Asymptotic behaviour of a nonlinear elliptic equation with critical Sobolev exponent: the radial case. (Q5954492)

The author studies the following problem NEWLINE\[NEWLINE (I)\qquad\begin{cases} \Delta u +a(| x| )u=N(N-2)f(| x| )u^p &\quad \text{ in }\;B\cr u>0 \quad \text{ in}\;B, \qquad\qquad u=0 &\quad \text{ on } \;\partial B,\end{cases} NEWLINE\]NEWLINE where \(B\) is the unit ball in \({\mathbb R}^N,\) \(N\geq 3,\) \(p\) is the critical Sobolev exponent, \(a(| x| )\) and \(f(| x| )\) are locally in \(C^{0,\alpha},\) \(\alpha\in(0,1).\) The author is interested in the existence of conditions on \(a\) and \(f\) for \((I)\) to have a solution. The problem NEWLINE\[NEWLINE (I_\varepsilon)\qquad\begin{cases} \Delta u_\varepsilon +a(| x| )u_\varepsilon=N(N-2)f(| x| )u_\varepsilon^{p-\varepsilon} &\quad in \;B\cr u_\varepsilon>0 \quad \text{ in}\;B, \qquad\qquad u_\varepsilon=0 &\quad on \;\partial B\end{cases} NEWLINE\]NEWLINE is solvable with a solution \(u_\varepsilon\in C^2(\bar B)\) for all \(\varepsilon\in(0,p-1).\) He studies the asymptotic behaviour of \(u_\varepsilon\) as \(\varepsilon\to 0\) when \(I\) does not have a solution.NEWLINENEWLINEPart II, cf. NoDEA, Nonlinear Differ. Equ. Appl. 9, No. 3, 361--384 (2002; Zbl 1088.35020).