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May 2018, 17(3): 1219-1253. doi: 10.3934/cpaa.2018059

Remarks on minimizers for (p, q)-Laplace equations with two parameters

1. 

Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8,306 14 Plzeň, Czech Republic

2. 

Department of Mathematics, Tokyo University of Science, Kagurazaka 1-3, Shinjyuku-ku, Tokyo 162-8601, Japan

* Corresponding author: V. Bobkov.

Received  June 2017 Revised  June 2017 Published  January 2018

We study in detail the existence, nonexistence and behavior of global minimizers, ground states and corresponding energy levels of the $(p, q)$ -Laplace equation $-Δ_p u -Δ_q u = α |u|^{p-2}u + β |u|^{q-2}u$ in a bounded domain $Ω \subset \mathbb{R}^N$ under zero Dirichlet boundary condition, where $p > q > 1$ and $α, β ∈ \mathbb{R}$ . A curve on the $(α, β)$ -plane which allocates a set of the existence of ground states and the multiplicity of positive solutions is constructed. Additionally, we show that eigenfunctions of the p-and q-Laplacians under zero Dirichlet boundary condition are linearly independent.

Citation: Vladimir Bobkov, Mieko Tanaka. Remarks on minimizers for (p, q)-Laplace equations with two parameters. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1219-1253. doi: 10.3934/cpaa.2018059
References:
[1]

S. AizicoviciN. S. Papageorgiou and V. Staicu, Nodal solutions for (p, 2)-equations, Transactions of the American Mathematical Society, 367 (2015), 7343-7372. doi: 10.1090/S0002-9947-2014-06324-1.

[2]

M. J. AlvesR. B. Assunç ao and O. H. Miyagaki, Existence result for a class of quasilinear elliptic equations with (pq)-Laplacian and vanishing potentials, Illinois Journal of Mathematics, 59 (2015), 545-575.

[3]

A. Anane, Simplicité et isolation de la premiere valeur propre du p-laplacien avec poids, Comptes Rendus de l'Académie des Sciences-Series I-Mathematics, 305 (1987), 725-728.

[4]

J. Bellazzini and N. Visciglia, Max-min characterization of the mountain pass energy level for a class of variational problems, Proceedings of the American Mathematical Society, 138 (2010), 3335-3343. doi: 10.1090/S0002-9939-10-10415-8.

[5]

V. BenciP. D'AveniaD. Fortunato and L. Pisani, Solitons in several space dimensions: Derrick's problem and infinitely many solutions, Archive for Rational Mechanics and Analysis, 154 (2000), 297-324. doi: 10.1007/s002050000101.

[6]

V. Bobkov and M. Tanaka, On positive solutions for (p, q)-Laplace equations with two parameters, Calculus of Variations and Partial Differential Equations, 54 (2015), 3277-3301. doi: 10.1007/s00526-015-0903-5.

[7]

V. Bobkov and M. Tanaka, On sign-changing solutions for (p, q)-Laplace equations with two parameters Advances in Nonlinear Analysis. doi: 10.1515/anona-2016-0172.

[8]

P. J. Bushell and D. E. Edmunds, Remarks on generalised trigonometric functions, Rocky Mountain Journal of Mathematics, 42 (2012), 25-57. doi: 10.1216/rmj-2012-42-1-25.

[9]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, The Journal of chemical physics, 28 (1958), 258-267. doi: 10.1063/1.1744102.

[10]

L. Cherfils and Y. Il'yasov, On the stationary solutions of generalized reaction diffusion equations with p&q-laplacian, Communications on Pure and Applied Mathematics, 4 (2005), 9-22. doi: 10.3934/cpaa.2005.4.9.

[11]

I. Chueshov and I. Lasiecka, Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models, Discrete and Continuous Dynamical Systems, 15 (2006), 777-809. doi: 10.3934/dcds.2006.15.777.

[12]

M. Colombo and M. Colombo, Regularity for double phase variational problems, Archive for Rational Mechanics and Analysis, 215 (2015), 443-496. doi: 10.1007/s00205-014-0785-2.

[13]

M. CuestaD. de Figueiredo and J.-P. Gossez, The beginning of the Fučik spectrum for the p-Laplacian, Journal of Differential Equations, 159 (1999), 212-238. doi: 10.1006/jdeq.1999.3645.

[14]

L. Damascelli and B. Sciunzi, Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations, Journal of Differential Equations, 206 (2004), 483-515. doi: 10.1016/j.jde.2004.05.012.

[15]

P. Drábek, Geometry of the energy functional and the Fredholm alternative for the p-Laplacian in higher dimensions, 2001-Luminy Conference on Quasilinear Elliptic and Parabolic Equations and System, Electronic Journal of Differential Equations, Conference 08 (2002), 103-120.

[16]

P. Drábek, A. Kufner and F. Nicolosi, Quasilinear Elliptic Equations with Degenerations and Singularities, Walter de Gruyter, 1997.

[17]

P. Drábek and J. Milota, Methods of Nonlinear Analysis: Applications to Differential Equations, 2nd edition, Springer, 2013. doi: 10.1007/978-3-0348-0387-8.

[18]

L. F. FariaO. H. Miyagaki and D. Motreanu, Comparison and positive solutions for problems with the (p, q)-Laplacian and a convection term, Proceedings of the Edinburgh Mathematical Society (Series 2), 57 (2014), 687-698. doi: 10.1017/S0013091513000576.

[19]

G. M. Figueiredo, Existence of positive solutions for a class of p&q elliptic problems with critical growth on $\mathbb{R}^N$, Journal of Mathematical Analysis and Applications, 378 (2011), 507-518. doi: 10.1016/j.jmaa.2011.02.017.

[20]

J. García-Melián, On the behavior of the first eigenfunction of the p-Laplacian near its critical points, Bulletin of the London Mathematical Society, 35 (2003), 391-400. doi: 10.1112/S0024609303001966.

[21]

J. Fleckinger-Pellé and P. Takáč, An improved Poincaré inequality and the p-Laplacian at resonance for p>2, Advances in Differential Equations, 7 (2002), 951-971.

[22]

Y. S. Il'yasov, Bifurcation calculus by the extended functional method, Functional Analysis and Its Applications, 41 (2007), 18-30. doi: 10.1007/s10688-007-0002-2.

[23]

Y. Il'yasov and K. Silva, On branches of positive solutions for p-Laplacian problems at the extreme value of Nehari manifold method, preprint,arXiv: 1704.02477.

[24]

L. Jeanjean and K. Tanaka, A remark on least energy solutions in $\mathbb{R}^N$, Proceedings of the American Mathematical Society, 131 (2003), 2399-2408.

[25]

R. KajikiyaM. Tanaka and S. Tanaka, Bifurcation of positive solutions for the one dimensional (p, q)-Laplace equation, Electronic Journal of Differential Equations, 2017 (2017), 1-37.

[26]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Analysis: Theory, Methods & Applications, 12 (1988), 1203-1219. doi: 10.1016/0362-546X(88)90053-3.

[27]

G. M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations, Communications in Partial Differential Equations, 16 (1991), 311-361. doi: 10.1080/03605309108820761.

[28]

S. Marano and S. Mosconi, Some recent results on the Dirichlet problem for (p, q)-Laplace equations, Discrete and Continuous Dynamical Systems -Series S, 11 (2018), 279-291. doi: 10.3934/dcdss.2018015.

[29]

S. A. Marano and N. S. Papageorgiou, Constant-sign and nodal solutions of coercive $(p, q)$-Laplacian problems, Nonlinear Analysis: Theory, Methods & Applications, 77 (2013), 118-129. doi: 10.1016/j.na.2012.09.007.

[30]

D. Motreanu and M. Tanaka, On a positive solution for $(p, q)$-Laplace equation with indefinite weight, Minimax Theory and its Applications, 1 (2016), 1-20.

[31]

S. I. Pohozaev, Nonlinear variational problems via the fibering method, in Handbook of differential equations: stationary partial differential equations (ed. M. Chipot), 5 (2008), 49–209. doi: 10.1016/S1874-5733(08)80009-5.

[32]

P. Pucci and J. Serrin, The Maximum Principle, Springer, 2007. doi: 10.1007/978-3-7643-8145-5.

[33]

J. Sun, J. Chu and T. F. Wu, Existence and multiplicity of nontrivial solutions for some biharmonic equations with p-Laplacian Journal of Differential Equations, 262 (2017), 945-977. doi: 10.016/j.jde.2016.10.001.

[34]

P. Takáč, On the Fredholm alternative for the $p$-Laplacian at the first eigenvalue, Indiana University Mathematics Journal, 51 (2002), 187-238. doi: 10.1512/iumj.2002.51.2156.

[35]

M. Tanaka, Uniqueness of a positive solution and existence of a sign-changing solution for (p, q)-Laplace equation, Journal of Nonlinear Functional Analysis, 2014 (2014), 1–15.

[36]

M. Tanaka, Generalized eigenvalue problems for (p, q)-Laplacian with indefinite weight, Journal of Mathematical Analysis and Applications, 419 (2014), 1181-1192. doi: 10.1016/j.jmaa.2014.05.044.

[37]

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, Journal of Differential equations, 51 (1984), 126-150. doi: 10.1016/0022-0396(84)90105-0.

[38]

H. Yin and Z. Yang, A class of p-q-Laplacian type equation with concave-convex nonlinearities in bounded domain, Journal of Mathematical Analysis and Applications, 382 (2011), 843-855. doi: 10.1016/j.jmaa.2011.04.090.

[39]

V. E. Zakharov, Collapse of Langmuir waves, Soviet Journal of Experimental and Theoretical Physics, 35 (1972), 908-914.

show all references

References:
[1]

S. AizicoviciN. S. Papageorgiou and V. Staicu, Nodal solutions for (p, 2)-equations, Transactions of the American Mathematical Society, 367 (2015), 7343-7372. doi: 10.1090/S0002-9947-2014-06324-1.

[2]

M. J. AlvesR. B. Assunç ao and O. H. Miyagaki, Existence result for a class of quasilinear elliptic equations with (pq)-Laplacian and vanishing potentials, Illinois Journal of Mathematics, 59 (2015), 545-575.

[3]

A. Anane, Simplicité et isolation de la premiere valeur propre du p-laplacien avec poids, Comptes Rendus de l'Académie des Sciences-Series I-Mathematics, 305 (1987), 725-728.

[4]

J. Bellazzini and N. Visciglia, Max-min characterization of the mountain pass energy level for a class of variational problems, Proceedings of the American Mathematical Society, 138 (2010), 3335-3343. doi: 10.1090/S0002-9939-10-10415-8.

[5]

V. BenciP. D'AveniaD. Fortunato and L. Pisani, Solitons in several space dimensions: Derrick's problem and infinitely many solutions, Archive for Rational Mechanics and Analysis, 154 (2000), 297-324. doi: 10.1007/s002050000101.

[6]

V. Bobkov and M. Tanaka, On positive solutions for (p, q)-Laplace equations with two parameters, Calculus of Variations and Partial Differential Equations, 54 (2015), 3277-3301. doi: 10.1007/s00526-015-0903-5.

[7]

V. Bobkov and M. Tanaka, On sign-changing solutions for (p, q)-Laplace equations with two parameters Advances in Nonlinear Analysis. doi: 10.1515/anona-2016-0172.

[8]

P. J. Bushell and D. E. Edmunds, Remarks on generalised trigonometric functions, Rocky Mountain Journal of Mathematics, 42 (2012), 25-57. doi: 10.1216/rmj-2012-42-1-25.

[9]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, The Journal of chemical physics, 28 (1958), 258-267. doi: 10.1063/1.1744102.

[10]

L. Cherfils and Y. Il'yasov, On the stationary solutions of generalized reaction diffusion equations with p&q-laplacian, Communications on Pure and Applied Mathematics, 4 (2005), 9-22. doi: 10.3934/cpaa.2005.4.9.

[11]

I. Chueshov and I. Lasiecka, Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models, Discrete and Continuous Dynamical Systems, 15 (2006), 777-809. doi: 10.3934/dcds.2006.15.777.

[12]

M. Colombo and M. Colombo, Regularity for double phase variational problems, Archive for Rational Mechanics and Analysis, 215 (2015), 443-496. doi: 10.1007/s00205-014-0785-2.

[13]

M. CuestaD. de Figueiredo and J.-P. Gossez, The beginning of the Fučik spectrum for the p-Laplacian, Journal of Differential Equations, 159 (1999), 212-238. doi: 10.1006/jdeq.1999.3645.

[14]

L. Damascelli and B. Sciunzi, Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations, Journal of Differential Equations, 206 (2004), 483-515. doi: 10.1016/j.jde.2004.05.012.

[15]

P. Drábek, Geometry of the energy functional and the Fredholm alternative for the p-Laplacian in higher dimensions, 2001-Luminy Conference on Quasilinear Elliptic and Parabolic Equations and System, Electronic Journal of Differential Equations, Conference 08 (2002), 103-120.

[16]

P. Drábek, A. Kufner and F. Nicolosi, Quasilinear Elliptic Equations with Degenerations and Singularities, Walter de Gruyter, 1997.

[17]

P. Drábek and J. Milota, Methods of Nonlinear Analysis: Applications to Differential Equations, 2nd edition, Springer, 2013. doi: 10.1007/978-3-0348-0387-8.

[18]

L. F. FariaO. H. Miyagaki and D. Motreanu, Comparison and positive solutions for problems with the (p, q)-Laplacian and a convection term, Proceedings of the Edinburgh Mathematical Society (Series 2), 57 (2014), 687-698. doi: 10.1017/S0013091513000576.

[19]

G. M. Figueiredo, Existence of positive solutions for a class of p&q elliptic problems with critical growth on $\mathbb{R}^N$, Journal of Mathematical Analysis and Applications, 378 (2011), 507-518. doi: 10.1016/j.jmaa.2011.02.017.

[20]

J. García-Melián, On the behavior of the first eigenfunction of the p-Laplacian near its critical points, Bulletin of the London Mathematical Society, 35 (2003), 391-400. doi: 10.1112/S0024609303001966.

[21]

J. Fleckinger-Pellé and P. Takáč, An improved Poincaré inequality and the p-Laplacian at resonance for p>2, Advances in Differential Equations, 7 (2002), 951-971.

[22]

Y. S. Il'yasov, Bifurcation calculus by the extended functional method, Functional Analysis and Its Applications, 41 (2007), 18-30. doi: 10.1007/s10688-007-0002-2.

[23]

Y. Il'yasov and K. Silva, On branches of positive solutions for p-Laplacian problems at the extreme value of Nehari manifold method, preprint,arXiv: 1704.02477.

[24]

L. Jeanjean and K. Tanaka, A remark on least energy solutions in $\mathbb{R}^N$, Proceedings of the American Mathematical Society, 131 (2003), 2399-2408.

[25]

R. KajikiyaM. Tanaka and S. Tanaka, Bifurcation of positive solutions for the one dimensional (p, q)-Laplace equation, Electronic Journal of Differential Equations, 2017 (2017), 1-37.

[26]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Analysis: Theory, Methods & Applications, 12 (1988), 1203-1219. doi: 10.1016/0362-546X(88)90053-3.

[27]

G. M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations, Communications in Partial Differential Equations, 16 (1991), 311-361. doi: 10.1080/03605309108820761.

[28]

S. Marano and S. Mosconi, Some recent results on the Dirichlet problem for (p, q)-Laplace equations, Discrete and Continuous Dynamical Systems -Series S, 11 (2018), 279-291. doi: 10.3934/dcdss.2018015.

[29]

S. A. Marano and N. S. Papageorgiou, Constant-sign and nodal solutions of coercive $(p, q)$-Laplacian problems, Nonlinear Analysis: Theory, Methods & Applications, 77 (2013), 118-129. doi: 10.1016/j.na.2012.09.007.

[30]

D. Motreanu and M. Tanaka, On a positive solution for $(p, q)$-Laplace equation with indefinite weight, Minimax Theory and its Applications, 1 (2016), 1-20.

[31]

S. I. Pohozaev, Nonlinear variational problems via the fibering method, in Handbook of differential equations: stationary partial differential equations (ed. M. Chipot), 5 (2008), 49–209. doi: 10.1016/S1874-5733(08)80009-5.

[32]

P. Pucci and J. Serrin, The Maximum Principle, Springer, 2007. doi: 10.1007/978-3-7643-8145-5.

[33]

J. Sun, J. Chu and T. F. Wu, Existence and multiplicity of nontrivial solutions for some biharmonic equations with p-Laplacian Journal of Differential Equations, 262 (2017), 945-977. doi: 10.016/j.jde.2016.10.001.

[34]

P. Takáč, On the Fredholm alternative for the $p$-Laplacian at the first eigenvalue, Indiana University Mathematics Journal, 51 (2002), 187-238. doi: 10.1512/iumj.2002.51.2156.

[35]

M. Tanaka, Uniqueness of a positive solution and existence of a sign-changing solution for (p, q)-Laplace equation, Journal of Nonlinear Functional Analysis, 2014 (2014), 1–15.

[36]

M. Tanaka, Generalized eigenvalue problems for (p, q)-Laplacian with indefinite weight, Journal of Mathematical Analysis and Applications, 419 (2014), 1181-1192. doi: 10.1016/j.jmaa.2014.05.044.

[37]

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, Journal of Differential equations, 51 (1984), 126-150. doi: 10.1016/0022-0396(84)90105-0.

[38]

H. Yin and Z. Yang, A class of p-q-Laplacian type equation with concave-convex nonlinearities in bounded domain, Journal of Mathematical Analysis and Applications, 382 (2011), 843-855. doi: 10.1016/j.jmaa.2011.04.090.

[39]

V. E. Zakharov, Collapse of Langmuir waves, Soviet Journal of Experimental and Theoretical Physics, 35 (1972), 908-914.

Figure 1.  The global minimum $m$ of ${E_{\alpha, \beta }}$ on $ W_0^{1, p}$.
Figure 2.  The least energy $d$ on ${\mathcal{N}_{\alpha, \beta }}$.
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