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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.
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참고문헌 (14)
참고문헌 신청
J. Bourgain / 1999 / Global solutions of Nonlinear Schr¨odinger Equations / Providence
J. Byeon / 2009 / Symmetry and monotonicity of least energy solutions / Calc. Var. PDEs 36(4):481~492
T. Cazenave / 2003 / Semilinear Schr¨odinger equations / New York University, Courant Institute of Mathematical Sciences
W. Chen / 1991 / Classification of solutions of some nonlinear elliptic equations / Duke Math. J 63(3):615~622
W. Chen / 2009 / Classification of positive solutions for nonlinear differential and integral systems with critical exponents / Acta Math. Sci. Ser. B Engl. Ed 29(4):949~960
J. Byeon / 2009 / Symmetry and monotonicity of least energy solutions / Calc. Var. PDEs 36(4):481~492
T. Cazenave / 2003 / Semilinear Schr¨odinger equations / New York University, Courant Institute of Mathematical Sciences
W. Chen / 1991 / Classification of solutions of some nonlinear elliptic equations / Duke Math. J 63(3):615~622
W. Chen / 2009 / Classification of positive solutions for nonlinear differential and integral systems with critical exponents / Acta Math. Sci. Ser. B Engl. Ed 29(4):949~960
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