A Picone’s identity for the \(p\)-Laplacian and applications. (English) Zbl 0930.35053

From the introduction: It is the purpose of this paper to present a Picone identity for the \(p\)-Laplacian, which is an extension of the classic identity for the ordinary Laplacian. As immediate consequence, a variety of results is obtained by elementary means. These include simplicity of principal eigenvalues; Barta’s inequalities; nonexistence of positive solutions; Hardy’s inequalities; Sturmian comparison and oscillation/nonoscillation results for solutions which need not be radial. Besides its simplicity, this approach enables us to avoid postulating regularity conditions on the boundary of the domain under consideration. In particular, the cases of bounded or unbounded \(\Omega\) can be treated in a unified fashion. Detailed references and comparison with earlier results are presented after each application.
Finally, this paper is not directly concerned with existence/regularity results. While we occasionally mention some such criteria, we always assume that the solutions \(u\) are at least in \(C^{1+\alpha}\) and \(\Delta_pu\in C\). The reader interested in these questions will find detailed conditions in some of the cited references.


35J30 Higher-order elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35P99 Spectral theory and eigenvalue problems for partial differential equations
35B99 Qualitative properties of solutions to partial differential equations
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