Weak Eigenfunctions for the Linearization of Extremal Elliptic Problems

Abstract We consider the semilinear elliptic problem[formula]where λ is a nonnegative parameter and g is a positive, nondecreasing, convex nonlinearity. There exists a value λ * of the parameter which is extremal in terms of existence of solution. We study the linearization of the semilinear problem at the extremal weak solution corresponding to the parameter λ = λ *. In some cases, this linearized problem has discrete and positive H 1 0 -spectrum. However, we prove that there always exists a positive weak eigenfunction in L 1 ( Ω ) with eigenvalue zero for this linearized problem. The zero L 1 -eigenvalue is coherent with the nonexistence of solutions of the semilinear problem for λ > λ *. Finally, we find all weak eigenfunctions and eigenvalues for the linearization of the extremal problem when Ω is the unit ball and g ( u )= e u or g ( u )=(1+ u ) p .