Let B1 С RN be a unit ball centered at the origin. The main purpose of this paper is to discuss the critical dimension phenomenon for radial solutions of the following quasilinear elliptic problem involving critical ...Let B1 С RN be a unit ball centered at the origin. The main purpose of this paper is to discuss the critical dimension phenomenon for radial solutions of the following quasilinear elliptic problem involving critical Sobolev exponent and singular coefficients：{-div（｜△u｜p-2△u）=｜x｜s｜u｜p＊（s）-2u＋λ｜x｜t｜u｜p-2u, x∈B1, u｜σB1 =0, where t, s〉-p, 2≤p〈N, p＊（s）= （N＋s）pN-p andλ is a real parameter. We show particularly that the above problem exists infinitely many radial solutions if the space dimension N 〉p（p-1）t＋p（p2-p＋1） andλ∈（0,λ1,t）, whereλ1,t is the first eigenvalue of-△p with the Dirichlet boundary condition. Meanwhile, the nonexistence of sign-changing radial solutions is proved if the space dimension N ≤ （ps＋p） min{1, p＋t/p＋s}＋p2p-（p-1） min{1, p＋tp＋s} andλ〉0 is small.展开更多
In this paper N-dimensional singular, p-Laplace equations of the following form △pu：=N↑∑↑i=1Di（｜Du｜^p-2Diu）=f（｜x｜,u,｜Du｜u^-β,x∈R^N（N≥3） are considered, where p≥N,β〉0,and f：[0,∞）×[0,∞）&#...In this paper N-dimensional singular, p-Laplace equations of the following form △pu：=N↑∑↑i=1Di（｜Du｜^p-2Diu）=f（｜x｜,u,｜Du｜u^-β,x∈R^N（N≥3） are considered, where p≥N,β〉0,and f：[0,∞）×[0,∞）×[0,∞）is a continuous tunctlon. Some sufficient conditions are obtained for the existence of infinitely many radially positive entire solutions of the equation which are asymptotic to positive constant multiples of ｜x｜^（p-N）/（p-1） for p〉N or log｜x｜ for N-p as ｜x｜→∞.展开更多
基金supported by the National Natural Science Foundation of China(11326139,11326145)Tian Yuan Foundation(KJLD12067)Hubei Provincial Department of Education(Q20122504)
文摘Let B1 С RN be a unit ball centered at the origin. The main purpose of this paper is to discuss the critical dimension phenomenon for radial solutions of the following quasilinear elliptic problem involving critical Sobolev exponent and singular coefficients：{-div（｜△u｜p-2△u）=｜x｜s｜u｜p＊（s）-2u＋λ｜x｜t｜u｜p-2u, x∈B1, u｜σB1 =0, where t, s〉-p, 2≤p〈N, p＊（s）= （N＋s）pN-p andλ is a real parameter. We show particularly that the above problem exists infinitely many radial solutions if the space dimension N 〉p（p-1）t＋p（p2-p＋1） andλ∈（0,λ1,t）, whereλ1,t is the first eigenvalue of-△p with the Dirichlet boundary condition. Meanwhile, the nonexistence of sign-changing radial solutions is proved if the space dimension N ≤ （ps＋p） min{1, p＋t/p＋s}＋p2p-（p-1） min{1, p＋tp＋s} andλ〉0 is small.
基金The work is supported by the National Natural Science Foundation of China (10271056)the Natural Science Foundation of Fujian Province (F00018).
文摘In this paper N-dimensional singular, p-Laplace equations of the following form △pu：=N↑∑↑i=1Di（｜Du｜^p-2Diu）=f（｜x｜,u,｜Du｜u^-β,x∈R^N（N≥3） are considered, where p≥N,β〉0,and f：[0,∞）×[0,∞）×[0,∞）is a continuous tunctlon. Some sufficient conditions are obtained for the existence of infinitely many radially positive entire solutions of the equation which are asymptotic to positive constant multiples of ｜x｜^（p-N）/（p-1） for p〉N or log｜x｜ for N-p as ｜x｜→∞.
