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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 and 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.

[2]

S. AizicoviciN. 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.

[3]

S. AizicoviciN. S. Papageorgiou and V. Staicu, Multiplicity of solutions for a class of nonlinear nonhomogeneous elliptic equations, J. Nonlinear Convex Anal., 15 (2014), 7-34. 

[4]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional. Anal., 14 (1973), 349-381. 

[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.

[6]

V. BenciP. D'AveniaD. 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.

[7]

K. C. Chang, Methods in Nonlinear Analysis, Springer, Berlin, 2005.

[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. 

[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.

[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.

[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.

[12]

M. FilippakisA. 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.

[13]

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

[14]

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

[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.

[16] O. Ladyzhenskaya and N. Uraltseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. 
[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.

[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.

[19]

R. Palais, Homotopy theory of infinite dimensional manifolds, Topology, 5 (1966), 1-16.  doi: 10.1016/0040-9383(66)90002-4.

[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.

[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.

[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.

[23]

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

[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.

[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.

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, Mem. Amer. Math. Soc. , 196 (2008). doi: 10.1090/memo/0915.

[2]

S. AizicoviciN. 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.

[3]

S. AizicoviciN. S. Papageorgiou and V. Staicu, Multiplicity of solutions for a class of nonlinear nonhomogeneous elliptic equations, J. Nonlinear Convex Anal., 15 (2014), 7-34. 

[4]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional. Anal., 14 (1973), 349-381. 

[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.

[6]

V. BenciP. D'AveniaD. 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.

[7]

K. C. Chang, Methods in Nonlinear Analysis, Springer, Berlin, 2005.

[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. 

[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.

[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.

[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.

[12]

M. FilippakisA. 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.

[13]

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

[14]

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

[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.

[16] O. Ladyzhenskaya and N. Uraltseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. 
[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.

[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.

[19]

R. Palais, Homotopy theory of infinite dimensional manifolds, Topology, 5 (1966), 1-16.  doi: 10.1016/0040-9383(66)90002-4.

[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.

[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.

[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.

[23]

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

[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.

[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.

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