Exact multiplicity results for boundary value problems with nonlinearities generalising cubic

Abstract

Using techniques of bifurcation theory we present two exact multiplicity results for boundary value problems of the typeThe first result concerns the case when the nonlinearity is independent of x and behaves like a cubic in u. The second one deals with a class of nonlinearities with explicit x dependence.

Computing the location and the direction of bifurcation

“…(We will see later on that the approach in [8] could not possibly cover the general case.) Then P. Korman, Y. Li and T. Ouyang [6], [7] used bifurcation theory to attack the problem, but again some restrictions were necessary (all of the above mentioned papers covered more general f (u), behaving like cubic). We shall now recall the results of [7], and describe what is the desired optimal result.…”

“…the problem is open, but the results of P. Korman, Y. Li and T. Ouyang [6] still help. Indeed to justify the picture as in Figure 1, one needs to prove that only turns to the right are possible on the upper curve, and hence there is only one turn (the properties of the lower curve follow easily, since f (u) < 0 for 0 < u < a).…”

“…Indeed to justify the picture as in Figure 1, one needs to prove that only turns to the right are possible on the upper curve, and hence there is only one turn (the properties of the lower curve follow easily, since f (u) < 0 for 0 < u < a). Turns can only occur at singular u(x), and it was proved in [6] and [7], that u(0) > β at any singular solution, and only the turns to the right are possible if u(0) > γ. This means there is a "danger" of a wrong turn only when u(0) = α ∈ (β, γ).…”

“…We recall that solutions of ( 1.1) are even functions, with u (x) < 0 for x > 0, and hence any solution is uniquely identified by α = u(0), see [6]. Actually, even more is true: the value of u(0) = α uniquely identifies both λ and u(x), as follows easily by scaling λ out of ( 1.1), and using uniqueness for initial value problems, see E. N. Dancer [4].…”

Multiplicity of Positive Radial Solutions to a Higher Dimensional Scalar-Field Equation Involving the Critical Sobolev Exponent under the Robin Condition

“…Local structures has been well investigated, however, the global ones were not so. Recently, for example, Korman, Li and Ouyang [10,11] and Ouyang and Shi [13,14] considered various nonlinearity under which the exact multiplicity of solutions and the global bifurcation diagrams are studied. In this context, we mainly deal with the non-existence and the multiplicity of positive solutions to the scalar-field type equation with the critical Sobolev exponent under the Robin condition as a subsequent paper of Kabeya [6].…”