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 \(\mathbb{R}^n\).


35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J70 Degenerate elliptic equations
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[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
[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, () · 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 New York · Zbl 0191.09904
[11] Swanson, C.A., Picone’s identity, Rend. mat., 8, 373-397, (1975) · Zbl 0327.34028
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