Structure of the positive radial solutions for the supercritical Neumann problem \(\varepsilon^2\Delta u-u+u^p=0\) in a ball (Q2786743)

The author investigates the structure of positive radial solutions for the equation NEWLINENEWLINE\[NEWLINE \varepsilon^2\Delta u-u+u^p=0\quad\text{in }B_1(0)\setminus\{0\}\subset{\mathbb R}^N, \quad N\geq 3 NEWLINE\]NEWLINE NEWLINEsubject to homogeneous Neumann boundary conditions, where \(p>(N+2)/(N-2)\). It is shown that there exists a sequence \(\{\varepsilon_n\}\) of positive numbers that converges to zero, for which the above problem has infinitely many singular solutions. Further, asymptotic and bifurcation properties with respect to \(\varepsilon\) are investigated.