On the Fredholm alternative for the \(p\)-Laplacian
The author deals with the boundary value problem \[ (p-1)^{-1} ((\varphi_p(u'))'+ \alpha\varphi_p (u^+)- \beta\varphi_p (u^-))= f(\varphi_p(u))+ h(t) \text{ in }(0,T), \quad u(0)= u(T)= 0, \] where \(p>1\), \(\varphi_p(u)=| u|^{p-2}u\), \(T= (p-1)^{1/p} \alpha^{-1/p} \pi_p\), \(\pi_p= \frac {2\pi}{p}\sin (\frac \pi p)\), \(h\in L^\infty (0,T)\), \(f\in C(\mathbb{R}, \mathbb{R})\) is bounded, \(\lim_{x\to+\infty} f(x)= f(+\infty)\) exists, \(\alpha>0\), \(\beta>0\) and \((p-1)^{1/p} (\alpha^{-1/p}+ \beta^{-1/p})= 2\). To this end the author uses a generalized polar coordinates transformation and the methods of nonlinear functional analysis.
scientific article; zbMATH DE number 2082070