×

A Picone-type identity and Sturmian comparison and oscillation theorems for a class of half-linear partial differential equations of second order. (English) Zbl 0954.35018

The well-known Picone’s identity plays an important role in the study of qualitative properties of solutions of the second-order linear homogeneous differential equations. It has been recently generalized to the half-linear differential operators \[ \begin{aligned} l_\alpha[y]&=(r(t)\left|y'\right|^{\alpha -1}y')'+q(t)\left|y\right|^{\alpha -1}y,\\ L_{\alpha }[y] &=(R(t)\left|z'\right|^{\alpha -1}z')'+Q(t)\left|z\right|^{\alpha -1}z, \end{aligned} \] where \(\alpha>0\) is a constant, and \(r,q,R,Q\) are real-valued continuous functions on an interval. Using a generalization of Picone’s identity to the linear elliptic operators \[ \begin{aligned} p[u]&=\nabla \cdot (a(x)\nabla u)+c(x)u,\\ P[v] & =\nabla \cdot (A(x)\nabla v)+C(x)v, \end{aligned} \] a number of authors developed Sturmian theory for second order linear elliptic equations. In this paper, the authors generalized Picone’s identity to the half-linear partial differential operators \[ \begin{aligned} p_\alpha[u]&=\nabla \cdot (a(x)\left|\nabla u\right|^{\alpha -1}\nabla u)'+c(x)\left|u\right|^{\alpha -1}u,\\ P_{\alpha }[v] & =\nabla \cdot (A(x)\left|\nabla v\right|^{\alpha -1}\nabla v)'+C(x)\left|v\right|^{\alpha-1}v,\end{aligned} \] where \(\alpha>0\) is a constant, and \(a,c,A,C\) are continuous (continuously differentiable) functions defined in a domain \(G\subset\mathbb{R}^n\). Then the obtained Picone-type identity is applied to prove Sturmian comparison and oscillation theorems for second-order half-linear degenerate elliptic equations of the form \(p_\alpha[u]=0\) or \(P_\alpha[v]=0\) in an unbounded domain in \(\mathbb{R}^n\).

MSC:

35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J70 Degenerate elliptic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Dunninger, D. R., A Sturm comparison theorem for some degenerate quasilinear elliptic operators, Boll. Un. Mat. Ital. A, 9, 7, 117-121 (1995) · Zbl 0834.35011
[2] Hardy, G.; Littlewood, J. E.; Pólya, G., Inequalities (1988), Cambridge University Press: Cambridge University Press Cambridge
[3] Jarǒs, J.; Kusano, T., On second-order half-linear differential equations with forcing term, Sūrikaisekikenkyūsho Kōkyūroku, 984, 191-197 (1997) · Zbl 0925.34039
[4] Jarǒs, J.; Kusano, T., A Picone type identity for second order half-linear differential equations, Acta Math. Univ. Comenian, 68, 117-121 (1999) · Zbl 0926.34023
[5] Kreith, K., Oscillation Theory, (Lecture Notes in Mathematics, Vol. 324 (1973), Springer: Springer Berlin) · Zbl 0155.46301
[6] Kreith, K., Picone’s identity and generalizations, Rend. Mat., 8, 251-261 (1975) · Zbl 0341.34002
[7] Kusano, T.; Naito, Y., Oscillation and nonoscillation criteria for second order quasilinear differential equations, Acta Math. Hungar., 76, 81-99 (1997) · Zbl 0906.34024
[8] Kusano, T.; Naito, Y.; Ogata, A., Strong oscillation and nonoscillation of quasilinear differential equations of second order, Differential Equations Dyn. Systems, 2, 1-10 (1994) · Zbl 0869.34031
[9] Picone, M., Sui valori eccezionali di un parametro da cui dipende un’equazione differenziale lineare ordinaria del second’ordine, Ann. Scuola Norm. Sup. Pisa, 11, 1-141 (1909) · JFM 41.0351.01
[10] Swanson, C. A., Comparison and Oscillation Theory of Linear Differential Equations (1968), Academic Press: Academic Press New York · Zbl 0191.09904
[11] Swanson, C. A., Picone’s identity, Rend. Mat., 8, 373-397 (1975) · Zbl 0327.34028
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.