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

Vladimir Bobkov; Mieko Tanaka

doi:10.3934/cpaa.2018059

Article Contents

2018, Volume 17, Issue 3: 1219-1253. Doi: 10.3934/cpaa.2018059

This issue Previous Article Multiplicity and concentration of solutions for nonlinear fractional elliptic equations with steep potential Next Article A loop type component in the non-negative solutions set of an indefinite elliptic problem

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.
* Corresponding author: V. Bobkov.

Received: May 31, 2017

Revised: May 31, 2017

Published: May 2018

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Abstract

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.

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

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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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References

	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	P. Drábek, A. Kufner and F. Nicolosi, Quasilinear Elliptic Equations with Degenerations and Singularities, Walter de Gruyter, 1997.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	P. Pucci and J. Serrin, The Maximum Principle, Springer, 2007. doi: 10.1007/978-3-7643-8145-5.
	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.
	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.
	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.
	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.
	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.
	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.
	V. E. Zakharov, Collapse of Langmuir waves, Soviet Journal of Experimental and Theoretical Physics, 35 (1972), 908-914.