On the Fredholm alternative for the \(p\)-Laplacian at higher eigenvalues (in one dimension) (Q847359)

The existence of a weak solution of the following Dirichlet problem for a \(p\)-Laplacian equation with \(p\in (1,2)\) is considered: \[ -(|u'|^{p-2}u')' = \lambda_k |u|^{p-2}u + f(x),\qquad u(0)=u(a)=0. \] It is assumed that \(f\) is a non-zero given function in \(L^\infty(0,a)\), with \(0<a<\infty\). The number \(\lambda_k\) stands for the \(k\)-th eigenvalue of the \(p\)-Laplacian. For \(k=1\), it is well-known that the problem has a solution when \(f\) is orthogonal (in the \(L^2\) sense) to \(\sin_p(k\pi_px/a)\), the eigenvalue associated to \(\lambda_1\). In this work, the authors prove that if moreover \(f\) satisfies some extra assumptions, then the existence of solutions can be proved for \(k\geq 2\).