A fixed point operator for systems of vector \(p\)-Laplacian with singular weights (Q276368)

The \(p\)-Laplacian (which is the second derivative for \(p=2\)) induces a strongly coupled nonlinear differential system \[ -(|u'(t)|^{p-2} u_i'(t))' = \lambda h_i(t) f_i(u) \] for \(i=1,\dots,N\) on \((0,1)\) subject to the boundary conditions \[ u(0) = 0 = u(1). \] Here we have \(\lambda > 0, \; p > 1, \; u(t) = (u_1(t), \dots, u_N(t))\) and \(|\cdot|\) is the Euclidean norm in \({\mathbb R}^N\). The function \(f=(f_1, \dots, f_N) : \; {\mathbb R}^N \rightarrow {\mathbb R}^N\) is assumed to be continuous. For the vector valued weight function \(h=(h_1, \dots, h_N) \in L^1_{\mathrm{loc}}((0,1), \; {\mathbb R}^N)\) there are no sign restrictions imposed but certain weak conditions on the singularities at the boundaries. These conditions on \(h\) seem to be the main generalizations of the present setting compared with earlier \(L^1\)-approaches, see e.g. [\textit{R. Manásevich} and \textit{J. Mawhin}, J. Differ. Equations 145, No. 2, 367--393, Art. No. DE983425 (1998; Zbl 0910.34051); J. Korean Math. Soc. 37, No. 5, 665--685 (2000; Zbl 0976.34013); the second and the third author, Abstr. Appl. Anal. 2012, Article ID 243740, 15 p. (2012; Zbl 1253.34030)]. A fixed point solution operator \(T: \; C([0,1], {\mathbb R}^N) \rightarrow C([0,1], {\mathbb R}^N)\) is constructed (which allows to rewrite the problem for \(\lambda = 1\) as \(Tu = u\)) and it is shown that \(T\) is compact. For the main result additional conditions on \(f\) are imposed (excluding the linear case): \(f_i(0) > 0\) and \(\lim_{|s| \rightarrow \infty} f_i(s)/|s|^{p-1} = 0\) for \(i=1, \dots, N, \; s \in {\mathbb R}^N\). Then, using the compactness of \(T\) and the global continuation theorem it is proved that the boundary value problem has at least one nontrivial solution for all \(\lambda > 0\). Finally, a number of explicit examples are discussed.