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Constrained topological degree and positive solutions of fully nonlinear boundary value problems. (English) Zbl 1188.35088

This article consists of two parts: in the first one the authors define a “constrained degree”: Let \(E\) be a Banach space and let \(M\subset E\) (the “set of constraints”) be an \(\mathcal{L}\)-retract, i.e., there is an \(\eta>0\) and an \(L\geq1\) and a retraction \(r:B:=\{x\in E\mid d(x,M)<\eta\}\to M\) such that \(\|r(x)-x\|\leq Ld(x,M)\) whenever \(x\in B\). Let then \(U\subset M\) be open, let \(A:D(A)\to E\) be a densely defined \(m\)-accretive operator and \(F:U\to E\) be a continuous map such that (1) there is a \(\lambda_0>0\) such that \((I+\lambda A)^{-1}(M)\subset M\) whenever \(0<\lambda\leq\lambda_0\), (2) for positive \(\lambda\) the resolvent \((I+\lambda A)^{-1}\) is completely continuous when restricted to \(M\), (3) \(F(x)\in \{u\in E\mid \liminf_{h\to0+}\frac1h d(x+hu,M)=0\}\) for \(x\in U\), (4) the zero set \(\{x\in U\cap D(A)\mid 0\in -A(x)+F(x)\}\) is compact. In this situation, the authors define a degree, \(\deg_M((A,F),U)\), that besides the existence, additivity, and homotopy property satisfies the following normalization property: \(\deg_M((A,F),M)=\chi(M)\) provided \(M\) and \(F(M)\) are bounded.
In the second part, the authors apply their degree theory to problems which can be cast in the form \(0\in -Ax+F(x)\), \(x\in M\). In this context they deal with stationary reaction-diffusion problems, one-dimensional non-homogeneous \(p\)-Laplacian problems, and controlled Neumann-like problems.

MSC:

35J66 Nonlinear boundary value problems for nonlinear elliptic equations
47H04 Set-valued operators
47H06 Nonlinear accretive operators, dissipative operators, etc.
47H11 Degree theory for nonlinear operators
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35A16 Topological and monotonicity methods applied to PDEs
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