Anisotropic Robin problems with logistic reaction
The paper deals with existence results of positive solutions for elliptic equations driven by the \(p(x)\)-Laplacian (so-called anisotropic \(p\)-Laplacian) involving a nonnegative potential and a right-hand side depending on a parametric term with exponent \(q(x)\) and a perturbation \(f\). Here, the boundary condition is of Robin-type. The authors discuss in a very precise way the cases when the equation is superdiffusive, subdiffusive and equidiffusive. In the superdiffusive case the existence of a bifurcation point \(\lambda^*\) is shown so that for large parameters, the equation has at least two positive smooth solution, at least one positive smooth solution when the parameter is equal \(\lambda^*\) and no positive solution when the parameter is less than \(\lambda^*\). In the subdiffusive case the authors obtain uniqueness of the positive solutions and for the equidiffusive case uniqueness and nonexistence is shown. In this last case the results are only obtained in the isotropic case which is due to the fact, that is not known if the first eigenvalue of the anisotropic \(p\)-Laplacian with Neumann boundary condition is positive.
multiplicity and uniqueness of positive solutions
scientific article; zbMATH DE number 7349705
scientific article