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Solvability of the normal derivative problem for nonlinear nonvariational elliptic systems. (English) Zbl 0872.35034
Summary: Let $$\Omega$$ be a bounded open convex set of class $$C^2$$. Let $$a(x,H(u))$$ be a nonlinear operator satisfying some condition (A). We show that $$\forall\lambda>0$$ and $$\forall f\in L^2(\Omega,\mathbb{R}^N)$$ the problem $a(x,H(u))-\lambda u=f(x)\quad\text{a.e. in }\Omega,\quad{\partial u\over\partial\vec n}=0\quad\text{on }\partial\Omega$ with $$\vec n$$ the unit outward normal to $$\partial\Omega$$, admits a unique solution in a suitable Sobolev class. As a consequence, the operator $$a(x,H(u))$$ has possibly eigenvalues which are not positive.
##### MSC:
 35J60 Nonlinear elliptic equations
##### Keywords:
nonlinear nonvariational elliptic systems