The Fredholm alternative for the one-dimensional \(p\)-Laplacian
This paper deals with a Dirichlet BVP of the form \[ \phi_p(u')'+\lambda_1 \phi_p(u)=f(u)+h(t), \quad u(0)=0=u(T), \] where \(\phi_p\) denotes the one-dimensional \(p\)-Laplacian, \(f\) is continuous and bounded, the limit \(\lim_{u \to +\infty} f(u)\) exists and \(h \in L^{\infty}(0,T)\). The following result is proved: Assume that \(H(t):=f(+\infty)+h(t) \not\equiv 0\). Suppose that \(\int_0^T H(t) \sin({{\pi_p t}\over{T}})dt =0,\) where \(\pi_p=2\pi/(p\sin(\pi/p)),\) and \(\lim_{u \to +\infty} | x| ^{p-1}[f(x)-f(+\infty)]=0\). Then the given problem has at least one solution. Moreover, if \(p\neq 2\) then the set of all possible solutions is bounded in \(C^1 [0,T]\). The proof is performed in the framework of the Leray-Scheuder continuation method; a generalized polar coordinates transformation is used. This result is strictly related to the work by \textit{R. F. Manásevich} and \textit{R. Takáč} [Proc. Lond. Math. Soc., III. Ser. 84, No. 2, 324--342 (2002; Zbl 1024.34010)].
scientific article; zbMATH DE number 2126902