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 & 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. AmbrosettiH. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Functional Anal., 122 (1994), 519-543.

[3]

V. BenciD. 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. FragnelliD. Mugnai and N. S. Papageorgiou, The Brezis-Oswald result for quasilinear Robin problems, Adv. Nonlin. Studies, 16 (2016), 403-422.

[11]

G. FragnelliD. 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 AzoreroJ. 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. MaranoS. 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. PapageorgiouV. 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.

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.

[2]

A. AmbrosettiH. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Functional Anal., 122 (1994), 519-543.

[3]

V. BenciD. 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. FragnelliD. Mugnai and N. S. Papageorgiou, The Brezis-Oswald result for quasilinear Robin problems, Adv. Nonlin. Studies, 16 (2016), 403-422.

[11]

G. FragnelliD. 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 AzoreroJ. 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. MaranoS. 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. PapageorgiouV. 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.

[1]

C. Fabry, Raul Manásevich. Equations with a $p$-Laplacian and an asymmetric nonlinear term. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 545-557. doi: 10.3934/dcds.2001.7.545

[2]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4155-4182. doi: 10.3934/dcds.2014.34.4155

[3]

Boris Andreianov, Nicolas Seguin. Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 1939-1964. doi: 10.3934/dcds.2012.32.1939

[4]

Igor Chueshov, Irena Lasiecka, Daniel Toundykov. Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 459-509. doi: 10.3934/dcds.2008.20.459

[5]

Yuri V. Rogovchenko, Fatoş Tuncay. Interval oscillation of a second order nonlinear differential equation with a damping term. Conference Publications, 2007, 2007 (Special) : 883-891. doi: 10.3934/proc.2007.2007.883

[6]

Tae Gab Ha. On viscoelastic wave equation with nonlinear boundary damping and source term. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1543-1576. doi: 10.3934/cpaa.2010.9.1543

[7]

María Rosa, María de los Santos Bruzón, María de la Luz Gandarias. Lie symmetries and conservation laws of a Fisher equation with nonlinear convection term. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1331-1339. doi: 10.3934/dcdss.2015.8.1331

[8]

Jong-Shenq Guo, Bei Hu. Blowup rate estimates for the heat equation with a nonlinear gradient source term. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 927-937. doi: 10.3934/dcds.2008.20.927

[9]

V. V. Motreanu. Multiplicity of solutions for variable exponent Dirichlet problem with concave term. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 845-855. doi: 10.3934/dcdss.2012.5.845

[10]

Jerrold E. Marsden, Alexey Tret'yakov. Factor analysis of nonlinear mappings: p-regularity theory. Communications on Pure & Applied Analysis, 2003, 2 (4) : 425-445. doi: 10.3934/cpaa.2003.2.425

[11]

Nassif Ghoussoub. A variational principle for nonlinear transport equations. Communications on Pure & Applied Analysis, 2005, 4 (4) : 735-742. doi: 10.3934/cpaa.2005.4.735

[12]

Bruno Sixou, Tom Hohweiller, Nicolas Ducros. Morozov principle for Kullback-Leibler residual term and Poisson noise. Inverse Problems & Imaging, 2018, 12 (3) : 607-634. doi: 10.3934/ipi.2018026

[13]

Arrigo Cellina. The regularity of solutions to some variational problems, including the p-Laplace equation for 3≤p < 4. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4071-4085. doi: 10.3934/dcds.2018177

[14]

Giuseppe Maria Coclite, Mario Michele Coclite. Positive solutions of an integro-differential equation in all space with singular nonlinear term. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 885-907. doi: 10.3934/dcds.2008.22.885

[15]

Ferhat Mohamed, Hakem Ali. Energy decay of solutions for the wave equation with a time-varying delay term in the weakly nonlinear internal feedbacks. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 491-506. doi: 10.3934/dcdsb.2017024

[16]

Daniela Giachetti, Maria Michaela Porzio. Global existence for nonlinear parabolic equations with a damping term. Communications on Pure & Applied Analysis, 2009, 8 (3) : 923-953. doi: 10.3934/cpaa.2009.8.923

[17]

Mourad Choulli, El Maati Ouhabaz, Masahiro Yamamoto. Stable determination of a semilinear term in a parabolic equation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 447-462. doi: 10.3934/cpaa.2006.5.447

[18]

Maria Francesca Betta, Rosaria Di Nardo, Anna Mercaldo, Adamaria Perrotta. Gradient estimates and comparison principle for some nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2015, 14 (3) : 897-922. doi: 10.3934/cpaa.2015.14.897

[19]

Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1175-1185. doi: 10.3934/dcdss.2017064

[20]

Markus Grasmair. Well-posedness and convergence rates for sparse regularization with sublinear $l^q$ penalty term. Inverse Problems & Imaging, 2009, 3 (3) : 383-387. doi: 10.3934/ipi.2009.3.383

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (51)
  • HTML views (97)
  • Cited by (0)

Other articles
by authors

[Back to Top]