The existence and nonexistence of entire positive solutions of semilinear elliptic systems with gradient term

Abstract We show the existence and nonexistence of entire positive solutions for semilinear elliptic system with gradient term Δ u + | ∇ u | = p ( | x | ) f ( u , v ) , Δ v + | ∇ v | = q ( | x | ) g ( u , v ) on R N , N ⩾ 3 , provided that nonlinearities f and g are positive and continuous, the potentials p and q are continuous, c-positive and satisfy appropriate growth conditions at infinity. We find that entire large positive solutions fail to exist if f and g are sublinear and p and q have fast decay at infinity, while if f and g satisfy some growth conditions at infinity, and p, q are of slow decay or fast decay at infinity, then the system has infinitely many entire solutions, which are large or bounded.

[1]  Yongli Song,et al.  Existence of entire large positive solutions of a semilinear elliptic system , 2004, Appl. Math. Comput..

[2]  Shaohua Chen,et al.  Existence and nonexistence of positive radial solutions for a class of semilinear elliptic system , 1999 .

[3]  Laurent Véron,et al.  Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations , 1997 .

[4]  P. Quittner.  Blow-up for semilinear parabolic equations with a gradient term , 1991 .

[5]  Ester Giarrusso,et al.  On Blow Up Solutions of a Quasilinear Elliptic Equation , 2000 .

[6]  Constantin P. Niculescu,et al.  Explosive solutions of elliptic equations with absorption and nonlinear gradient term , 2002 .

[7]  Marius Ghergu,et al.  Explosive solutions of semilinear elliptic systems with gradient term. , 2003 .

[8]  D. G. Figueiredo,et al.  Decay, symmetry and existence of solutions of semilinear elliptic systems , 1998 .

[9]  Cecilia S. Yarur.  Existence of continuous and singular ground states for semilinear elliptic systems , 1998 .

[10]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[11]  M. García-Huidobro,et al.  Existence and Nonexistence of Positive Singular Solutions for a Class of Semilinear Elliptic Systems , 1996 .

[12]  J. Serrin,et al.  Non-existence of positive solutions of Lane-Emden systems , 1996, Differential and Integral Equations.

[13]  James Serrin,et al.  Existence of positive entire solutions of elliptic hamiltonian systems , 1998 .

[14]  R. Dalmasso.  Existence and uniqueness of positive solutions of semilinear elliptic system , 2000 .

[15]  Catherine Bandle,et al.  Boundary blow up for semilinear elliptic equations with nonlinear gradient terms , 1996, Advances in Differential Equations.

[16]  Yihong Du,et al.  Blow-Up Solutions for a Class of Semilinear Elliptic and Parabolic Equations , 1999, SIAM J. Math. Anal..

[17]  P. Lions,et al.  Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints , 1989 .

[18]  Vicenţiu D. Rădulescu,et al.  Entire solutions blowing up at infinity for semilinear elliptic systems , 2002 .

[19]  Vicentiu D. Rădulescu,et al.  Singular elliptic problems : bifurcation and asymptotic analysis , 2008 .

[20]  Alan V. Lair,et al.  Existence of Entire Large Positive Solutions of Semilinear Elliptic Systems , 2000 .

[21]  Philippe Clément,et al.  Positive solutions for a quasilinear system via blow up , 1993 .

[22]  Alan V. Lair,et al.  Entire Solution of a Singular Semilinear Elliptic Problem , 1996 .

[23]  Xuefeng Wang,et al.  Existence and Nonexistence of Entire Positive Solutions of Semilinear Elliptic Systems , 2002 .

[24]  P. Clément,et al.  Department of technical mathematics and informatics university of technology , 1993 .