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July  2017, 16(4): 1147-1168. doi: 10.3934/cpaa.2017056

## Nonlinear Dirichlet problems with double resonance

 1 Department of Mathematics, Ohio University, Athens, OH 45701, USA 2 Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece 3 CIDMA and Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

* Corresponding author

Received  June 2016 Revised  February 2017 Published  April 2017

We study a nonlinear Dirichlet problem driven by the sum of a $p-$Laplacian ($p>2$) and a Laplacian and which at $\pm\infty$ is resonant with respect to the spectrum of $\left(-\triangle_{p}, W_{0}^{1, p}\left(\Omega\right)\right)$ and at zero is resonant with respect to the spectrum of $\left(-\triangle, H_{0}^{1}\left(\Omega\right) \right)$ (double resonance). We prove two multiplicity theorems providing three and four nontrivial solutions respectivelly, all with sign information. Our approach uses critical point theory together with truncation and comparison techniques and Morse theory.

Citation: Sergiu Aizicovici, Nikolaos S. Papageorgiou, Vasile Staicu. Nonlinear Dirichlet problems with double resonance. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1147-1168. doi: 10.3934/cpaa.2017056
##### References:
 [1] S. Aizicovici, N. S. Papageorgiou and V. Staicu, Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints, Mem. Amer. Math. Soc. , 196 (2008). doi: 10.1090/memo/0915. Google Scholar [2] S. Aizicovici, N. S. Papageorgiou and V. Staicu, Nodal solutions for (p. 2)-equations, Trans. Amer. Math. Soc., 367 (2015), 7343-7372. doi: 10.1090/S0002-9947-2014-06324-1. Google Scholar [3] S. Aizicovici, N. S. Papageorgiou and V. Staicu, Multiplicity of solutions for a class of nonlinear nonhomogeneous elliptic equations, J. Nonlinear Convex Anal., 15 (2014), 7-34. Google Scholar [4] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional. Anal., 14 (1973), 349-381. Google Scholar [5] D. Arcoya and D. Ruiz, The Ambrosetti-Prodi problem for the p-Laplace operator, Comm. Partial Differential Equations, 31 (2006), 849-865. doi: 10.1080/03605300500394447. Google Scholar [6] V. Benci, P. D'Avenia, D. Fortunato and L. Pisani, Solitons in several space dimensions: Derrick's problem and infinitely many solutions, Arch. Rational Mech. Anal., 154 (2000), 297-324. doi: 10.1007/s002050000101. Google Scholar [7] K. C. Chang, Methods in Nonlinear Analysis, Springer, Berlin, 2005. Google Scholar [8] L. Cherfils and Y. Ilyasov, On the stationary solutions of generalized reaction-diffusion equations with p & q Laplacian, Commun. Pure Appl. Anal., 4 (2005), 9-22. Google Scholar [9] S. Cingolani and M. Degiovanni, Nontrivial solutions for p-Laplace equations with right-hand side having p-linear growth at infinity, Comm. Partial Differential Equations, 30 (2005), 1191-1203. doi: 10.1080/03605300500257594. Google Scholar [10] S. Cingolani and G. Vannella, Marino-Prodi perturbation type results and Morse indices of minimax critical points for a class of functionals in Banach spaces, Annali Mat. Pura Appl., 186 (2007), 155-183. doi: 10.1007/s10231-005-0176-2. Google Scholar [11] J. N. Corvellec and A. Hantoute, Homotopy stability of isolated critical points of continuous functionals, Set-Valued Anal., 10 (2002), 143-164. doi: 10.1023/A:1016544301594. Google Scholar [12] M. Filippakis, A. Kristaly and N. S. Papageorgiou, Existence of five nonzero solutions with exact sign for a p-Laplacian equation, Discrete Contin. Dyn. Syst., 24 (2009), 405-440. doi: 10.3934/dcds.2009.24.405. Google Scholar [13] L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Chapman & Hall/ CRC Press, Boca Raton, 2006. Google Scholar [14] L. Gasinski and N. S. Papageorgiou, Dirichlet (p, q)-equations at resonance, Discrete Contin. Dyn. Syst., 34 (2014), 2037-2060. Google Scholar [15] C. He and G. Li, The existence of a nontrivial solution to the p & q Laplacian problem with nonlinearity asymptotic to up-1 at infinity in $\mathbb{R}^N$, Nonlinear Anal., 65 (2006), 1110-1119. doi: 10.1016/j.na.2006.12.008. Google Scholar [16] O. Ladyzhenskaya and N. Uraltseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. Google Scholar [17] G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219. doi: 10.1016/0362-546X(88)90053-3. Google Scholar [18] D. Motreanu, V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Boundary Value Problems, Springer, New York, 2014. doi: 10.1007/978-1-4614-9323-5. Google Scholar [19] R. Palais, Homotopy theory of infinite dimensional manifolds, Topology, 5 (1966), 1-16. doi: 10.1016/0040-9383(66)90002-4. Google Scholar [20] N. S. Papageorgiou and V. D. Radulescu, Resonant (p, 2)-equations with asymmetric reaction, Anal. Appl., 13 (2015), 481-506. doi: 10.1142/S0219530514500134. Google Scholar [21] N. S. Papageorgiou and V. D. Radulescu, Bifurcation of positive solutions for nonlinear nonhomogeneous Robin and Neumann problems with competing nonlinearities, Discrete Contin. Dyn. Syst., 35 (2015), 5003-5036. doi: 10.3934/dcds.2015.35.5003. Google Scholar [22] N. S. Papageorgiou and P. Winkert, Resonant (p, 2)-equations with concave terms, Appl. Anal., 94 (2015), 342-360. doi: 10.1080/00036811.2014.895332. Google Scholar [23] P. Pucci and J. Serrin, The Maximum Principle, Birkhauser, Basel, 2007. Google Scholar [24] J. Su, Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues, Nonlinear Anal., 48 (2002), 881-895. doi: 10.1016/S0362-546X(00)00221-2. Google Scholar [25] M. Sun, Multiplicity of solutions for a class of quasilinear elliptic equations at resonance, J. Math. Anal. Appl., 386 (2012), 661-668. doi: 10.1016/j.jmaa.2011.08.030. Google Scholar

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##### References:
 [1] S. Aizicovici, N. S. Papageorgiou and V. Staicu, Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints, Mem. Amer. Math. Soc. , 196 (2008). doi: 10.1090/memo/0915. Google Scholar [2] S. Aizicovici, N. S. Papageorgiou and V. Staicu, Nodal solutions for (p. 2)-equations, Trans. Amer. Math. Soc., 367 (2015), 7343-7372. doi: 10.1090/S0002-9947-2014-06324-1. Google Scholar [3] S. Aizicovici, N. S. Papageorgiou and V. Staicu, Multiplicity of solutions for a class of nonlinear nonhomogeneous elliptic equations, J. Nonlinear Convex Anal., 15 (2014), 7-34. Google Scholar [4] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional. Anal., 14 (1973), 349-381. Google Scholar [5] D. Arcoya and D. Ruiz, The Ambrosetti-Prodi problem for the p-Laplace operator, Comm. Partial Differential Equations, 31 (2006), 849-865. doi: 10.1080/03605300500394447. Google Scholar [6] V. Benci, P. D'Avenia, D. Fortunato and L. Pisani, Solitons in several space dimensions: Derrick's problem and infinitely many solutions, Arch. Rational Mech. Anal., 154 (2000), 297-324. doi: 10.1007/s002050000101. Google Scholar [7] K. C. Chang, Methods in Nonlinear Analysis, Springer, Berlin, 2005. Google Scholar [8] L. Cherfils and Y. Ilyasov, On the stationary solutions of generalized reaction-diffusion equations with p & q Laplacian, Commun. Pure Appl. Anal., 4 (2005), 9-22. Google Scholar [9] S. Cingolani and M. Degiovanni, Nontrivial solutions for p-Laplace equations with right-hand side having p-linear growth at infinity, Comm. Partial Differential Equations, 30 (2005), 1191-1203. doi: 10.1080/03605300500257594. Google Scholar [10] S. Cingolani and G. Vannella, Marino-Prodi perturbation type results and Morse indices of minimax critical points for a class of functionals in Banach spaces, Annali Mat. Pura Appl., 186 (2007), 155-183. doi: 10.1007/s10231-005-0176-2. Google Scholar [11] J. N. Corvellec and A. Hantoute, Homotopy stability of isolated critical points of continuous functionals, Set-Valued Anal., 10 (2002), 143-164. doi: 10.1023/A:1016544301594. Google Scholar [12] M. Filippakis, A. Kristaly and N. S. Papageorgiou, Existence of five nonzero solutions with exact sign for a p-Laplacian equation, Discrete Contin. Dyn. Syst., 24 (2009), 405-440. doi: 10.3934/dcds.2009.24.405. Google Scholar [13] L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Chapman & Hall/ CRC Press, Boca Raton, 2006. Google Scholar [14] L. Gasinski and N. S. Papageorgiou, Dirichlet (p, q)-equations at resonance, Discrete Contin. Dyn. Syst., 34 (2014), 2037-2060. Google Scholar [15] C. He and G. Li, The existence of a nontrivial solution to the p & q Laplacian problem with nonlinearity asymptotic to up-1 at infinity in $\mathbb{R}^N$, Nonlinear Anal., 65 (2006), 1110-1119. doi: 10.1016/j.na.2006.12.008. Google Scholar [16] O. Ladyzhenskaya and N. Uraltseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. Google Scholar [17] G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219. doi: 10.1016/0362-546X(88)90053-3. Google Scholar [18] D. Motreanu, V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Boundary Value Problems, Springer, New York, 2014. doi: 10.1007/978-1-4614-9323-5. Google Scholar [19] R. Palais, Homotopy theory of infinite dimensional manifolds, Topology, 5 (1966), 1-16. doi: 10.1016/0040-9383(66)90002-4. Google Scholar [20] N. S. Papageorgiou and V. D. Radulescu, Resonant (p, 2)-equations with asymmetric reaction, Anal. Appl., 13 (2015), 481-506. doi: 10.1142/S0219530514500134. Google Scholar [21] N. S. Papageorgiou and V. D. Radulescu, Bifurcation of positive solutions for nonlinear nonhomogeneous Robin and Neumann problems with competing nonlinearities, Discrete Contin. Dyn. Syst., 35 (2015), 5003-5036. doi: 10.3934/dcds.2015.35.5003. Google Scholar [22] N. S. Papageorgiou and P. Winkert, Resonant (p, 2)-equations with concave terms, Appl. Anal., 94 (2015), 342-360. doi: 10.1080/00036811.2014.895332. Google Scholar [23] P. Pucci and J. Serrin, The Maximum Principle, Birkhauser, Basel, 2007. Google Scholar [24] J. Su, Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues, Nonlinear Anal., 48 (2002), 881-895. doi: 10.1016/S0362-546X(00)00221-2. Google Scholar [25] M. Sun, Multiplicity of solutions for a class of quasilinear elliptic equations at resonance, J. Math. Anal. Appl., 386 (2012), 661-668. doi: 10.1016/j.jmaa.2011.08.030. Google Scholar
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