scholarly journals Upper and Lower Solutions for a Homogeneous Dirichlet Problem with Nonlinear Diffusion and the Principle of Linearized Stability

2000 ◽  
Vol 30 (4) ◽  
pp. 1229-1236 ◽  
Author(s):  
Robert Stephen Cantrell ◽  
Chris Cosner
Keyword(s):  
Dirichlet Problem ◽  
Nonlinear Diffusion ◽  
Upper And Lower Solutions ◽  
Linearized Stability ◽  
Homogeneous Dirichlet Problem

scholarly journals Eigencurves of the p(·)-Biharmonic operator with a Hardy-type term

2020 ◽  
Vol 6 (2) ◽  
pp. 198-209
Author(s):  
Mohamed Laghzal ◽  
Abdelouahed El Khalil ◽  
My Driss Morchid Alaoui ◽  
Abdelfattah Touzani
Keyword(s):  
Dirichlet Problem ◽  
Variational Approach ◽  
Type Inequality ◽  
Biharmonic Operator ◽  
Nondecreasing Sequence ◽  
Hardy Type Inequality ◽  
Principal Curve ◽  
Hardy Type ◽  
Homogeneous Dirichlet Problem

AbstractThis paper is devoted to the study of the homogeneous Dirichlet problem for a singular nonlinear equation which involves the p(·)-biharmonic operator and a Hardy-type term that depend on the solution and with a parameter λ. By using a variational approach and min-max argument based on Ljusternik-Schnirelmann theory on C1-manifolds [13], we prove that the considered problem admits at least one nondecreasing sequence of positive eigencurves with a characterization of the principal curve μ1(λ) and also show that, the smallest curve μ1(λ) is positive for all 0 ≤ λ < CH, with CH is the optimal constant of Hardy type inequality.

scholarly journals On Dirichlet's boundary value problem for some formally hypoelliptic differntial operators

1988 ◽  
Vol 44 (3) ◽  
pp. 324-349 ◽  
Author(s):  
Niels Jacob
Keyword(s):  
Hilbert Space ◽  
Dirichlet Problem ◽  
Differential Operator ◽  
Boundary Value Problem ◽  
Differential Operators ◽  
Boundary Value ◽  
Divergence Form ◽  
Alternative Theorem ◽  
Sesquilinear Form ◽  
Homogeneous Dirichlet Problem

AbstractFor a class of formally hypoelliptic differential operators in divergence form we prove a generalized Gårding inequality. Using this inequality and further properties of the sesquilinear form generated by the differential operator a generalized homogeneous Dirichlet problem is treated in a suitable Hilbert space. In particular Fredholm's alternative theorem is proved to be valid.

scholarly journals Regions of existence for a class of nonlinear diffusion type problems

2020 ◽  
Vol 14 (1) ◽  
pp. 106-121 ◽  
Author(s):  
Amit Verma ◽  
Mandeep Singh ◽  
Ravi Agarwal
Keyword(s):  
Biological System ◽  
Nonlinear Diffusion ◽  
Applied Mathematics ◽  
Upper And Lower Solutions ◽  
Sufficient Conditions ◽  
Real Life ◽  
Modern Science ◽  
Diffusion Type ◽  
Theoretical Methods ◽  
Type Boundary

The regions of existence are established for a class of two point nonlinear diffusion type boundary value problems (NDBVP) ?S??(x)- ns?(x)- m x s?(x) = f(x,s), m > 0, n ? R, x ? (0,1), s?(0) = 0, a1s(1) + a2s?(1) = C, where a1 > 0, a2 ? 0, C ? R. These problems arise very frequently in many branches of engineering, applied mathematics, astronomy, biological system and modern science (see the existing literature of this paper). By using the concept of upper and lower solutions with monotone constructive technique, we derive some sufficient conditions for existence in the regions where ?f/?s ? 0 and ?f/?s ? 0. Theoretical methods are applied for a set of problems which arise in real life.

scholarly journals ON A DIRICHLET PROBLEM WITH p-LAPLACIAN AND SET-VALUED NONLINEARITY

2011 ◽  
Vol 86 (1) ◽  
pp. 83-89 ◽  
Author(s):  
S. A. MARANO
Keyword(s):  
Fixed Point ◽  
Dirichlet Problem ◽  
Banach Spaces ◽  
Differential Inclusion ◽  
Existence Of Solutions ◽  
Type Theorem ◽  
Special Cases ◽  
Homogeneous Dirichlet Problem ◽  
Point Type

AbstractThe existence of solutions to a homogeneous Dirichlet problem for a p-Laplacian differential inclusion is studied via a fixed-point type theorem concerning operator inclusions in Banach spaces. Some meaningful special cases are then worked out.

scholarly journals Existence and uniqueness of solutions for a semilinear elliptic system

2005 ◽  
Vol 2005 (10) ◽  
pp. 1507-1523 ◽  
Author(s):  
Robert Dalmasso
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equations ◽  
Existence And Uniqueness ◽  
Biharmonic Equation ◽  
Nonlinear Elliptic Equations ◽  
Uniqueness Of Solutions ◽  
Nonlinear Elliptic ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic System ◽  
Homogeneous Dirichlet Problem

We consider the existence, the nonexistence, and the uniqueness of solutions of some special systems of nonlinear elliptic equations with boundary conditions. In a particular case, the system reduces to the homogeneous Dirichlet problem for the biharmonic equationΔ2u=|u|pin a ball withp>0.

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