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On sign-changing solutions for \((p,q)\)-Laplace equations with two parameters. (English) Zbl 1419.35071
Summary: We investigate the existence of nodal (sign-changing) solutions to the Dirichlet problem for a two-parametric family of partially homogeneous \((p,q)\)-Laplace equations \(-\Delta_{p}u-\Delta_{q}u=\alpha\vert u\vert^{p-2}u+\beta\vert u\vert^{q-2}u\) where \(p\neq q\). By virtue of the Nehari manifolds, the linking theorem, and descending flow, we explicitly characterize subsets of the \((\alpha,\beta)\)-plane which correspond to the existence of nodal solutions. In each subset the obtained solutions have prescribed signs of energy and, in some cases, exactly two nodal domains. The nonexistence of nodal solutions is also studied. Additionally, we explore several relations between eigenvalues and eigenfunctions of the \(p\)- and \(q\)-Laplacians in one dimension.

35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35A15 Variational methods applied to PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
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[1] S. Aizicovici, N. S. Papageorgiou and V. Staicu, Nodal solutions for {(p,2)}-equations, Trans. Amer. Math. Soc. 367 (2015), 7343-7372. · Zbl 1357.35165
[2] A. Anane, Simplicité et isolation de la premiere valeur propre du p-Laplacien avec poids, C. R. Acad. Sci. Paris Sér. I 305 (1987), no. 16, 725-728. · Zbl 0633.35061
[3] G. E. Andrews, R. Askeyand and R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999.
[4] S. Barile and G. M. Figueiredo, Some classes of eigenvalues problems for generalized {p&q}-Laplacian type operators on bounded domains, Nonlinear Anal. 119 (2015), 457-468. · Zbl 1328.35137
[5] T. Bartsch and Z. Liu, On a superlinear elliptic p-Laplacian equation, J. Differential Equations 198 (2004), no. 1, 149-175. · Zbl 1087.35034
[6] T. Bartsch, T. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math. 96 (2005), 1-18. · Zbl 1206.35086
[7] V. Benci, P. D’Avenia, D. Fortunato and L. Pisani, Solitons in several space dimensions: Derrick’s problem and infinitely many solutions, Arch. Ration. Mech. Anal. 154 (2000), no. 4, 297-324. · Zbl 0973.35161
[8] V. Benci, D. Fortunato and L. Pisani, Soliton like solutions of a Lorentz invariant equation in dimension 3, Rev. Math. Phys. 10 (1998), no. 3, 315-344. · Zbl 0921.35177
[9] V. Bobkov, Least energy nodal solutions for elliptic equations with indefinite nonlinearity, Electron. J. Qual. Theory Differ. Equ. 2014 (2014), Paper No. 56. · Zbl 1324.35047
[10] V. Bobkov and M. Tanaka, On positive solutions for {(p,q)}-Laplace equations with two parameters, Calc. Var. Partial Differential Equations 54 (2015), no. 3, 3277-3301. · Zbl 1328.35052
[11] P. J. Bushell and D. E. Edmunds, Remarks on generalised trigonometric functions, Rocky Mountain J. Math. 42 (2012), no. 1, 25-57. · Zbl 1246.33001
[12] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys. 28 (1958), no. 2, 258-267.
[13] P. Candito, S. A. Marano and K. Perera, On a class of critical {(p,q)}-Laplacian problems, NoDEA Nonlinear Differential Equations Appl. 22 (2015), no. 6, 1959-1972. · Zbl 1328.35053
[14] K. C. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser, Boston, 1993.
[15] L. Cherfils and Y. Il’yasov, On the stationary solutions of generalized reaction diffusion equations with {p&q}-Laplacian, Comm. Pure Appl. Math. 4 (2005), no. 1, 9-22. · Zbl 1210.35090
[16] M. Cuesta, D. de Figueiredo and J.-P. Gossez, The beginning of the Fučik spectrum for the p-Laplacian, J. Differential Equations 159 (1999), no. 1, 212-238. · Zbl 0947.35068
[17] P. Drábek and R. Manásevich, On the closed solution to some nonhomogeneous eigenvalue problems with p-Laplacian, Differential Integral Equations 12 (1999), no. 6, 773-788. · Zbl 1015.34071
[18] P. Drábek and S. B. Robinson, Resonance problems for the p-Laplacian, J. Funct. Anal. 169 (1999), no. 1, 189-200. · Zbl 0940.35087
[19] T. He, H. Yan, Z. Sun and M. Zhang, On nodal solutions for nonlinear elliptic equations with a nonhomogeneous differential operator, Nonlinear Anal. 118 (2015), 41-50. · Zbl 1317.35031
[20] R. Kajikiya, M. Tanaka and S. Tanaka, Bifurcation of positive solutions for the one dimensional {(p,q)}-Laplace equation, submitted. · Zbl 1370.34038
[21] G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), no. 11, 1203-1219. · Zbl 0675.35042
[22] G. M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations, Comm. Partial Differential Equations 16 (1991), no. 2-3, 311-361. · Zbl 0742.35028
[23] Z. L. Liu and J. X. Sun, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations, J. Differential Equations 172 (2001), 257-299. · Zbl 0995.58006
[24] S. A. Marano and N. S. Papageorgiou, Constant-sign and nodal solutions of coercive {(p,q)}-Laplacian problems, Nonlinear Anal. 77 (2013), 118-129. · Zbl 1260.35036
[25] S. Miyajima, D. Motreanu and M. Tanaka, Multiple existence results of solutions for the Neumann problems via super-and sub-solutions, J. Funct. Anal. 262 (2012), no. 4, 1921-1953. · Zbl 1276.35086
[26] R. Molle and D. Passaseo, Variational properties of the first curve of the Fučík spectrum for elliptic operators, Calc. Var. Partial Differential Equations 54 (2015), no. 4, 3735-3752. · Zbl 1336.35271
[27] D. Motreanu and M. Tanaka, Multiple existence results of solutions for quasilinear elliptic equations with a nonlinearity depending on a parameter, Ann. Mat. Pura Appl. (4) 193 (2014), no. 5, 1255-1282. · Zbl 1305.35070
[28] D. Motreanu and M. Tanaka, On a positive solution for {(p,q)}-Laplace equation with indefinite weight, Minimax Theory Appl. 1 (2016), no. 1, 1-20. · Zbl 1334.35069
[29] P. Pucci and J. Serrin, The maximum Principle, Progr. Nonlinear Differential Equations Appl. 73, Birkhäuser, Basel, 2007.
[30] H. R. Quoirin, An indefinite type equation involving two p-Laplacians, J. Math. Anal. Appl. 387 (2012), no. 1, 189-200. · Zbl 1229.35074
[31] M. Tanaka, Generalized eigenvalue problems for {(p,q)}-Laplacian with indefinite weight, J. Math. Anal. Appl. 419 (2014), no. 2, 1181-1192. · Zbl 1294.35051
[32] M. Tanaka, Uniqueness of a positive solution and existence of a sign-changing solution for {(p,q)}-Laplace equation, J. Nonlinear Funct. Anal. 2014 (2014), 1-15.
[33] V. E. Zakharov, Collapse of Langmuir waves, Sov. J. Exp. Theoret. Phys. 35 (1972), no. 5, 908-914.
[34] A. A. Zerouali and B. Karim, Existence and nonexistence of a positive solution for {(p,q)}-Laplacian with singular weight, Bol. Soc. Parana. Mat. (3) 34 (2016), 147-167. · Zbl 1424.35139
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