Let m be a positive integer and B be the unit ball of Rn （n≥2）. We investigate the existence, uniqueness and the asymptotic behavior of a positive continuous solution to the following semilinear polyharmonic bounda...Let m be a positive integer and B be the unit ball of Rn （n≥2）. We investigate the existence, uniqueness and the asymptotic behavior of a positive continuous solution to the following semilinear polyharmonic boundary value problem （-△）mu=a1（x）uα1＋a2（x）uα2 , lim｜x｜→1 u（x） （1-｜x｜）m-1 =0, where α1,α2∈（-1, 1） and a1, a2 are two nonnegative measurable functions on B satisfying some appropriate assumptions related to Karamata regular variation theory.展开更多
In this paper, the authors aim at proving two existence results of fractional differential boundary value problems of the form （Pa,bα）{D^au（x）＋f（x,u（x））=0,x∈（0,1）,u（0）=u（1）=0,D^a-3u（0）=a,u^（1）=-6w...In this paper, the authors aim at proving two existence results of fractional differential boundary value problems of the form （Pa,bα）{D^au（x）＋f（x,u（x））=0,x∈（0,1）,u（0）=u（1）=0,D^a-3u（0）=a,u^（1）=-6where 3 ≤ a 〈 4, D^ is the standard Riemann-Liouville fractional derivative and a, b are nonnegative constants. First the authors suppose that f（x, t） = -p（x）t^σ, with cr ~ （-1, 1） and p being a nonnegative continuous function that may be singular at x - 0 or x - 1 and satisfies some conditions related to the Karamata regular variation theory. Combining sharp estimates on some potential functions and the Sch^uder fixed point theorem, the authors prove the existence of a unique positive continuous solution to problem （P0,0）. Global estimates on such a solution are also obtained. To state the second existence result, the authors assume that a, b are nonnegative constants such that a ＋ b 〉 0 and f（x, t） -= tφ（x, t）, with φ（x, t） being a nonnegative continuous function in （0, 1） × [0, ∞） that is required to satisfy some suitable integrability condition. Using estimates on the Green＇s function and a perturbation argument, the authors prove the existence and uniqueness of a positive continuous solution u to problem （Pa,b）, which behaves like the unique solution of the homogeneous problem corresponding the existence results. to （Pa,b）. Some examples are given to illustrate the existence results.,展开更多
文摘Let m be a positive integer and B be the unit ball of Rn （n≥2）. We investigate the existence, uniqueness and the asymptotic behavior of a positive continuous solution to the following semilinear polyharmonic boundary value problem （-△）mu=a1（x）uα1＋a2（x）uα2 , lim｜x｜→1 u（x） （1-｜x｜）m-1 =0, where α1,α2∈（-1, 1） and a1, a2 are two nonnegative measurable functions on B satisfying some appropriate assumptions related to Karamata regular variation theory.
基金funded by the National Plan for Science,Technology and Innovation(MAARIFAH),King Abdulaziz City for Science and Technology,Kingdom of Saudi Arabia,Award Number(No.13-MAT2137-02)
文摘In this paper, the authors aim at proving two existence results of fractional differential boundary value problems of the form （Pa,bα）{D^au（x）＋f（x,u（x））=0,x∈（0,1）,u（0）=u（1）=0,D^a-3u（0）=a,u^（1）=-6where 3 ≤ a 〈 4, D^ is the standard Riemann-Liouville fractional derivative and a, b are nonnegative constants. First the authors suppose that f（x, t） = -p（x）t^σ, with cr ~ （-1, 1） and p being a nonnegative continuous function that may be singular at x - 0 or x - 1 and satisfies some conditions related to the Karamata regular variation theory. Combining sharp estimates on some potential functions and the Sch^uder fixed point theorem, the authors prove the existence of a unique positive continuous solution to problem （P0,0）. Global estimates on such a solution are also obtained. To state the second existence result, the authors assume that a, b are nonnegative constants such that a ＋ b 〉 0 and f（x, t） -= tφ（x, t）, with φ（x, t） being a nonnegative continuous function in （0, 1） × [0, ∞） that is required to satisfy some suitable integrability condition. Using estimates on the Green＇s function and a perturbation argument, the authors prove the existence and uniqueness of a positive continuous solution u to problem （Pa,b）, which behaves like the unique solution of the homogeneous problem corresponding the existence results. to （Pa,b）. Some examples are given to illustrate the existence results.,
