Perturbation of the \(p\)-Laplacian by vanishing nonlinearities (in one dimension) (Q413612)

The authors investigate the solvability of the quasi-linear spectral problem NEWLINE\[NEWLINE -(|u'|^{p-2}u')'=\lambda |u|^{p-2}u+h(x,u),\;\;x\in (0,a); \;\;u(0)=u(a)=0NEWLINE\]NEWLINE at resonance with vanishing nonlinearity \(h(x,u)=g(u)+f(x),\,g(u)\rightarrow 0\) as \(u\rightarrow \pm \infty\), and \(f\in L^{\infty}(0,a)\,f\not \equiv 0\) satisfying certain orthogonality-related hypotheses, \(1<p<3, p\neq 2\). When the rate of decay \(g(u)\rightarrow 0\) as \(u\rightarrow \pm \infty\) is sufficient fast, they obtain an existence result by means of Leray-Schauder degree theory. The key of the proof is to find a boundedness of the set of all weak solutions in the Sobolev space \(W_0^{1,p}(0,a)\).