##
**Neumann problems of semilinear elliptic equations involving critical Sobolev exponents.**
*(English)*
Zbl 0766.35017

This paper deals with the existence of solutions of the problem
\[
\Delta u+u^ p+f(x,u)=0\quad\text{ in } \Omega, \qquad u>0 \quad\text{ in } \Omega, \qquad D_ \gamma u+\alpha(x)u=0 \quad\text{ on } \partial\Omega,\tag{*}
\]
where \(\Omega\) is a \(C^ 1\) bounded domain in \(\mathbb{R}^ n\), \(n\geq 3\), \(\gamma\) is the outer unit normal to \(\partial\Omega\), \(p=(n+2)/(n-2)\) is the critical Sobolev exponent, \(f\) is a lower order perturbation of \(u^ p\) with \(f(x,0)=0\), and \(\alpha\) is a nonnegative function on \(\partial\Omega\). The Dirichlet counterpart of (*) was studied by H. Brezis and L. Nirenberg [Commun. Pure Appl. Math. 36, 437-477 (1983; Zbl 0541.35029)]. They showed that the existence of solutions depends strongly on the behaviour of \(f\).

Here the author shows that for the Neumann boundary condition the situation is somewhat better, in the sense that (*) has a solution for a large class of \(f\). For the case \(\alpha\equiv 0\) conditions on \(f\) are found which guarantee the existence of a solution of (*) for arbitrary \(C^ 1\) bounded \(\Omega\), while for general \(\alpha\geq 0\) an additional assumption on \(\Omega\) is required. The author also proves some results for equations with variable coefficients, and gives a number of interesting examples.

Neumann problems for equations involving critical Sobolev exponents have been studied recently by a number of authors e.g., M. Comte and M. C. Knaap [Differ. Integral Equ. 4, No. 6, 1133-1146 (1991)].

Here the author shows that for the Neumann boundary condition the situation is somewhat better, in the sense that (*) has a solution for a large class of \(f\). For the case \(\alpha\equiv 0\) conditions on \(f\) are found which guarantee the existence of a solution of (*) for arbitrary \(C^ 1\) bounded \(\Omega\), while for general \(\alpha\geq 0\) an additional assumption on \(\Omega\) is required. The author also proves some results for equations with variable coefficients, and gives a number of interesting examples.

Neumann problems for equations involving critical Sobolev exponents have been studied recently by a number of authors e.g., M. Comte and M. C. Knaap [Differ. Integral Equ. 4, No. 6, 1133-1146 (1991)].

Reviewer: J.Urbas (Canberra)

### MSC:

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

35J20 | Variational methods for second-order elliptic equations |

35J25 | Boundary value problems for second-order elliptic equations |

35D05 | Existence of generalized solutions of PDE (MSC2000) |

### Citations:

Zbl 0541.35029
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\textit{X. Wang}, J. Differ. Equations 93, No. 2, 283--310 (1991; Zbl 0766.35017)

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### References:

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