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Radial solutions for the Brezis-Nirenberg problem involving large nonlinearities. (English) Zbl 1154.35050

The author establishes existence of radial solutions of the Brezis-Nirenberg problem, \[ -\Delta u + a(| x| ) u = u^p, \]
in the unit ball, \({\mathbb B} \subset {\mathbb R}^n\), \(n \geq 3\), with \(u\) positive and satisfying boundary data \(u =0\) on \(\partial {\mathbb B}\). If \(p\) is large enough – and under some additional assumption on the regular part of the Green’s function for the corresponding ordinary differential equation
\[ -u''(r) -\frac{n-1}{r}u^\prime + a(r) u(r) =u^p, \]
with \(u >0\), \(u'(0) =0\), and \(u(1)=0\) – it is shown that there exists at least one solution of the Brezis-Nirenberg problem. While for subcritical exponents, \(1 < p < (n+2)/(n-2)\), existence and uniqueness of positive solutions with vanishing boundary data always hold, the situation for (super)critical exponents is more complicated, and related to the domain under investigation.
In order to establish existence, the author linearizes around a solution which is in some sense close to a solution of the so-called “limit problem”,
\[ -u^{\prime\prime} = \exp(u). \]
Much effort is devoted to expanding this approximate solution with the help of the Green’s function and its properties. It is proved that the linear operator is invertible. A fixed point argument then yields the existence of the exact solution. Finally, an example is given, showing that an enough concentrated mollifier would satisfy the requirements on the function \(a\).

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35A35 Theoretical approximation in context of PDEs
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