zbMATH — the first resource for mathematics

Remarks on minimizers for \((p,q)\)-Laplace equations with two parameters. (English) Zbl 1400.35145
Summary: We study in detail the existence, nonexistence and behavior of global minimizers, ground states and corresponding energy levels of the \((p, q)\)-Laplace equation \(-\Delta_p u -\Delta_q u = \alpha |u|^{p-2}u + \beta |u|^{q-2}u\) in a bounded domain \(\Omega \subset \mathbb{R}^N\) under zero Dirichlet boundary condition, where \(p > q > 1\) and \(\alpha, \beta \in \mathbb{R}\). A curve on the \((\alpha, \beta)\)-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.

35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35J40 Boundary value problems for higher-order elliptic equations
35J35 Variational methods for higher-order elliptic equations
Full Text: DOI
[1] S. Aizicovici; N. S. Papageorgiou; V. Staicu, Nodal solutions for (p, 2)-equations, Transactions of the American Mathematical Society, 367, 7343, (2015) · Zbl 1357.35165
[2] M. J. Alves; R. B. Assunç ao; O. H. Miyagaki, Existence result for a class of quasilinear elliptic equations with (pq)-Laplacian and vanishing potentials, Illinois Journal of Mathematics, 59, 545, (2015) · Zbl 1352.35061
[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, 725, (1987) · Zbl 0633.35061
[4] J. Bellazzini; N. Visciglia, MAX-MIN characterization of the mountain pass energy level for a class of variational problems, Proceedings of the American Mathematical Society, 138, 3335, (2010) · Zbl 1202.58009
[5] V. Benci; P. D’Avenia; D. Fortunato; L. Pisani, Solitons in several space dimensions: Derrick’s problem and infinitely many solutions, Archive for Rational Mechanics and Analysis, 154, 297, (2000) · Zbl 0973.35161
[6] V. Bobkov; M. Tanaka, On positive solutions for (p, q)-Laplace equations with two parameters, Calculus of Variations and Partial Differential Equations, 54, 3277, (2015) · Zbl 1328.35052
[7] V. Bobkov and M. Tanaka, On sign-changing solutions for (p, q)-Laplace equations with two parameters Advances in Nonlinear Analysis.
[8] P. J. Bushell; D. E. Edmunds, Remarks on generalised trigonometric functions, Rocky Mountain Journal of Mathematics, 42, 25, (2012) · Zbl 1246.33001
[9] J. W. Cahn; J. E. Hilliard, Free energy of a nonuniform system. I. interfacial free energy, The Journal of chemical physics, 28, 258, (1958)
[10] L. Cherfils; Y. Il’yasov, On the stationary solutions of generalized reaction diffusion equations with p&q-Laplacian, Communications on Pure and Applied Mathematics, 4, 9, (2005) · Zbl 1210.35090
[11] I. Chueshov; 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, 777, (2006) · Zbl 1220.35014
[12] M. Colombo; M. Colombo, Regularity for double phase variational problems, Archive for Rational Mechanics and Analysis, 215, 443, (2015) · Zbl 1322.49065
[13] M. Cuesta; D. de Figueiredo; J.-P. Gossez, The beginning of the fučik spectrum for the p-Laplacian, Journal of Differential Equations, 159, 212, (1999) · Zbl 0947.35068
[14] L. Damascelli; B. Sciunzi, Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations, Journal of Differential Equations, 206, 483, (2004) · Zbl 1108.35069
[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, 103, (2002) · Zbl 1114.35318
[16] P. Drábek, A. Kufner and F. Nicolosi, Quasilinear Elliptic Equations with Degenerations and Singularities, Walter de Gruyter, 1997. · Zbl 0894.35002
[17] P. Drábek and J. Milota, Methods of Nonlinear Analysis: Applications to Differential Equations, 2\^{}{nd} edition, Springer, 2013.
[18] L. F. Faria; O. H. Miyagaki; 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, 687, (2014) · Zbl 1315.35114
[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, 507, (2011) · Zbl 1211.35114
[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, 391, (2003) · Zbl 1023.35040
[21] J. Fleckinger-Pellé; P. Takáč, An improved Poincaré inequality and the p-Laplacian at resonance for p>2, Advances in Differential Equations, 7, 951, (2002) · Zbl 1208.35049
[22] Y. S. Il’yasov, Bifurcation calculus by the extended functional method, Functional Analysis and Its Applications, 41, 18, (2007) · Zbl 1124.35307
[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. · Zbl 1435.35185
[24] L. Jeanjean; K. Tanaka, A remark on least energy solutions in \(\mathbb{R}^N\), Proceedings of the American Mathematical Society, 131, 2399, (2003) · Zbl 1094.35049
[25] R. Kajikiya; M. Tanaka; S. Tanaka, Bifurcation of positive solutions for the one dimensional (p, q)-Laplace equation, Electronic Journal of Differential Equations, 2017, 1, (2017) · Zbl 1370.34038
[26] G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Analysis: Theory, Methods & Applications, 12, 1203, (1988) · Zbl 0675.35042
[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, 311, (1991) · Zbl 0742.35028
[28] S. Marano; S. Mosconi, Some recent results on the Dirichlet problem for (p, q)-Laplace equations, Discrete and Continuous Dynamical Systems -Series S, 11, 279, (2018) · Zbl 1374.35137
[29] S. A. Marano; N. S. Papageorgiou, Constant-sign and nodal solutions of coercive \((p, q)\)-Laplacian problems, Nonlinear Analysis: Theory, Methods & Applications, 77, 118, (2013) · Zbl 1260.35036
[30] D. Motreanu; M. Tanaka, On a positive solution for \((p, q)\)-Laplace equation with indefinite weight, Minimax Theory and its Applications, 1, 1, (2016) · Zbl 1334.35069
[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. · Zbl 1184.35001
[32] P. Pucci and J. Serrin, The Maximum Principle, Springer, 2007. · Zbl 1134.35001
[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. · Zbl 1354.35045
[34] P. Takáč, On the Fredholm alternative for the \(p\)-Laplacian at the first eigenvalue, Indiana University Mathematics Journal, 51, 187, (2002) · Zbl 1035.35046
[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, 1181, (2014) · Zbl 1294.35051
[37] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, Journal of Differential equations, 51, 126, (1984) · Zbl 0488.35017
[38] H. Yin; Z. Yang, A class of p-q-Laplacian type equation with concave-convex nonlinearities in bounded domain, Journal of Mathematical Analysis and Applications, 382, 843, (2011) · Zbl 1222.35083
[39] V. E. Zakharov, Collapse of Langmuir waves, Soviet Journal of Experimental and Theoretical Physics, 35 (1972), 908-914.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.