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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)].
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)
##### Keywords:
positive solutions; existence; Neumann boundary condition
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##### References:
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