Nodal solutions of elliptic equations with critical Sobolev exponents. (English) Zbl 0702.35099
The eigenvalue problem \(\Delta U=-\lambda U+| U|^{p-1}U\) in \(B=\{| x| <1\}\), \(U\neq 0\), \(U=0\) on \(\partial B\) is considered, where \(p=(N+1)/(N-2)\) and \(x=(x_ 1,...,x_ N)\) and radial solutions of variable sign are investigated. It is shown that for \(4\leq N\leq 6\) and small \(\lambda\), no radial solutions exist. Then the behaviour of the “eigenvalues” \(\lambda_ n\) corresponding to a solutions with \(n-1\) zeros are determined as \(| u_ n|_{\infty}\to \infty.\) A surprising phenomenon occurs, the \(\lambda_ n\) tend to well-defined numbers depending on the dimension N. The discussion is based on a subtle analysis of the ordinary differential equation \(y''+t^{-k}f(y)=0\).
Reviewer: C.Bandle
MSC:
| 35J65 | Nonlinear boundary value problems for linear elliptic equations |
| 35P30 | Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs |
Keywords:
nodal solutions; critical Sobolev exponents; nonexistence; eigenvalue problem; radial solutionsReferences:
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