PICONE-TYPE IDENTITY AND COMPARISON RESULTS FOR A CLASS OF PARTIAL DIFFERENTIAL EQUATIONS OF ORDER 4m
نویسندگان
چکیده
Recently, an identity of the Picone type for the weighted p-biharmonic operator was established by the present author [5] and employed in proving comparison theorems and other qualitative results for a pair of fourth-order elliptic partial differential equations of the form ∆ ( a(x)|∆u|p−2∆u ) − c(x)|u|p−2u = 0, (1.1) and ∆ ( A(x)|∆v|p−2∆v ) − C(x)|v|p−2v = 0, (1.2) where p > 1, a,A ∈ C(Ḡ,R+), c, C ∈ C(Ḡ,R), G is a bounded domain in R with a piecewise smooth boundary ∂G, ∆ = (∂/∂x1) + . . .+ (∂/∂xn) is the usual Laplace operator and | · | denotes the Euclidean length of a vector in R. Roughly speaking, the identity from [5] states that if u and v are (classical) solutions of (1.1) and (1.2), respectively, and v(x) > 0 in the domain under consideration, then:
منابع مشابه
Picone’s Identity for the P-biharmonic Operator with Applications
In this article, a Picone-type identity for the weighted p-biharmonic operator is established and comparison results for a class of half-linear partial differential equations of fourth order based on this identity are derived.
متن کاملComparison Theorems for Half-linear Differential Equations of the Fourth Order
An identity of the Picone type for fourth-order half-linear ordinary differential operators of the form lα[x] ≡ (pφ(x′′))′′ − (rφ(x′))′ + qφ(x) and Lα[y] ≡ (Pφ(y′′))′′ − (Rφ(y′))′ +Qφ(y). where φ(u) := |u|α−1u, α > 0, u ∈ R, and p, q, r, P,Q and R are continuous functions on a given interval I is derived and then Sturmian comparison theory for the corresponding fourth-order equations lα[x] = 0 ...
متن کاملComparison Results for Two-dimensional Half-linear Cyclic Differential Systems
Two identities of the Picone type for a class of half-linear differential systems in the plane are established and the Sturmian comparison theory for such systems is developed with the help of these new formulas.
متن کاملNonlinear eigenvalue problems for a class of ordinary differential equations
originates in the works of Mirzov [17] and Elbert [5], who named these equations `half linear’. This theory extends various aspects of oscillation theory, such as the Picone identity [10], Sturm comparison theorem [16, 17], oscillation and nonoscillation criteria of Kneser type [16] and Hille type [14], and other oscillation criteria [3]. One branch of this research is the study of the eigenval...
متن کاملNonlinear Picone identities to Pseudo $p$-Laplace operator and applications
In this paper, we derive a nonlinear Picone identity to the pseudo p-Laplace operator, which contains some known Picone identities and removes a condition used in many previous papers. Some applications are given including a Liouville type theorem to the singular pseudo p-Laplace system, a Sturmian comparison principle to the pseudo p-Laplace equation, a new Hardy type inequality with weight an...
متن کامل