This paper gives several criteria on the oscillatory behavior of solutions of the forced secondorder equation x″+ a(t)i(x) = g(t), where g(t) is oscillatory) by using a geometric idea. Asspecial cases these results i...This paper gives several criteria on the oscillatory behavior of solutions of the forced secondorder equation x″+ a(t)i(x) = g(t), where g(t) is oscillatory) by using a geometric idea. Asspecial cases these results include and improve some recent results, given by J. S. Wong. Thecriteria also solve the problem posed by H. Onose in Mathematical Reviews, 1986.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
We consider the quasilinear Schrdinger equations of the form-ε~2?u + V(x)u- ε~2?(u2)u = g(u), x ∈ R^N,where ε 〉 0 is a small parameter, the nonlinearity g(u) ∈ C^1(R) is an odd function with subcrit...We consider the quasilinear Schrdinger equations of the form-ε~2?u + V(x)u- ε~2?(u2)u = g(u), x ∈ R^N,where ε 〉 0 is a small parameter, the nonlinearity g(u) ∈ C^1(R) is an odd function with subcritical growth and V(x) is a positive Hlder continuous function which is bounded from below, away from zero, and infΛV(x) 0 such that for all ε∈(0, ε0],the above mentioned problem possesses a sign-changing solution uε which exhibits concentration profile around the local minimum point of V(x) as ε→ 0~+.展开更多
文摘This paper gives several criteria on the oscillatory behavior of solutions of the forced secondorder equation x″+ a(t)i(x) = g(t), where g(t) is oscillatory) by using a geometric idea. Asspecial cases these results include and improve some recent results, given by J. S. Wong. Thecriteria also solve the problem posed by H. Onose in Mathematical Reviews, 1986.
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.,
基金supported by National Natural Science Foundation of China(Grant Nos.11371160 and 11328101)the Program for Changjiang Scholars and Innovative Research Team in University(Grant No.#IRT13066)
文摘We consider the quasilinear Schrdinger equations of the form-ε~2?u + V(x)u- ε~2?(u2)u = g(u), x ∈ R^N,where ε 〉 0 is a small parameter, the nonlinearity g(u) ∈ C^1(R) is an odd function with subcritical growth and V(x) is a positive Hlder continuous function which is bounded from below, away from zero, and infΛV(x) 0 such that for all ε∈(0, ε0],the above mentioned problem possesses a sign-changing solution uε which exhibits concentration profile around the local minimum point of V(x) as ε→ 0~+.