# American Institute of Mathematical Sciences

November  2018, 17(6): 2639-2656. doi: 10.3934/cpaa.2018125

## Positive solutions for resonant (p, q)-equations with concave terms

 1 College of Mathematics, Shandong Normal University, Jinan, Shandong, China 2 Department of Mathematics, Missouri State University, Springfield, MO 65804, USA 3 Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece

Received  November 2017 Revised  April 2018 Published  June 2018

We consider a parametric (p, q)-equation with competing nonlinearities in the reaction. There is a parametric concave term and a resonant Caratheordory perturbation. The resonance is with respect to the principal eigenvalue and occurs from the right. So the energy functional of the problem is indefinite. Using variational tools and truncation and comparison techniques we show that for all small values of the parameter the problem has at least two positive smooth solutions.

Citation: Shouchuan Hu, Nikolas S. Papageorgiou. Positive solutions for resonant (p, q)-equations with concave terms. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2639-2656. doi: 10.3934/cpaa.2018125
##### 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. [2] A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Functional Anal., 122 (1994), 519-543. [3] V. Benci, D. Fortunato and L. Pisani, Soliton like solutions of a Lorentz invariant equation in dimension 3, Rev. Math. Physics, 10 (1998), 315-344. [4] G. Bonanno and G. D'Agu, Mixed elliptic problems involving the p-Laplacian with nonhomogeneous boundary conditions, Discrete Contin. Dynam. Systems, 37 (2017), 5797-5817. [5] 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. [6] F. Colasuonno and B. Noris, A p-Laplacian supercritical Neumann problem, Discrete Contin. Dynam. Systems, 37 (2017), 3025-3057. [7] G. Dai, Bifurcation and one-sign solutions of the p-Laplacian involving a nonlinearity with zeros, Discrete Contin. Dynam. Systems, 36 (2016), 5323-5345. [8] J. I. Dia and J. E. Saa, Existence et unicité de solutions positives pour certaines equations elliptiques quasilineaires, CRAS Paris, 305 (1987), 521-524. [9] M. Filippakis and N. S. Papageorgiou, Papageorgiou, Nodal solutions for Neumann problems with a nonhomogeneous differential operator, Funkc. Ekv., 56 (2013), 63-79. [10] G. Fragnelli, D. Mugnai and N. S. Papageorgiou, The Brezis-Oswald result for quasilinear Robin problems, Adv. Nonlin. Studies, 16 (2016), 403-422. [11] G. Fragnelli, D. Mugnai and N. S. Papageorgiou, Positive and nodal solutions for parametric nonlinear Robin problems with indefinite potential, Discrete Contin. Dynam. Systems, 36 (2016), 6133-6166. [12] J. Garcia Azorero, J. Manfredi and J. Peral Alonso, Sobolev versus H¨older local minimizers and global multiplicity for some quasilinear elliptic equations, Comm. Contemp. Math., 2 (2000), 385-404. [13] L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, hapman & Hall/CRC, Boca Raton, 2006. [14] L. Gasinski and N. S. Papageorgiou, Positive solutions for the Robin p-Laplacian problem with competing nonlinearities, Adv. Calc. Var., http://doi.org/10.1515/acv-2016-0039. [15] L. Gasinski and N. S. Papageorgiou, Dirichlet (p, q)-equations at resonance, Discrete Contin. Dynam. Systems, 36 (2014), 2037-2060. [16] Z. Guo and Z. Zhang, W1, p versus C1 local minimizers and multiplicity results for quasilinear elliptic equations, J. Math. Anal. Appl., 286 (2003), 32-50. [17] Shouchuan Hu and N. S. Papageorgiou, Multipcility of solutions for parametric p-Laplacian equations with nonlinearity concave near the origin, Tohoku Math. J., 62 (2010), 137-162. [18] G. Lieberman, The natural generalization of the natural conditions of Ladyzhenskoya and Uraltseva for elliptic equations, Comm. Partial Diff. Equ., 16 (1991), 311-361. [19] S. A. Marano and S. Mosconi, Some recent results on the Dirichlet problem for (p, q)-Laplace equations, Discrete Cont Dyn Systems - S, 11 (2018), 279-291. [20] S. A. Marano, S. Mosconi and N. S. Papageorgiou, Multiple solutions to (p, q)-Laplacian problems with resonant concave nonlinearity, Adv. Nonlin. Studies, 16 (2016), 51-65. [21] S. Marano and N. S. Papageorgiou, Positive solutions to a Dirichlet problem with p-Laplacian and concave-convex nonlinearity depending on a parameter, Comm. Pure Appl. Anal., 12 (2013), 815-829. [22] D. Motreanu, V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014. [23] D. Mugnai and N. S. Papageorgiou, Resonant nonlinear Neumann problems with indefinite weight, Ann. Sc. Norm. Super Pisa Cl. SCI., 11 (2012), 729-788. [24] N. S. Papageorgiou and V. D. Radulescu, Multiple solutions with precise sign for nonlinear parametric Robin problems, J. Diff. Equ., 256 (2014), 2449-2479. [25] N. S. Papageorgiou and V. D. Radulescu, Nonlinear nonhomogeneous Robin problems with superlinear reaction term, Adv. Nonlin. Studies, 16 (2016), 737-764. [26] N. S. Papageorgiou and V. D. Radulescu, Bifurcation near infinity for the Robin p-Laplacian, Manusc. Math., 148 (2015), 415-433. [27] N. S. Papageorgiou, V. D. Radulescu and D. Repovs, Positive solutions for nonlinear nonhomogeneous parametric Robin problems, Forum Math., DOI: 10-15115/forum-20170124. [28] N. S. Papageorgiou, V. D. Radulescu and D. Repovs, Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential, Discrete Contin. Dynam. Systems, 37 (2017), 2589-2618. [29] N. S. Papageorgiou and P. Winkert, Positive solutions for nonlinear nonhomogeneous Dirichlet problems with concave-convex nonlinearities, Positivity, 20 (2016), 945-979. [30] P. Pucci and J. Serrin, The Maximum Principle, Birkhauser, Basel, 2007. [31] H. Wilhelmsson, Explosive instabilities of reaction-diffusion equations, Phys. Rev. A, 36 (1987), 965-966.

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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, Memoirs, AMS, vol. 196, no. 905, 2008. [2] A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Functional Anal., 122 (1994), 519-543. [3] V. Benci, D. Fortunato and L. Pisani, Soliton like solutions of a Lorentz invariant equation in dimension 3, Rev. Math. Physics, 10 (1998), 315-344. [4] G. Bonanno and G. D'Agu, Mixed elliptic problems involving the p-Laplacian with nonhomogeneous boundary conditions, Discrete Contin. Dynam. Systems, 37 (2017), 5797-5817. [5] 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. [6] F. Colasuonno and B. Noris, A p-Laplacian supercritical Neumann problem, Discrete Contin. Dynam. Systems, 37 (2017), 3025-3057. [7] G. Dai, Bifurcation and one-sign solutions of the p-Laplacian involving a nonlinearity with zeros, Discrete Contin. Dynam. Systems, 36 (2016), 5323-5345. [8] J. I. Dia and J. E. Saa, Existence et unicité de solutions positives pour certaines equations elliptiques quasilineaires, CRAS Paris, 305 (1987), 521-524. [9] M. Filippakis and N. S. Papageorgiou, Papageorgiou, Nodal solutions for Neumann problems with a nonhomogeneous differential operator, Funkc. Ekv., 56 (2013), 63-79. [10] G. Fragnelli, D. Mugnai and N. S. Papageorgiou, The Brezis-Oswald result for quasilinear Robin problems, Adv. Nonlin. Studies, 16 (2016), 403-422. [11] G. Fragnelli, D. Mugnai and N. S. Papageorgiou, Positive and nodal solutions for parametric nonlinear Robin problems with indefinite potential, Discrete Contin. Dynam. Systems, 36 (2016), 6133-6166. [12] J. Garcia Azorero, J. Manfredi and J. Peral Alonso, Sobolev versus H¨older local minimizers and global multiplicity for some quasilinear elliptic equations, Comm. Contemp. Math., 2 (2000), 385-404. [13] L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, hapman & Hall/CRC, Boca Raton, 2006. [14] L. Gasinski and N. S. Papageorgiou, Positive solutions for the Robin p-Laplacian problem with competing nonlinearities, Adv. Calc. Var., http://doi.org/10.1515/acv-2016-0039. [15] L. Gasinski and N. S. Papageorgiou, Dirichlet (p, q)-equations at resonance, Discrete Contin. Dynam. Systems, 36 (2014), 2037-2060. [16] Z. Guo and Z. Zhang, W1, p versus C1 local minimizers and multiplicity results for quasilinear elliptic equations, J. Math. Anal. Appl., 286 (2003), 32-50. [17] Shouchuan Hu and N. S. Papageorgiou, Multipcility of solutions for parametric p-Laplacian equations with nonlinearity concave near the origin, Tohoku Math. J., 62 (2010), 137-162. [18] G. Lieberman, The natural generalization of the natural conditions of Ladyzhenskoya and Uraltseva for elliptic equations, Comm. Partial Diff. Equ., 16 (1991), 311-361. [19] S. A. Marano and S. Mosconi, Some recent results on the Dirichlet problem for (p, q)-Laplace equations, Discrete Cont Dyn Systems - S, 11 (2018), 279-291. [20] S. A. Marano, S. Mosconi and N. S. Papageorgiou, Multiple solutions to (p, q)-Laplacian problems with resonant concave nonlinearity, Adv. Nonlin. Studies, 16 (2016), 51-65. [21] S. Marano and N. S. Papageorgiou, Positive solutions to a Dirichlet problem with p-Laplacian and concave-convex nonlinearity depending on a parameter, Comm. Pure Appl. Anal., 12 (2013), 815-829. [22] D. Motreanu, V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014. [23] D. Mugnai and N. S. Papageorgiou, Resonant nonlinear Neumann problems with indefinite weight, Ann. Sc. Norm. Super Pisa Cl. SCI., 11 (2012), 729-788. [24] N. S. Papageorgiou and V. D. Radulescu, Multiple solutions with precise sign for nonlinear parametric Robin problems, J. Diff. Equ., 256 (2014), 2449-2479. [25] N. S. Papageorgiou and V. D. Radulescu, Nonlinear nonhomogeneous Robin problems with superlinear reaction term, Adv. Nonlin. Studies, 16 (2016), 737-764. [26] N. S. Papageorgiou and V. D. Radulescu, Bifurcation near infinity for the Robin p-Laplacian, Manusc. Math., 148 (2015), 415-433. [27] N. S. Papageorgiou, V. D. Radulescu and D. Repovs, Positive solutions for nonlinear nonhomogeneous parametric Robin problems, Forum Math., DOI: 10-15115/forum-20170124. [28] N. S. Papageorgiou, V. D. Radulescu and D. Repovs, Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential, Discrete Contin. Dynam. Systems, 37 (2017), 2589-2618. [29] N. S. Papageorgiou and P. Winkert, Positive solutions for nonlinear nonhomogeneous Dirichlet problems with concave-convex nonlinearities, Positivity, 20 (2016), 945-979. [30] P. Pucci and J. Serrin, The Maximum Principle, Birkhauser, Basel, 2007. [31] H. Wilhelmsson, Explosive instabilities of reaction-diffusion equations, Phys. Rev. A, 36 (1987), 965-966.
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