Weak eigenfunctions for the linearization of extremal elliptic problems (Q1270230)

A semilinear elliptic problem \[ -\Delta u= \lambda g(u) \quad\text{in } \Omega,\qquad u=0 \quad\text{on } \partial \Omega \] is considered. Here \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(\lambda\) is a nonnegative parameter, \(g\in C^1\) is a positive, nondecreasing, convex function, \(\lim_{u\to \infty} g(u)/u= \infty\). It is known that there is an extremal parameter \(\lambda^* \in (0,\infty)\) such that there is a minimal classical solution for any \(\lambda \in [0,\lambda^*)\), there is a weak solution \(u^*\) for \(\lambda= \lambda^*\) and there is no solution for \(\lambda \in (\lambda^*,\infty)\). The linearization of this problem with \(\lambda= \lambda^*\) at the extremal solution \(u^*\) is studied. It is proved that this problem has always a positive weak eigenfunction in \(L_1(\Omega)\) corresponding to the zero eigenvalue, even if it has a discrete and positive \(H_0^1\)-spectrum. The zero eigenvalue is coherent with the nonexistence of solutions of the original semilinear problem for \(\lambda > \lambda^*\) and therefore the \(L_1\)-spectrum seems to be more relevant than the \(H_0^1\)-spectrum. It is shown that also a continuous spectrum of the linearization mentioned can occure. For the case of the unit ball and \(g(u)= e^u\) or \(g(u)= (1+u)^p\), \(p>1\), all weak eigenfunctions and eigenvalues are found.