Nonlinear resonant periodic problems with concave terms
The authors continue their study of periodic BVPs for quasilinear equations involving the one-dimensional \(p\)-Laplacian. The problem under consideration reads: \[ -(|u'(t)|^{p-2} u'(t))' = \lambda |u(t)|^{q-2} u(t) + f(t, u(t)) \text{ on } [0,b], \] \[ u(0) = u(b), \quad u'(0) = u'(b), \] where \(1 < q < p < \infty\), with parameter \(\lambda > 0\). \(f\) is Carathéodory and satisfies some technical conditions. In particular, if \((\lambda_m)_{m \geq 1}\) is the sequence of eigenvalues of the negative periodic \(p\)-Laplacian on \([0,b]\), it is required that \(f(t,x)/(|x|^{p-2}x)\) belongs asymptotically, as \(|x| \to \infty\), to the spectral interval \([\lambda_m, \lambda_{m+1}]\) for some \(m \geq 2\). More precisely, the problem is permitted to be resonant at \(\lambda_{m+1}\) and non-uniformly nonresonant at \(\lambda_m\). This together with the appearance of the `concave' term \(\lambda |u|^{q-2} u\) constitutes the new feature compared with earlier work. The authors establish the existence of three distinct non-trivial solutions provided \(\lambda\) is not too large. One is of mountain pass type, one is a local minimizer, and the third one is detected using the Morse inequalities and the calculation of the critical groups at zero and at infinity.
scientific article; zbMATH DE number 5835305
scientific article