Boundary effect for an elliptic Neumann problem with critical nonlinearity.

*(English)*Zbl 0891.35040Consider the Neumann problem
\[
-\Delta u+\mu u=u^p\quad\text{in }\Omega,\quad u>0\quad\text{in }\Omega,\quad {\partial u\over\partial\nu}= 0\quad\text{on }\partial\Omega,
\]
where \(\mu>0\), \(\Omega\) is smooth and bounded in \(\mathbb{R}^n\), \(n\geq 3\), and \(p=(n+2)/(n-2)\) is the critical exponent. This paper describes some interactions between the boundary behavior of solutions to the Neumann problem and the mean curvature \(H\) of \(\partial\Omega\) [see also Adimurthi, F. Pacella, and S. L. Yadava, J. Funct. Anal. 113, No. 2, 318-350 (1993; Zbl 0793.35033)], adapting the methods developed by the author for similar Dirichlet problems [J. Funct. Anal. 89, No. 1, 1-52 (1990; Zbl 0786.35059)].

Denote by \(H^b\) the level set of \(H\) to the level \(b\) and assume that the relative topology \((H^{a+\delta},H^{a-\delta})\) is non-trivial for a positive critical value \(a\) of \(H\) and for any \(\delta>0\) sufficiently small. If \(n\geq 5\), the author then proves the existence of a solution \(u_\mu\) to the Neumann problem which concentrates, as \(\mu\) tends to infinity, at some point \(y\in \partial\Omega\) such that \(H(y)= a\). Moreover, if \(n\geq 6\) and \(y^1,\dots,y^k\) are nondegenerate critical points of \(H\) with \(H(y^j)>0\), there exists, for \(\mu\) large enough, a solution \(u_\mu\) which concentrates at \(y^1,\dots, y^k\) (in the sense \(|\nabla u_\mu|^2\rightharpoonup{1\over 2} S^{n/2} \sum^k_{i= 1}\delta_{y^i}\), \(S\) being the best Sobolev constant for the embedding \(H^1_0(\Omega)\hookrightarrow L^{2n/n- 2}(\Omega)\)). In this case, it is proved additionally that, for \(\mu\) large enough, the Neumann problem admits at least \(2^k- 1\) nonconstant solutions.

Denote by \(H^b\) the level set of \(H\) to the level \(b\) and assume that the relative topology \((H^{a+\delta},H^{a-\delta})\) is non-trivial for a positive critical value \(a\) of \(H\) and for any \(\delta>0\) sufficiently small. If \(n\geq 5\), the author then proves the existence of a solution \(u_\mu\) to the Neumann problem which concentrates, as \(\mu\) tends to infinity, at some point \(y\in \partial\Omega\) such that \(H(y)= a\). Moreover, if \(n\geq 6\) and \(y^1,\dots,y^k\) are nondegenerate critical points of \(H\) with \(H(y^j)>0\), there exists, for \(\mu\) large enough, a solution \(u_\mu\) which concentrates at \(y^1,\dots, y^k\) (in the sense \(|\nabla u_\mu|^2\rightharpoonup{1\over 2} S^{n/2} \sum^k_{i= 1}\delta_{y^i}\), \(S\) being the best Sobolev constant for the embedding \(H^1_0(\Omega)\hookrightarrow L^{2n/n- 2}(\Omega)\)). In this case, it is proved additionally that, for \(\mu\) large enough, the Neumann problem admits at least \(2^k- 1\) nonconstant solutions.

Reviewer: Ralf Beyerstedt (Triest)

##### MSC:

35J65 | Nonlinear boundary value problems for linear elliptic equations |

35J67 | Boundary values of solutions to elliptic equations and elliptic systems |

35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |

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\textit{O. Rey}, Commun. Partial Differ. Equations 22, No. 7--8, 1055--1139 (1997; Zbl 0891.35040)

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