. Poincare Analyse Nonlin., 20 (2003), 271-292. doi: 10.1016/S0294-1449(02)00011-2.

[8]

M. Cuesta, D. deFigueiredo and J. P. Gossez, The beginning of the Fučik spectrum for the $p$-Laplacian, J. Diff. Equas., 159 (1999), 212-238. doi: 10.1006/jdeq.1999.3645.

[9]

Y. Deng, S. Peng and L. Wang, Existence of multiple solutions for a nonhomogeneous semilinear elliptic equatio involving critical exponent, Discrete Contin. Dynam. Systems, 32 (2012), 795-826.

[10]

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

[11]

L. Gasinski and N. S. Papageorgiou, Multiple solutions for nonlinear Neumann problems with asymmetric reaction via Morse theory, Adv. Nonlin. Studies, 11 (2011), 781-808.

[12]

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

[13]

Z. Guo and Z. Liu, Perturbed elliptic equations with oscillatory nonlinearities, Discrete Cont Dyn Systems, 32 (2012), 3567-3585. doi: 10.3934/dcds.2012.32.3567.

[14]

Z. Guo, Z. Liu, J. Wei and F. Zhou, Bifurcations of some elliptic problems with a singular nonlinearity via Morse index, Comm. Pure. Appl. Anal., 10 (2011), 507-525. doi: 10.3934/cpaa.2011.10.507.

[15]

Shouchuan Hu and N. S. Papageorgiou, Multiple nontrivial solutions for p-Laplacian equations with an asymmetric nonlinearity, Diff. Integ. Equas., 19 (2006), 1371-1390.

[16]

Shouchuan Hu and N. S. Papageorgiou, Double resonance for Dirichlet problems with unbounded indefinite potential and combined nonlinearities, Comm. Pure Applied Anal., 11 (2012), 2005-2021. doi: 10.3934/cpaa.2012.11.2005.

[17]

J. Garcia Melian, J. Rossi and J. Sabina de Lis, A convex-concave problem with a parameter on the boundary condition, Discrete Cont. Dyn. Systems, 32 (2012), 1095-1124.

[18]

Q. Jiu and J. Su, Existence and multiplicity results for Dirichlet problems with $p$-Laplacian, J. Math. Anal. Appl., 281 (2003), 587-601. doi: 10.1016/S0022-247X(03)00165-3.

[19]

S. Kyritsi and N. S. Papageorgiou, Multiple solutions for nonlinear elliptic equations with asymmetric reaction term, Discrete Cont Dyn Systems, 33 (2013), 2469-2494.

[20]

O. Ladyzhenskaya and N. Uraltseva, Linear and Quasilinear Elliptic Equations, Acad. Press, New York, 1968.

[21]

G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlin. Anal., 12 (1988), 1203-1219. doi: 10.1016/0362-546X(88)90053-3.

[22]

Z. Liang and J. Su, Multiple solutions for semilinar elliptic boundary value problems with double resonance, J. Math. Anal. Appl., 354 (2009), 147-158. doi: 10.1016/j.jmaa.2008.12.053.

[23]

Jean Mawhin, Multiplicity of solutions of variational systems involving $p$-Laplacians with singular $\phi$ and periodic nonlinearities, Discrete Cont Dyn Systems, 32 (2012), 4015-4026. doi: 10.3934/dcds.2012.32.4015.

[24]

D. Motreanu, V. Motreanu and N. S. Papageorgiou, On $p$-Laplacian equations with concave terms and asymmetric perturbations, Proc. Royal Soc. Edinburgh, 141A (2011), 171-192.

[25]

D. Motreanu, V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014. doi: 10.1007/978-1-4614-9323-5.

[26]

N. S. Papageorgiou and S. Th. Kyritsi, Handbook of Applied Analysis, Springer, New York, 2008. doi: 10.1007/b120946.

[27]

N. S. Papageorgiou and G. Smyrlis, On nonlinear nonhomogeneous resonant Dirichlet equations, Pacific J. Math., 264 (2013), 421-453. doi: 10.2140/pjm.2013.264.421.

[28]

P. Pucci and J. Serrin, The Maximum Principle, Birkhauser, Basel, 2007.

[29]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Cont Dyn Systems, 33 (2013), 2105-2137.

[30]

M. Tanaka, Existence of the Fučik type spectrum for the generalized $p$-Laplacian operators, Nonlin. Anal., 75 (2012), 3407-3435. doi: 10.1016/j.na.2012.01.006.

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]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, On $p$-superlinear equations with nonhomogeneous differential operator,, \emph{Nolin. Diff. Equa. Appl.}., (). 

[3]

W. Allegretto and Y. X. Huang, A Picone's identity for the $p$-Laplacian and applications, Nonlin. Anal., 32 (1998), 819-830. doi: 10.1016/S0362-546X(97)00530-0.

[4]

S. Cano-Casanova, Coercivity of elliptic mixed boundary value problems in annulus of $\mathbbR^N$, Discrete and Continuous Dynamical Systems, 32 (2012), 3819-3839. doi: 10.3934/dcds.2012.32.3819.

[5]

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

[6]

L. Cherfils and V. Ilyasov, On the stationary solutions of generalized reaction diffusion equations with $p$ & $q$ Laplacian, Commun. Pure Appl. Anal., 4 (2005), 9-22.

[7]

S. Cingolani and G. Vannella, Critical groups computations on a class of Sobolev Banach spaces via Morse index, Ann. Inst. H. Poincare Analyse Nonlin., 20 (2003), 271-292. doi: 10.1016/S0294-1449(02)00011-2.

[8]

M. Cuesta, D. deFigueiredo and J. P. Gossez, The beginning of the Fučik spectrum for the $p$-Laplacian, J. Diff. Equas., 159 (1999), 212-238. doi: 10.1006/jdeq.1999.3645.

[9]

Y. Deng, S. Peng and L. Wang, Existence of multiple solutions for a nonhomogeneous semilinear elliptic equatio involving critical exponent, Discrete Contin. Dynam. Systems, 32 (2012), 795-826.

[10]

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

[11]

L. Gasinski and N. S. Papageorgiou, Multiple solutions for nonlinear Neumann problems with asymmetric reaction via Morse theory, Adv. Nonlin. Studies, 11 (2011), 781-808.

[12]

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

[13]

Z. Guo and Z. Liu, Perturbed elliptic equations with oscillatory nonlinearities, Discrete Cont Dyn Systems, 32 (2012), 3567-3585. doi: 10.3934/dcds.2012.32.3567.

[14]

Z. Guo, Z. Liu, J. Wei and F. Zhou, Bifurcations of some elliptic problems with a singular nonlinearity via Morse index, Comm. Pure. Appl. Anal., 10 (2011), 507-525. doi: 10.3934/cpaa.2011.10.507.

[15]

Shouchuan Hu and N. S. Papageorgiou, Multiple nontrivial solutions for p-Laplacian equations with an asymmetric nonlinearity, Diff. Integ. Equas., 19 (2006), 1371-1390.

[16]

Shouchuan Hu and N. S. Papageorgiou, Double resonance for Dirichlet problems with unbounded indefinite potential and combined nonlinearities, Comm. Pure Applied Anal., 11 (2012), 2005-2021. doi: 10.3934/cpaa.2012.11.2005.

[17]

J. Garcia Melian, J. Rossi and J. Sabina de Lis, A convex-concave problem with a parameter on the boundary condition, Discrete Cont. Dyn. Systems, 32 (2012), 1095-1124.

[18]

Q. Jiu and J. Su, Existence and multiplicity results for Dirichlet problems with $p$-Laplacian, J. Math. Anal. Appl., 281 (2003), 587-601. doi: 10.1016/S0022-247X(03)00165-3.

[19]

S. Kyritsi and N. S. Papageorgiou, Multiple solutions for nonlinear elliptic equations with asymmetric reaction term, Discrete Cont Dyn Systems, 33 (2013), 2469-2494.

[20]

O. Ladyzhenskaya and N. Uraltseva, Linear and Quasilinear Elliptic Equations, Acad. Press, New York, 1968.

[21]

G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlin. Anal., 12 (1988), 1203-1219. doi: 10.1016/0362-546X(88)90053-3.

[22]

Z. Liang and J. Su, Multiple solutions for semilinar elliptic boundary value problems with double resonance, J. Math. Anal. Appl., 354 (2009), 147-158. doi: 10.1016/j.jmaa.2008.12.053.

[23]

Jean Mawhin, Multiplicity of solutions of variational systems involving $p$-Laplacians with singular $\phi$ and periodic nonlinearities, Discrete Cont Dyn Systems, 32 (2012), 4015-4026. doi: 10.3934/dcds.2012.32.4015.

[24]

D. Motreanu, V. Motreanu and N. S. Papageorgiou, On $p$-Laplacian equations with concave terms and asymmetric perturbations, Proc. Royal Soc. Edinburgh, 141A (2011), 171-192.

[25]

D. Motreanu, V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014. doi: 10.1007/978-1-4614-9323-5.

[26]

N. S. Papageorgiou and S. Th. Kyritsi, Handbook of Applied Analysis, Springer, New York, 2008. doi: 10.1007/b120946.

[27]

N. S. Papageorgiou and G. Smyrlis, On nonlinear nonhomogeneous resonant Dirichlet equations, Pacific J. Math., 264 (2013), 421-453. doi: 10.2140/pjm.2013.264.421.

[28]

P. Pucci and J. Serrin, The Maximum Principle, Birkhauser, Basel, 2007.

[29]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Cont Dyn Systems, 33 (2013), 2105-2137.

[30]

M. Tanaka, Existence of the Fučik type spectrum for the generalized $p$-Laplacian operators, Nonlin. Anal., 75 (2012), 3407-3435. doi: 10.1016/j.na.2012.01.006.

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