Bobkov, Vladimir; Tanaka, Mieko On sign-changing solutions for resonant \((p,q)\)-Laplace equations. (English) Zbl 1401.35127 Differ. Equ. Appl. 10, No. 2, 197-208 (2018). Summary: We provide two existence results for sign-changing solutions to the Dirichlet problem for the family of equations \(-\Delta_pu-\Delta_qu=\alpha|u|^{p-2}u+\beta|u|^{q-2} u\), where \(1<q<p\) and \(\alpha\), \(\beta\) are parameters. First, we show the existence in the resonant case \(\alpha\in\sigma(-\Delta_p)\) for sufficiently large \(\beta\), thereby generalizing previously known results. The obtained solutions have negative energy. Second, we show the existence for any \(\beta\geqslant \lambda_1(q)\) and sufficiently large \(\alpha\) under an additional nonresonant assumption, where \(\lambda_1(q)\) is the first eigenvalue of the \(q\)-Laplacian. The obtained solutions have positive energy. MSC: 35J92 Quasilinear elliptic equations with \(p\)-Laplacian 35J35 Variational methods for higher-order elliptic equations 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs Keywords:\((p,q)\)-Laplacian; generalized eigenvalue problem; nodal solutions × Cite Format Result Cite Review PDF Full Text: DOI References: [1] V. BOBKOV ANDM. TANAKA, On positive solutions for(p,q)-Laplace equations with two parame- ters, Calculus of Variations and Partial Differential Equations 54, 3 (2015), 3277–3301. · Zbl 1328.35052 [2] V. BOBKOV ANDM. TANAKA, On sign-changing solutions for(p,q)-Laplace equations with two parameters, Advances in Nonlinear Analysis, in press, · Zbl 1419.35071 [3] V. BOBKOV ANDM. TANAKA, Remarks on minimizers for(p,q)-Laplace equations with two param- eters, Communications on Pure and Applied Analysis 17, 3 (2018), 1219–1253. · Zbl 1400.35145 [4] P. CANDITO, S. A. MARANO,ANDK. PERERA, On a class of critical(p,q)-Laplacian problems, Nonlinear Differential Equations and Applications NoDEA 22, 6 (2015), 1959–1972. · Zbl 1328.35053 [5] K. C. CHANG, Infinite-Dimensional Morse Theory and Multiple Solution Problems, Birkh¨auser, Boston, 1993 · Zbl 0779.58005 [6] M. CUESTA, D. G. FIGUEIREDO ANDJ.P. GOSSEZ, A nodal domain property for the p -Laplacian, Comptes Rendus de l’Acad´emie des Sciences – Series I – Mathematics 330, 8 (2000), 669–673. · Zbl 0954.35124 [7] M. DEGIOVANNI ANDS. LANCELOTTI, Linking solutions for p -Laplace equations with nonlinearity at critical growth, Journal of Functional Analysis 256, 11 (2009), 3643–3659. · Zbl 1170.35418 [8] P. DRABEK AND´S. B. ROBINSON, Resonance problems for the p -Laplacian, Journal of Functional Analysis 169, 1 (1999), 189–200. · Zbl 0940.35087 [9] P. DRABEK AND´S. B. ROBINSON, On the generalization of the Courant nodal domain theorem, Journal of Differential Equations 181, 1 (2002), 58–71. · Zbl 1163.35449 [10] E. R. FADELL ANDP. H. RABINOWITZ, Generalized cohomological index theories for Lie group ac- tions with an application to bifurcation questions for Hamiltonian systems, Inventiones mathematicae 45, 2 (1978), 139–174. · Zbl 0403.57001 [11] R. KAJIKIYA, M. TANAKA,ANDS. TANAKA, Bifurcation of positive solutions for the one- dimensional(p,q)-Laplace equation, Electronic Journal of Differential Equations 2017, 107 (2017), 1–37. · Zbl 1370.34038 [12] S. MARANO ANDS. MOSCONI, Some recent results on the Dirichlet problem for(p,q)-Laplace equations, Discrete and Continuous Dynamical Systems – Series S 11, 2 (2018), 279–291. · Zbl 1374.35137 [13] K. PERERA, Nontrivial critical groups in p -Laplacian problems via the Yang index, Topological Methods in Nonlinear Analysis 21, 2 (2003), 301–309. · Zbl 1039.47041 [14] M. TANAKA, Generalized eigenvalue problems for(p,q)-Laplacian with indefinite weight, Journal of Mathematical Analysis and Applications 419, (2) (2014), 1181–1192. · Zbl 1294.35051 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.