Bifurcation Structure Of Semilinear Elliptic Equations

In this paper, we study all regularity positive solutions of Dirichlct problem for the semilinear elliptic equation:and corresponding perturbed equation:when a perturbation arises in variable of nonlinearity f, where B~n is the open unit ball in R~n with n≥1,λ> 0 is a bifurcation parameter, andε> 0 is a small constant.The problems mentioned above arise in many different physical, chemical and biologicsituations, for instance, in the combustion theory, in the quantum filed theory, in the population dynamics and so on (see [50] and the references therein), where the well-known equations such as the Gelfand equation (by rescaling argument, f(u) = exp(-1/u), see [29]), a reaction-diffusion equation modeling an autocatalytic chemical reaction (f(u) = u~pu~_q, 1 < p < q, see [73]) and Holling-Tanner biologic model (f(u) = ku - u~2 -lu/l9u, k >1 > 1, see [35]).By making use of bifurcation theory, the method of upper and lower solutions and continuation method, we discuss a class of these problems, where f has certain regularity,and all nonlinearities are concave and convex. For wilder application in practice, we will get the results of exact multiplicity and bifurcation structure of positive solutions for the equations mentioned above, through precise analysis on different dimension and different linearity of nonlinearity f.