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Remarks on critical exponents for Hessian operators. (English) Zbl 0715.35031
Let $$S_ k(\nabla^ 2u)$$, $$k=1,...,n$$, be the k-th Hessian operator, i.e., the k-th elementary symmetric function of the Hessian matrix of u, where $$n\geq 2$$ is the dimension. Thus, for $$k=1$$ we have the Laplacian operator and for $$k=n$$ the determinant of the Hessian matrix (Monge- Ampère operator). Consider the Dirichlet problem $S_ k(\nabla^ 2u)=(-u)^ p\text{ in } \Omega,\quad u=0\text{ on } \partial \Omega.$ The author proves the following result: Let $$\Omega$$ be a ball of $${\mathbb{R}}^ n$$ and let $\gamma (k)=(n+2k)/(n-2k)\quad if\quad 1\leq k<n/2,\quad \gamma (k)=\infty \quad if\quad n/2\leq k\leq n.$ Then (I) The above problem has no negative solution in $$C^ 1({\bar \Omega})\cap C^ 4(\Omega)$$ when $$p\geq \gamma (k)$$; and (II) It admits a negative solution which is radially symmetric and is in $$C^ 2({\bar \Omega})$$ when $$0<p<\gamma (k)$$ and $$p\neq k.$$
Hence, the critical exponent of the k-th Hessian operator is determined. Actually, $$(-u)^ p$$ is replaced throughout the paper by a more general nonlinearity of the form f(x,-u) and the nonexistence result is proved for any bounded star-shaped $$C^ 2$$-domain $$\Omega$$. (Negative solutions assure that the k-th Hessian operators are elliptic for $$k\geq 2).$$
By using the Newtonian tensor, the above equation is written as the Euler-Lagrange equation for a certain functional. Then a minimax method and the mountain pass lemma yield the existence result (for the ball). Under stronger hypotheses on f(x,-u) the solution is also an absolute minimum of the functional. Nonexistence is established by means of a Pohozaev-type argument.
Reviewer: F.Bernis

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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##### References:
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