On a nonlinear boundary value problem with a mixed nonhomogeneous condition: asymptotic behavior of solutions (Q2757032)

The authors study the nonlinear boundary value problem NEWLINE\[NEWLINE\begin{aligned}\frac{-1}{x^\gamma }\frac d{dx}(x^\gamma |u'(x)|^{p-2}u^{\prime }(x))+f(x,u(x))=F(x),\qquad 0<x<1, \\ \Bigl|\lim_{x\rightarrow 0+}\lim x^{\frac \gamma p}u^{\prime }(x)\Bigr|<+\infty ,\qquad |u^{\prime }(1)|^{p-2}u^{\prime }(1)+hu(1)=g,\end{aligned} \tag \(*\) NEWLINE\]NEWLINE where \(\gamma >0, p\geq 2, h>0, g \)are given constants and \(f, F\) are given functions. The Galerkin and compactness method in approximate Sobolev spaces with weight are used to prove the existence of a weak solution to problem \((*)\). The asymptotic behavior of the solution \(u_h\) depending on \(h\) as \(h\rightarrow 0+\) is studied. It is shown that the function \(h\rightarrow |u_h(1)|\) is non-increasing on \((0,+\infty).\)