Remarks on minimizers for (<i>p</i>, <i>q</i>)-Laplace equations with two parameters

Vladimir Bobkov; Mieko Tanaka

doi:10.3934/cpaa.2018059

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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

Vladimir Bobkov ^1,, and Mieko Tanaka ^2,

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.

Keywords: p-Laplacian, (p,q)-Laplacian, nonlinear eigenvalue problem, global minimizer, ground states, Nehari manifold, fibered functional, improved Poincaré inequality.

Mathematics Subject Classification: Primary: 35J60, 35J20; Secondary: 35P30.

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. Aizicovici, N. 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. Google Scholar
[2]	M. J. Alves, R. B. Assunç ao and O. H. Miyagaki, Existence result for a class of quasilinear elliptic equations with (p − q)-Laplacian and vanishing potentials, Illinois Journal of Mathematics, 59 (2015), 545-575. Google Scholar
[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. Google Scholar
[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. Google Scholar
[5]	V. Benci, P. D'Avenia, D. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[13]	M. Cuesta, D. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	P. Drábek, A. Kufner and F. Nicolosi, Quasilinear Elliptic Equations with Degenerations and Singularities, Walter de Gruyter, 1997. Google Scholar
[17]	P. Drábek and J. Milota, Methods of Nonlinear Analysis: Applications to Differential Equations, 2^nd edition, Springer, 2013. doi: 10.1007/978-3-0348-0387-8. Google Scholar
[18]	L. F. Faria, O. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[25]	R. Kajikiya, M. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[32]	P. Pucci and J. Serrin, The Maximum Principle, Springer, 2007. doi: 10.1007/978-3-7643-8145-5. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[39]	V. E. Zakharov, Collapse of Langmuir waves, Soviet Journal of Experimental and Theoretical Physics, 35 (1972), 908-914. Google Scholar

show all references

References:

[1]	S. Aizicovici, N. 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. Google Scholar
[2]	M. J. Alves, R. B. Assunç ao and O. H. Miyagaki, Existence result for a class of quasilinear elliptic equations with (p − q)-Laplacian and vanishing potentials, Illinois Journal of Mathematics, 59 (2015), 545-575. Google Scholar
[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. Google Scholar
[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. Google Scholar
[5]	V. Benci, P. D'Avenia, D. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[13]	M. Cuesta, D. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	P. Drábek, A. Kufner and F. Nicolosi, Quasilinear Elliptic Equations with Degenerations and Singularities, Walter de Gruyter, 1997. Google Scholar
[17]	P. Drábek and J. Milota, Methods of Nonlinear Analysis: Applications to Differential Equations, 2^nd edition, Springer, 2013. doi: 10.1007/978-3-0348-0387-8. Google Scholar
[18]	L. F. Faria, O. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[25]	R. Kajikiya, M. 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[32]	P. Pucci and J. Serrin, The Maximum Principle, Springer, 2007. doi: 10.1007/978-3-7643-8145-5. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[39]	V. E. Zakharov, Collapse of Langmuir waves, Soviet Journal of Experimental and Theoretical Physics, 35 (1972), 908-914. Google Scholar

Figure 1. The global minimum $m$ of ${E_{\alpha, \beta }}$ on $ W_0^{1, p}$.

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Figure 2. The least energy $d$ on ${\mathcal{N}_{\alpha, \beta }}$.

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