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Keywords:
Picone’s identity; half-linear PDE; $p$-Laplacian; variational technique
Picone’s identity; half-linear PDE; $p$-Laplacian; variational technique
Summary:
The Picone-type identity for the half-linear second order partial differential equation \[ \sum _{i=1}^n\frac{\partial }{\partial x_i} \Phi \bigg (\frac{\partial u}{\partial x_i}\bigg )+c(x)\Phi (u)=0,\quad \Phi (u):=|u|^{p-2}u,\ p>1, \] is established and some applications of this identity are suggested.
References:
[A-H-1] W. Allegretto, Y. X. Huang: A Picone’s identity for the $p$-Laplacian and applications. Nonlin. Anal. 32 (1998), 819–830. DOI 10.1016/S0362-546X(97)00530-0 | MR 1618334
[A-H-2] W. Allegretto, Y. X. Huang: Principal eigenvalues and Sturm comparison via Picone’s identity. J. Differ. Equations 156 (1999), 427–438. DOI 10.1006/jdeq.1998.3596 | MR 1705379
[B] M. Bohner: Linear Hamiltonian difference systems: disconjugacy and Jacobi-type conditions. J. Math. Anal. Appl. 199 (1996), 804–826. MR 1386607 | Zbl 0855.39018
[G-B-1] G. Bognár: A lower bound for smallest eigenvalue of some nonlinear eigenvalue problems on convex domains in two dimensions. Appl. Math. 51 (1993), 277–288. MR 1279006
[G-B-2] G. Bognár: On the solution of some nonlinear boundary value problem. Publ. Univ. Miskolc, Ser. D, Nat. Sci. Math. 36 (1995), 13–22.
[G-B-3] G. Bognár: Existence theorems for eigenvalues of a nonlinear eigenvalue problem. Commun. Appl. Nonlinear Anal. 4 (1997), 93–102.
[Diaz] J. I. Díaz: Nonlinear Partial Differential Equations and Free Boundaries. Vol I. Elliptic Equations, Pitman, London, 1985. MR 0853732
[D-1] O. Došlý: Oscillation criteria for half-linear second order differential equations. Hiroshima Math. J. 28 (1998), 507–521. DOI 10.32917/hmj/1206126680 | MR 1657543
[D-2] O. Došlý: Methods of oscillation theory of half-linear second order differential equations. Czechoslovak Math. J. 50 (2000), 657–671. DOI 10.1023/A:1022854131381 | MR 1777486
[D-M] O. Došlý, R. Mařík: Nonexistence of positive solutions of PDE’s with $p$-Laplacian. Acta Math. Hungar. 90 (2001), 89–107. DOI 10.1023/A:1006739909182 | MR 1910321
[D-K-N] P. Drábek, A. Kufner, F. Nicolosi: Nonlinear Elliptic Equations—Singular and Degenerate Case. University of West Bohemia Press, Plzeň, 1996.
[D] D. R. Dunninger: A Sturm comparison theorem for some degenerate quasilinear eliptic operators. Boll. UMI 9 (1995), 117–121. MR 1324611
[J-K] J. Jaroš, T. Kusano: A Picone type identity for second order half-linear differential equations. Acta Math. Univ. Comen. 68 (1999), 117–121. MR 1711081
[J-K-Y] J. Jaroš, T. Kusano, N. Yosida: A Picone-type identity and Sturmian comparison and oscillation theorems for a class of half-linear partial differential equations of the second order. Nonlinear Anal. TMA 40 (2000), 381–395. MR 1768900
[K-1] K. Kreith: Oscillation Theory. Lectures Notes in Math. No. 324, Springer, Berlin, 1973. Zbl 0258.35001
[K-2] K. Kreith: Picone’s identity and generalizations. Rend. Mat. 8 (1975), 251–261. MR 0374561 | Zbl 0341.34002
[K-P] W. C. Kelley, A. C. Peterson: Difference Equations: An Introduction with Applications. Acad. Press, San Diego, 1991. MR 1142573
[M-1] R. Mařík: Oscillation criteria for PDE with $p$-Laplacian via the Riccati technique. J. Math. Anal. Appl. 248 (2000), 290–308. DOI 10.1006/jmaa.2000.6901 | MR 1772598
[M-2] R. Mařík: Oscillation criteria for the Schrödinger PDE. Adv. Math. Sci. Appl. 10 (2000), 495–511. MR 1807439
[M] D. J. Mirzov: Asymptotic Properties of Solutions of Nonlinear Nonautonomous Ordinary Differential Systems. Adygea, Maikop, 1993.
[M-P] W. F. Moss, J. Piepenbrick: Positive solutions of elliptic equations. Pacific J. Math. 75 (1978), 219–226. DOI 10.2140/pjm.1978.75.219 | MR 0500041
[M-Pf-1] E. Müller-Pfeiffer: On the existence of nodal domain for elliptic differential operators. Proc. Roy. Soc. Edinburgh 94A (1983), 287–299. MR 0709722
[M-Pf-2] E. Müller-Pfeiffer: An extension of the Sturm-Picone theorem to elliptic differential equations. Proc. Roy. Soc. Edinburgh 97A (1984), 209–215. MR 0751193
[Pic] M. Picone: Sui valori eccezionalle di un parametro da cui dipende un’ equatione differenzialle lineare del secondo ordine. Ann. Scuola Norm. Sup. Pisa 11 (1910), 1–141.
[R] R. T. Rockafeller: Convex Analysis. Princeton, 1970.
[Sch] U. W. Schminke: The lower spectrum of Schrödinger operators. Arch. Rat. Mech. Anal. 75 (1981), 147–155. DOI 10.1007/BF00250476 | MR 0605526
[Sw] C. A. Swanson: Picone’s identity. Rend Mat. 8 (1975), 373–397. MR 0402188 | Zbl 0327.34028
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