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We establish an exact multiplicity result of positive solutions of u + f(u) = 0 in B n ; u = 0 on @B n , where B n is the unit ball, f satisses f 00 changes sign only once and asymptotically sublinear or linear. The nonlinearities which we concern here include f(u) = u(u ? b)(c ? u) for 0 < 2b < c and f(u) = u p ? u q for 1 < p < q. Precise global… (More)
where B” is the unit ball in R”. )I 2 I. and I. is a positive parameter. It is well known that iffis a smooth function, then any positive solution to the equation is radially symmetric, and all solutions can be pdrameterized by their maximum values. We develop a unitied approach to obtain the exact multiplicity of the positive solutions for a wide class ol’… (More)
For a class of Dirichlet problems in two dimensions, generalizing the model case ∆u+ λu(u− b)(c− u) = 0 in |x| < R,u = 0 on |x| = R, ∗Supported in part by the National Science Foundation.
Analytical methods are used to prove the existence of a periodic, symmetric solution in the 4-body problem with singularities. A numerical calculation and simulation are used to generate the orbit. The analytical method easily extends to any even number of bodies. Multiple simultaneous binary collisions are a key feature of the orbits generated.
We discuss the exact number of positive solutions of u+f(u) = 0 with homogeneous Dirichlet boundary condition and on the ball domain. The nonlinearity here includes f(u) = u q + u p for 0 < q < 1 < p n n?2 .
We study the radially symmetric solutions of a semilinear elliptic equation with a singular nonlinearity. Precise global bifurcation diagram of solutions is obtained, and the regularity of the solutions is also discussed.
For n-body problems, a central configuration (CC) plays an important role. In this paper, we establish the relation between the spatial pyramidal central configuration (PCC) and the planar central configuration. We prove that the base of PCC is also a CC and we also prove that for some given conditions a planar CC can be extended to a PCC. In particular, if… (More)