Solvability of the resonant 1-dimensional periodic \(p\)-Laplacian equations (Q984768)

The paper studies the Fredholm alternative of the equation \[ -(|x'|^{p-2}x')'=\lambda|x|^{p-2}x+f(t) \tag{1} \] with the periodic boundary conditions \[ x(0)=x(2\pi),\quad x'(0)=x'(2\pi) \tag{2} \] by the variational method. The authors denote by \(\Sigma_p\) for \(p>1\) the set of eigenvalues of problem (1), (2) with \(f\equiv 0\) and prove the following result which is the counterpart of the results concerning the Dirichlet problem of Del Pino, Drábek, Manásevich and Takáč. Theorem. Let \(\lambda=\lambda_n\in\Sigma_p\) with \(n\geq 1\) and let \(f\) be a continuous \(2\pi\)-periodic function such that for \(\theta\in[0,2\pi]\) \[ \int_0^{2\pi}f(t)\phi(t+\theta)\,dt=0,\qquad \int_0^{2\pi}f(t)\phi'(t+\theta)\left(\int_0^tf(s)\phi(s+\theta)\,ds\right)dt\not=0. \] Then (1) has a \(2\pi\)-periodic solution. Moreover, the set of \(2\pi\)-periodic solutions of (1) is bounded in \(C^1(S^1)\), where \(S^1=\mathbb R/2\pi Z\).