Perturbations of planar quasilinear differential systems

Abstract The quasilinear differential system x ′ = a x + b | y | p ⁎ − 2 y + k ( t , x , y ) , y ′ = c | x | p − 2 x + d y + l ( t , x , y ) is considered, where a, b, c and d are real constants with b 2 + c 2 > 0 , p and p ⁎ are positive numbers with ( 1 / p ) + ( 1 / p ⁎ ) = 1 , and k and l are continuous for t ≥ t 0 and small x 2 + y 2 . When p = 2 , this system is reduced to the linear perturbed system. It is shown that the behavior of solutions near the origin ( 0 , 0 ) is very similar to the behavior of solutions to the unperturbed system, that is, the system with k ≡ l ≡ 0 , near ( 0 , 0 ) , provided k and l are small in some sense. It is emphasized that this system can not be linearized at ( 0 , 0 ) when p ≠ 2 , because the Jacobian matrix can not be defined at ( 0 , 0 ) . Our result will be applicable to study radial solutions of the quasilinear elliptic equation with the differential operator r − ( γ − 1 ) ( r α | u ′ | β − a u ′ ) ′ , which includes p-Laplacian and k-Hessian.

[1]  E. Coddington,et al.  Theory of Ordinary Differential Equations , 1955 .

[2]  M. Bidaut-Véron.  Local and global behavior of solutions of quasilinear equations of Emden-Fowler type , 1989 .

[3]  M. Franca.  FOWLER TRANSFORMATION AND RADIAL SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS. PART 1: THE SUBCRITICAL AND THE SUPERCRITICAL CASE , 2008 .

[4]  M. Franca,et al.  Phase plane analysis for radial solutions to supercritical quasilinear elliptic equations in a ball , 2015 .

[5]  Satoshi Tanaka,et al.  Rectifiable and nonrectifiable solution curves of half-linear differential systems , 2018 .

[6]  Yasuhito Miyamoto,et al.  Generalized Joseph–Lundgren exponent and intersection properties for supercritical quasilinear elliptic equations , 2017 .

[7]  Z. Opluštil.  Oscillation criteria for two-dimensional system of non-linear ordinary differential equations , 2016 .

[8]  A. Tiryaki.  STURM-PICONE TYPE THEOREMS FOR NONLINEAR DIFFERENTIAL SYSTEMS , 2015 .

[9]  J. Jaros,et al.  Comparison results for two-dimensional half-linear cyclic differential systems , 2017 .

[10]  Á. Elbert,et al.  A half-linear second order differential equation , 1979 .

[11]  Z. Opluštil.  On non-oscillation for certain system of non-linear ordinary differential equations , 2017 .

[12]  I. Kiguradze,et al.  Solvability conditions for non-local boundary value problems for two-dimensional half-linear differe , 2011 .

[13]  J. D Mirzov,et al.  On some analogs of Sturm's and Kneser's theorems for nonlinear systems , 1976 .

[14]  M. Franca.  Classification of positive solutions of $p$-Laplace equation with a growth term , 2004 .

[15]  Ondřej Došlý,et al.  Half-linear differential equations , 2005 .

[16]  M. Franca.  Fowler transformation and radial solutions for quasilinear elliptic equations. Part 2: nonlinearities of mixed type , 2009 .

[17]  Nonoscillatory solutions of planar half-linear differential systems: a Riccati equation approach , 2018 .

[18]  J. Jaros,et al.  Elbert-type comparison theorems for a class of nonlinear Hamiltonian systems , 2016 .

[19]  J. Sugie,et al.  Global attractivity for half-linear differential systems with periodic coefficients , 2010 .

[20]  Florica C. Cîrstea,et al.  A Complete Classification of the Isolated Singularities for Nonlinear Elliptic Equations With Inverse Square Potentials , 2014 .

[21]  Shingo Takeuchi,et al.  Multiple-angle formulas of generalized trigonometric functions with two parameters , 2016, 1603.06709.

[22]  J. Sugie,et al.  Uniform global asymptotic stability for half-linear differential systems with time-varying coefficients , 2011, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[23]  J. Sugie,et al.  Global asymptotic stability for half-linear differential systems with coefficients of indefinite sign , 2008 .

[24]  Filippo Gazzola,et al.  Existence of ground states and free boundary problems for quasilinear elliptic operators , 2000, Advances in Differential Equations.

[25]  M. Garrione,et al.  Structure Results for Semilinear Elliptic Equations with Hardy Potentials , 2017 .

[26]  J. Sugie,et al.  ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF NONAUTONOMOUS HALF-LINEAR DIFFERENTIAL SYSTEMS , 2007 .

[27]  Satoshi Tanaka,et al.  Characteristic equation for autonomous planar half-linear differential systems , 2017 .

[28]  Yasuhito Miyamoto.  Intersection properties of radial solutions and global bifurcation diagrams for supercritical quasilinear elliptic equations , 2016 .