Asymptotics for some quasilinear elliptic equations
The author considers the semilinear Dirichlet problem in the unit ball \(B \subset \mathbb{R}^N\) \((N \geq 3)\) \[ \Delta u = N(N - 2) f \bigl( |x |\bigr) u^p, \quad u > 0 \text{ in } B, \;u = 0 \text{ on } \partial B, \tag{*} \] where \(p=(N + 2)/(N - 2)\) is the critical Sobolev exponent. He studies the asymptotic behaviour of the solutions \(u_\varepsilon\) to the corresponding problem \((*)_\varepsilon\), where \(p\) is replaced by \(p-\varepsilon\), as \(\varepsilon\to 0\). It is shown that if \(f^{(k)} (0) > 0\) and \(N > k + 2\) \((k \geq 2)\), the sequence \(u_\varepsilon\) tends to a strictly decreasing radially symmetric solution \(u\) of the critical problem \((*)\). In other cases, the sequence \(u_\varepsilon\) may blow up at the origin, as \(\varepsilon \to 0\). The blow-up behaviour is studied in detail. In the case \(f(|x |) \equiv 1\) in \((*)\), it is well known that there is no solution of \((*)\). The author also studies this problem by setting \(f(|x |) = 1 + \varepsilon_1 |x |^k\), in \((*)\). The blow-up behaviour of the solutions \(u_\varepsilon\) of this problem is also described in detail.
scientific article; zbMATH DE number 840302