Weak Eigenfunctions for the Linearization of Extremal Elliptic Problems

We consider the semilinear elliptic problem[formula]whereλis a nonnegative parameter andgis a positive, nondecreasing, convex nonlinearity. There exists a valueλ* 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 parameterλ=λ*. In some cases, this linearized problem has discrete and positiveH10-spectrum. However, we prove that there always exists a positive weak eigenfunction inL1(Ω) with eigenvalue zero for this linearized problem. The zeroL1-eigenvalue is coherent with the nonexistence of solutions of the semilinear problem forλ>λ*. Finally, we find all weak eigenfunctions and eigenvalues for the linearization of the extremal problem whenΩis the unit ball andg(u)=euorg(u)=(1+u)p.