×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite