×

Inverse nodal problems for the \(p\)-Laplacian with eigenparameter dependent energy functions. (English) Zbl 1383.34027

Summary: We study the inverse nodal problems for the \(p\)-Laplacian with two energy functions \[ \begin{cases} - (| y' | ^{p-2}y')'= (p-1) (\kappa^{2}-\kappa pq_{1}(x)-q_{0}(x)) | y| ^{p-2}y, \\ y(0)\sin_{p}'(\alpha)+y'(0)\sin_{p}(\alpha)=0, \\ y(1)\sin_{p}'(\beta)+y'(1)\sin_{p}(\beta)=0,\end{cases} \] where \(p>1\), \(\kappa\) is a spectral parameter, \(\alpha,\beta\in [0,\pi_{p}]\), \(\sin_{p}(x)\) is the generalized sine function and \(\pi_{p}\) is the generalized \(\pi\) constant. We use a Prüfer substitution derived by \(\sin _{p}(x)\) to find the asymptotic expansions of the eigenvalues and nodal lengths. Furthermore, we consider the inverse nodal problem and give the reconstruction formulas for the boundary conditions \(\alpha\), \(\beta\), and the energy functions \(q_{1}\), \(q_{0}\) by only using the information of nodal data.

MSC:

34A55 Inverse problems involving ordinary differential equations
34L05 General spectral theory of ordinary differential operators
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
47E05 General theory of ordinary differential operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bhuvaneswari, V, Lingeshwaran, S, Balachandran, K: Weak solutions for p-Laplacian equation. Adv. Nonlinear Anal. 1(4), 319-334 (2012) · Zbl 1277.35117
[2] Li, C, Agarwal, R, Tang, C-L: Infinitely many periodic solutions for ordinary p-Laplacian systems. Adv. Nonlinear Anal. 4(4), 251-261 (2015) · Zbl 1350.34038
[3] Radulescu, V: Finitely many solutions for a class of boundary value problems with superlinear convex nonlinearity. Arch. Math. (Basel) 84, 538-550 (2005) · Zbl 1075.34026 · doi:10.1007/s00013-005-1218-0
[4] Radulescu, V, Repovs, D: Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis. CRC Press/Taylor and Francis Group, Boca Raton (2015) · Zbl 1343.35003 · doi:10.1201/b18601
[5] Elbert, Á., A half-linear second order differential equation, Szeged, 1979, Amsterdam · Zbl 0511.34006
[6] Lindqvist, P: On the equation div(|∇u|p−2∇u)+λ|u|p−\(2u=0\operatorname{div}(|\nabla u|^{p-2}\nabla u)+\lambda |u|^{p-2}u=0\). Proc. Am. Math. Soc. 109(1), 157-164 (1990); Addendum: 116(2), 583-584 (1992) · Zbl 0714.35029
[7] Binding, PA, Drábek, P: Sturm-Liouville theory for the p-Laplacian. Studia Sci. Math. Hung. 40, 373-396 (2003) · Zbl 1048.34056
[8] Tamarkin, JD: On Some Problems of the Theory of Ordinary Linear Differential Equations. Petrograd (1917) · Zbl 1256.34013
[9] Keldysh, MV: On eigenvalues and eigenfunctions of some classes of nonselfadjoint equations. Dokl. Akad. Nauk SSSR 77, 11-14 (1951) · Zbl 0045.39402
[10] Jaulent, M, Jean, C: The inverse s-wave scattering problem for a class of potentials depending on energy. Commun. Math. Phys. 28, 177-220 (1972) · doi:10.1007/BF01645775
[11] Kostyuchenko, AG, Shkalikov, AA: Selfadjoint quadratic operator pencils and elliptic problems. Funkc. Anal. Prilozh. 17(2), 38-61 (1983); English transl.: Funct. Anal. Appl. 17, 109-128 (1983) · Zbl 0531.47017
[12] Markus, AS: Introduction to the Spectral Theory of Polynomial Operator Pencils. Shtinitsa, Kishinev (1986); English transl.: Am. Math. Soc., Providence (1988) · Zbl 0678.47006
[13] Showalter, RE: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. Mathematical Surveys and Monographs, vol. 49. Am. Math. Soc., Providence (1997) · Zbl 0870.35004
[14] Bensoussan, A, Frehse, J: Regularity Results for Nonlinear Elliptic Systems and Applications. Applied Mathematical Sciences, vol. 151. Springer, Berlin (2002) · Zbl 1055.35002
[15] Gasymov, MG, Guseinov, GS: Determination of diffusion operator on spectral data. Dokl. Akad. Nauk Azerb. SSR 37(2), 19-23 (1981) · Zbl 0479.34009
[16] Koyunbakan, H: Inverse problem for a quadratic pencil of Sturm-Liouville operator. J. Math. Anal. Appl. 378, 549-554 (2011) · Zbl 1221.34042 · doi:10.1016/j.jmaa.2011.01.069
[17] Hochstadt, H, Lieberman, B: An inverse Sturm-Liouville problem with mixed given data. SIAM J. Appl. Math. 34(4), 676-680 (1978) · Zbl 0418.34032 · doi:10.1137/0134054
[18] Yang, CF: A half-inverse problem for the coefficients for a diffusion equation. Chin. Ann. Math., Ser. A 32, 89-96 (2011) · Zbl 1240.35271 · doi:10.1007/s11401-010-0622-3
[19] Yang, CF, Guo, YX: Determination of a differential pencil from interior spectral data. J. Math. Anal. Appl. 375, 284-293 (2011) · Zbl 1210.34020 · doi:10.1016/j.jmaa.2010.09.011
[20] Buterin, SA, Shieh, CT: Incomplete inverse spectral and nodal problems for differential pencils. Results Math. 62, 167-179 (2012) · Zbl 1256.34010 · doi:10.1007/s00025-011-0137-6
[21] Yang, CF, Zettl, A: Half inverse problems for quadratic pencils of Sturm-Liouville operators. Taiwan. J. Math. 16(5), 1829-1846 (2012) · Zbl 1256.34013
[22] Buterin, SA, Shieh, CT: Inverse nodal problem for differential pencils. Appl. Math. Lett. 22, 1240-1247 (2009) · Zbl 1173.34304 · doi:10.1016/j.aml.2009.01.037
[23] Koyunbakan, H: Inverse nodal problem for p-Laplacian energy-dependent Sturm-Liouville equation. Bound. Value Probl. 2013, 272 (2013). doi:10.1186/1687-2770-2013-272 · Zbl 1291.34036 · doi:10.1186/1687-2770-2013-272
[24] Wang, WC, Cheng, YH, Lian, WC: Inverse nodal problems for the p-Laplacian with eigenparameter dependent boundary conditions. Math. Comput. Model. 54, 2718-2724 (2011) · Zbl 1235.34044 · doi:10.1016/j.mcm.2011.06.059
[25] Cheng, YH, Law, CK, Lian, WC, Wang, WC: An inverse nodal problem and Ambarzumyan problem for the periodic p-Laplacian operator with integrable potentials. Taiwan. J. Math. 19, 1305-1316 (2015) · Zbl 1357.34036
[26] Rudin, W: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987) · Zbl 0925.00005
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.