Symmetry of components for radial solutions of $\gamma$-Laplacian systems
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In this paper, we give several sufficient conditions ensuring that any positive radial solution $(u,v)$ of the following $\gamma$-Laplacian systems in the whole space ${\mathbb R^n}$ has the components symmetry property $u\equiv v$ $$ \left\{ \begin{array}{ll} &-{\rm div}(|\nabla u|^{\gamma-2}\nabla u) =f(u,v)\quad \text{in} \ \ {\mathbb R^n},\\ &-{\rm div}(|\nabla v|^{\gamma-2}\nabla v) =g(u,v)\quad \text{in} \ \ {\mathbb R^n}. \end{array} \right. $$ Here $n>\gamma$, $\gamma>1$. Thus, the systems will be reduced to a single $\gamma$-Laplacian equation: $$ -{\rm div}(|\nabla u|^{\gamma-2} \nabla u) =f(u) \quad \text{in} \ \ {\mathbb R^n}. $$ Our proofs are based on suitable comparation principle arguments, combined with properties of radial solutions.
Abstract
In this paper, we give several sufficient conditions ensuring that any positive radial solution $(u,v)$ of the following $\gamma$-Laplacian systems in the whole space ${\mathbb R^n}$ has the components symmetry property $u\equiv v$ $$ \left\{ \begin{array}{ll} &-{\rm div}(|\nabla u|^{\gamma-2}\nabla u) =f(u,v)\quad \text{in} \ \ {\mathbb R^n},\\ &-{\rm div}(|\nabla v|^{\gamma-2}\nabla v) =g(u,v)\quad \text{in} \ \ {\mathbb R^n}. \end{array} \right. $$ Here $n>\gamma$, $\gamma>1$. Thus, the systems will be reduced to a single $\gamma$-Laplacian equation: $$ -{\rm div}(|\nabla u|^{\gamma-2} \nabla u) =f(u) \quad \text{in} \ \ {\mathbb R^n}. $$ Our proofs are based on suitable comparation principle arguments, combined with properties of radial solutions.
참고문헌 (14)
- J. Bourgain, "Global solutions of Nonlinear Schr¨odinger Equations" (1999), Providence
- J. Byeon, "Symmetry and monotonicity of least energy solutions", Calc. Var. PDEs, 36(4), pp.481~492 (2009)
- T. Cazenave, "Semilinear Schr¨odinger equations" (2003), New York University, Courant Institute of Mathematical Sciences
- W. Chen, "Classification of solutions of some nonlinear elliptic equations", Duke Math. J, 63(3), pp.615~622 (1991)
- W. Chen, "Classification of positive solutions for nonlinear differential and integral systems with critical exponents", Acta Math. Sci. Ser. B Engl. Ed, 29(4), pp.949~960 (2009)
- M. Franca, "Classification of positive solutions of p-Laplace equation with a growth term", Arch. Math. (BRNO), 40(4), pp.415~434 (2004)
- N. Kawano, "Structure theorems for positive radial solutions to div(|Du|m−2Du) + K(|x|)u q = 0 in R n", J. Math. Soc. Japan, 45(4), pp.719~742 (1993)
- Y. Lei, "Integrability and asymptotics of positive solutions of a γ-Laplacian system", J. Differential Equations, 252(3), pp.2739~2758 (2012)
- Y. Lei, "Decay estimation for positive solutions of a γ-Laplace equation", Discrete Contin. Dyn. Syst, 30(2), pp.547~558 (2011)
- C. Li, "Uniqueness of positive bound states to Schr¨odinger systems with critical exponents", SIAM J. Math. Anal, 40(3), pp.1049~1057 (2008)
- T. Lin, "Ground state of N coupled nonlinear Schr¨odinger equations in Rn, n ≤ 3", Commun. Math. Phys, 255(3), pp.629~653 (2005)
- P. Quittner, "Symmetry of components for semilinear elliptic systems", SIAM J. Math. Anal, 44(4), pp.2545~2559 (2012)
- J. Serrin, "Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities", Acta Math, 189(1), pp.79~142 (2002)
- Ph. Souplet, "The proof of the Lane-Emden conjecture in four space dimensions", Adv. Math, 221(5), pp.1409~1427 (2009)