The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev exponent. (English) Zbl 0786.35059

Let \(\Omega\) be a smooth and bounded domain in \(\mathbb{R}^ N\), \(N \geq 4\), and \(p=(N+2)/(N-2)\) so that \(p+1\) is critical from the viewpoint of Sobolev embedding. The authors consider the nonlinear elliptic problems of the type \((P_ \varepsilon):-\Delta u=u^{(N+2)/(N-2)}+\varepsilon u\), \(u>0\) on \(\Omega\); \(u=0\) on \(\partial \Omega\), and \(\varepsilon>0\). They show that if the \(u_ \varepsilon\) are solutions of \((P_ \varepsilon)\) which concentrate around a point as \(\varepsilon \to 0\), then this point cannot be on the boundary of \(\Omega\) and is a critical point of the regular part of the Green’s function. Conversely, they show that for \(N \geq 5\) and any nondegenerate critical point \(x_ 0\) of the regular part of the Green’s function, there exist solutions of \((P_ \varepsilon)\) concentrating around \(x_ 0\) as \(\varepsilon \to 0\).


35J65 Nonlinear boundary value problems for linear elliptic equations
35B25 Singular perturbations in context of PDEs
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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