A bifurcation phenomenon for the critical Laplace and p-Laplace equation in the ball: part 2

Abstract In this paper we continue the discussion started in [12] concerning a bifurcation phenomenon for radial solutions of the following nonlinear eigenvalue problem: $$\begin{aligned} {\left\{ \begin{array}{ll} \Delta _p u(x) + \lambda {\mathcal {K}}(|x|) \,u(x) \, |u(x)|^{q-2} =0\,, \\ u(x)>0\,, & \quad |x|<1\,,\\ u(x)=0\,, & \quad |x|=1\,, \end{array}\right. } \end{aligned}$$ Δ p u ( x ) + λ K ( | x | ) u ( x ) | u ( x ) | q - 2 = 0 , u ( x ) > 0 , | x | < 1 , u ( x ) = 0 , | x | = 1 , where $$q= \frac{np}{n-p}$$ q = np n - p is the critical exponent and $$x \in {\mathbb {R}}^n$$ x ∈ R n . Our main purpose is to remove the restriction in the range of