# American Institute of Mathematical Sciences

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:
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Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlin. Anal., 12 (1988), 1203-1219. 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, J. Math. Anal. Appl., 354 (2009), 147-158. 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, Discrete Cont Dyn Systems, 32 (2012), 4015-4026. 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, Proc. Royal Soc. Edinburgh, 141A (2011), 171-192. Google Scholar [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.  Google Scholar [26] N. S. Papageorgiou and S. Th. Kyritsi, Handbook of Applied Analysis, Springer, New York, 2008. doi: 10.1007/b120946.  Google Scholar [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.  Google Scholar [28] P. Pucci and J. Serrin, The Maximum Principle, Birkhauser, Basel, 2007.  Google Scholar [29] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Cont Dyn Systems, 33 (2013), 2105-2137.  Google Scholar [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.  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, AMS, vol. 196, no. 905, 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, Nonlin. Anal., 32 (1998), 819-830. doi: 10.1016/S0362-546X(97)00530-0.  Google Scholar [4] S. Cano-Casanova, Coercivity of elliptic mixed boundary value problems in annulus of $\mathbbR^N$, Discrete and Continuous Dynamical Systems, 32 (2012), 3819-3839. 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, Discrete Cont Dyn Systems, 33 (2013), 123-140.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Chapman & Hall/CRC, Boca Raton, 2006.  Google Scholar [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.  Google Scholar [12] L. Gasinski and N. S. Papageorgiou, Dirichlet $(p,q)$-equations at resonance, Discrete Contin. Dynam. Systems, 34 (2014), 2037-2060.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] O. Ladyzhenskaya and N. Uraltseva, Linear and Quasilinear Elliptic Equations, Acad. Press, New York, 1968.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [26] N. S. Papageorgiou and S. Th. Kyritsi, Handbook of Applied Analysis, Springer, New York, 2008. doi: 10.1007/b120946.  Google Scholar [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.  Google Scholar [28] P. Pucci and J. Serrin, The Maximum Principle, Birkhauser, Basel, 2007.  Google Scholar [29] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Cont Dyn Systems, 33 (2013), 2105-2137.  Google Scholar [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.  Google Scholar
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