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Nonlinear boundary value problems for elliptic systems. (English) Zbl 0643.35038

Let X, Z be normed spaces with \(X,Z \hookrightarrow L_{\infty}(G,{\mathbb{R}}^ m)\), where \(\emptyset \neq G\subset {\mathbb{R}}^ n\) is a bounded domain and \(m,n\in {\mathbb{N}}\), let \(L: X\supset dom(L)\to Z\) be a Fredholm operator of index 0 and \(N: X\to Z\) be L-completely continuous. The author establishes a generalized continuation theorem of Leray-Schauder type in the spirit of J. Mawhin for semilinear operator equations \(Lu=Nu\), provided \[ \ker (L)=\{(a_ 1\phi_ 1,...,a_ m\phi_ m): (a_ 1,...,a_ m)\in {\mathbb{R}}^ m\}\quad with\quad \phi_ k\in C(\bar G,{\mathbb{R}}) \] positive on G for \(1\leq k\leq m\) and satisfying: \[ (\exists c>0)(\forall x\in \bar G)(\forall 1\leq k\leq m)(\forall (u_ 1,...,u_ m)\in X): | u_ k(x)| \leq c\phi_ k(x)\| (u_ j)\|_ X. \] This result is applied to second order elliptic systems of the form \[ A_ ku_ k-\lambda^ k_ 1 u_ k=g_ k(x,u_ 1,...,u_ m),\quad x\in G,\quad 1\leq k\leq m;\quad B_ ku_ k=0\quad on\quad \partial G, \] where \(\lambda^ k_ 1\) denotes the first eigenvalue of the linear problem \(A_ kw=\lambda w\) on G, \(B_ kw=0\) on \(\partial G\). \(B_ k\) may stand for the homogeneous Dirichlet or Neumann boundary conditions. General asymptotic conditions extending those of Landesman-Lazer or Kazdan-Warner in the case \(m=1\) are introduced, and a superlinear case is also treated.
Reviewer: G.Hetzer

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
47J05 Equations involving nonlinear operators (general)
35J25 Boundary value problems for second-order elliptic equations
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
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