Positive solutions for nonlinear periodic problems with the scalar \(p\)-Laplacian
The authors study the existence of positive solutions for the following nonlinear periodic problem with nonsmooth potential (hemivariational inequality): \[ -(|x'(t)|^{p-2}x'(t))'\in \partial j(t,x(t)), \text{ for a.a. } t \in T=[0,b], \] \[ x(0)=x(b), \qquad x'(0)=x'(b), \] where \(1<p<+\infty\). \(\partial j(t,x)\) denotes the generalized (Clarke) subdifferential with respect to the variable \(x\) of the function \(j:T\times \mathbb R\to\mathbb R\) which is locally Lipschitz and not necessarily \(C^1\) in the \(x\)-variable. They assume a nonuniform nonresonance condition at \(+\infty\) and a uniform nonresonance condition at \(0^+\) and, using degree theoretic argument based on a fixed point index for certain nonconvex valued multifunctions, they prove the existence of a strictly positive solution.
scientific article; zbMATH DE number 5529263