Positive solutions for the periodic scalar \(p\)-Laplacian: existence and uniqueness
The authors are concerned with the \(p\)-Laplacian nonlinear periodic boundary value problem (\(1<p<\infty\)) \[ \begin{aligned} -(| u'(t)|^{p-2}u'(t))'=f(t,u(t)),&\quad\text{ a.e. }\;t\in[0,b],\\ u(0)=u(b),\;u'(0)=u'(b),&\end{aligned} \] where the nonlinearity \(f\) is Carathéodory, satisfies a unilateral growth condition with respect to the second argument, and the ratio \(\frac{f(t,x)}{x^{p-1}}\) is assumed to be strictly monotone. Using variational methods, necessary and sufficient conditions are given to ensure the existence and the uniqueness of positive solutions in a Sobolev space.
existence and uniqueness of positive solutions
scientific article; zbMATH DE number 6137972