A Fredholm alternative for quasilinear elliptic equations with right-hand side measure (Q280967)

The authors consider a quasilinear elliptic equation involving a Leray-Lions operator which generalizes the \(p\)-Laplacian, a nonlinear term with the adequate homogeneity and a Radon measure which is absolutely continuous with respect to the \(p\)-capacity on the right-hand side.NEWLINENEWLINEThey prove a Fredholm-alternative like result, precisely they show that either the non-homogeneous problem has an entropic solution or a homogeneous problem related to the non-homogeneous one possesses a non trivial solution.NEWLINENEWLINEThe proof is obtained by constructing an adapted topological degree theory which is suited for the considered framework and by using its properties, which are in the line of the usual properties of the degree. Specifically, the original problem is transformed through an homotopy to the one without the nonlinear term. The authors then use some known and new properties of the latter to obtain their main result.