Existence and asymptotic behavior of solutions for a class of nonlinear elliptic equations with Neumann condition (Q1776926)

The author studies the existence of nonconstant solutions and their asymptotic behavior for the class of nonlinear elliptic equations in radial form \[ -\varepsilon^{2}(r^{\alpha}| u'| ^{\beta}u')'=r^{\gamma}f(u),\quad r\in(0,R),\quad u'(0)=u'(R)=0, \] where \(\alpha, \beta, \gamma\) are given real numbers and \(0<R<\infty\). Using the mountain pass theorem, the existence of nonconstant solutions is established.