Abstract
In this paper, we show the existence of an S-shaped connected component in the set of radial positive solutions of boundary value problem
$$\begin{array}{}
\displaystyle \left\{\,\begin{array}{}
-\text{ div}\big(\phi_N(\nabla y)\big)=\lambda a(|x|)f(y)\, \, \, \, \, \text{in}\, \, \mathcal{A},\\\frac{\partial y}{\partial \nu}=0\, \, \, \,\, \text{ on }\, \, {\it\Gamma}_1,\qquad y=0\, \, \, \, \text{ on}\, \, {\it\Gamma}_2,\\
\end{array}
\right.
\end{array} $$
where R2 ∈ (0, ∞) and R1 ∈ (0, R2) is a given constant, 𝓐 = {x ∈ ℝN : R1 < ∣x∣ < R2}, Γ1 = {x ∈ ℝN : ∣x∣ = R1}, Γ2 = {x ∈ ℝN : ∣x∣ = R2},
$\begin{array}{}
\phi_N(s)=\frac{s}{\sqrt{1-|s|^2}},
\end{array} $ s ∈ ℝN, λ is a positive parameter, a ∈ C[R1, R2], f ∈ C[0, ∞),
$\begin{array}{}
\frac{\partial y}{\partial \nu}
\end{array} $ denotes the outward normal derivative of y and ∣⋅∣ denotes the Euclidean norm in ℝN. The proof of main result is based upon bifurcation techniques.
Global Second Derivative Estimates for the Second Boundary Value Problem of the Prescribed Affine Mean Curvature and Abreu's Equations
