An index formula for the degree of (S)\(_+\)-mappings associated with one-dimensional \(p\)-Laplacian
This paper is devoted to the proof of the following result: Consider the quasilinear problem \(-\Delta_p u=- (|u'(x)|^{p-2} u'(x))'= \mu|u(x)|^{p-2} u(x)\) for \(x\in (0,1)\) with \(u(0)= u(1)= 0\), where \(p> 1\) is fixed and \(\mu\) is real. If \(\mu_k\) denotes the (simple) eigenvalues of \(-\Delta_p\), then we have \(d_{(S)_+}(T^p, B_r(0),0)= (-1)^k\) for \(\mu\in (\mu_k, \mu_{k+1})\) for \(r> 0\), where \(T^p: W^{1,p}_0(0,1)\to W^{-1,p'}(0,1)\) is the \((S)_+\) operator corresponding to the weak solution to the above problem. Here \(d_{(S)_+}\) denotes the degree for \((S)_+\) mappings. The proof uses a tricky argument involving different degrees in different Banach spaces, the degree for subdifferentials, Galerkin approximations, and a homotopy argument reducing the problem to the case \(p= 2\), which is reminiscent to the one used by del Pino and Manasevich for the classical Leray-Schauder degree. Applications to variational inequalities are given in order to prove existence of at least three solutions.
scientific article; zbMATH DE number 2163692