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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

L. Cherfils and V. Ilyasov, On the stationary solutions of generalized reaction diffusion equations with $p$ & $q$ Laplacian,, \emph{Commun. Pure Appl. Anal.}, 4 (2005), 9.   Google Scholar

[7]

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

[8]

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

[9]

Y. Deng, S. Peng and L. Wang, Existence of multiple solutions for a nonhomogeneous semilinear elliptic equatio involving critical exponent,, \emph{Discrete Contin. Dynam. Systems}, 32 (2012), 795.   Google Scholar

[10]

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

[11]

L. Gasinski and N. S. Papageorgiou, Multiple solutions for nonlinear Neumann problems with asymmetric reaction via Morse theory,, \emph{Adv. Nonlin. Studies}, 11 (2011), 781.   Google Scholar

[12]

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

[13]

Z. Guo and Z. Liu, Perturbed elliptic equations with oscillatory nonlinearities,, \emph{Discrete Cont Dyn Systems}, 32 (2012), 3567.  doi: 10.3934/dcds.2012.32.3567.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

S. Kyritsi and N. S. Papageorgiou, Multiple solutions for nonlinear elliptic equations with asymmetric reaction term,, \emph{Discrete Cont Dyn Systems}, 33 (2013), 2469.   Google Scholar

[20]

O. Ladyzhenskaya and N. Uraltseva, Linear and Quasilinear Elliptic Equations,, Acad. Press, (1968).   Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

N. S. Papageorgiou and S. Th. Kyritsi, Handbook of Applied Analysis,, Springer, (2008).  doi: 10.1007/b120946.  Google Scholar

[27]

N. S. Papageorgiou and G. Smyrlis, On nonlinear nonhomogeneous resonant Dirichlet equations,, \emph{Pacific J. Math.}, 264 (2013), 421.  doi: 10.2140/pjm.2013.264.421.  Google Scholar

[28]

P. Pucci and J. Serrin, The Maximum Principle,, Birkhauser, (2007).   Google Scholar

[29]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type,, \emph{Discrete Cont Dyn Systems}, 33 (2013), 2105.   Google Scholar

[30]

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

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

[3]

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

[4]

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

[5]

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

[6]

L. Cherfils and V. Ilyasov, On the stationary solutions of generalized reaction diffusion equations with $p$ & $q$ Laplacian,, \emph{Commun. Pure Appl. Anal.}, 4 (2005), 9.   Google Scholar

[7]

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

[8]

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

[9]

Y. Deng, S. Peng and L. Wang, Existence of multiple solutions for a nonhomogeneous semilinear elliptic equatio involving critical exponent,, \emph{Discrete Contin. Dynam. Systems}, 32 (2012), 795.   Google Scholar

[10]

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

[11]

L. Gasinski and N. S. Papageorgiou, Multiple solutions for nonlinear Neumann problems with asymmetric reaction via Morse theory,, \emph{Adv. Nonlin. Studies}, 11 (2011), 781.   Google Scholar

[12]

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

[13]

Z. Guo and Z. Liu, Perturbed elliptic equations with oscillatory nonlinearities,, \emph{Discrete Cont Dyn Systems}, 32 (2012), 3567.  doi: 10.3934/dcds.2012.32.3567.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

S. Kyritsi and N. S. Papageorgiou, Multiple solutions for nonlinear elliptic equations with asymmetric reaction term,, \emph{Discrete Cont Dyn Systems}, 33 (2013), 2469.   Google Scholar

[20]

O. Ladyzhenskaya and N. Uraltseva, Linear and Quasilinear Elliptic Equations,, Acad. Press, (1968).   Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

N. S. Papageorgiou and S. Th. Kyritsi, Handbook of Applied Analysis,, Springer, (2008).  doi: 10.1007/b120946.  Google Scholar

[27]

N. S. Papageorgiou and G. Smyrlis, On nonlinear nonhomogeneous resonant Dirichlet equations,, \emph{Pacific J. Math.}, 264 (2013), 421.  doi: 10.2140/pjm.2013.264.421.  Google Scholar

[28]

P. Pucci and J. Serrin, The Maximum Principle,, Birkhauser, (2007).   Google Scholar

[29]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type,, \emph{Discrete Cont Dyn Systems}, 33 (2013), 2105.   Google Scholar

[30]

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

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