A direct uniqueness proof for equations involving the p -Laplace operator

An alternative approach to the Faber–Krahn inequality for Robin problems

“…The proofs in [17] using the direct method in the calculus of variations are easily adapted to our situation (see [6,Section 2] or [16,Theorem 3.4]). Note that the very elegant and simple proof from [13,2] could be adapted.…”

On the asymptotics of solutions of the Lane-Emden problem for the p-Laplacian

“…Ω |∇u| p Ω |u| p and there exists only one eigenfunction e 1 (up to a multiplicative factor) which has constant sign (for a new and direct proof, see [5]). The higher eigenvalues can be obtained through the following minimax principle: let us define the Krasnoselskii genus of a set…”

Comparison principle and constrained radial symmetry for the subdiffusive $p$-Laplacian

“…Of course, there is no hope to obtain a comparison principle under assumption (3) alone: a counterexample is readily obtained with the equation −∆u = λ 1 u, where λ 1 is the first eigenvalue of the DirichletLaplacian in Ω, assuming that Ω is a bounded, smooth domain. A similar situation also occurs with the first eigenfunctions of the p-Laplace operator: see, for instance, [4,22] and the references therein.…”

“…Several related results are found in the literature: in particular, Díaz-Saá's inequality [5,10] and Picone's identity [2]. An elegant uniqueness result exploiting the strict convexity of the associated functional H p (u) with respect to the function u p (hidden convexity) is found in [4]. The fundamental comparison principle between a p-subharmonic and a p-superharmonic function is proved in [19,24].…”

“…From now on, assumptions (H 1 )-(H 3 ) and (4) are in effect. We make use of the known existence and uniqueness results [4,10] for bounded, smooth domains under homogeneous boundary conditions in order to ensure that equation (7) has a unique, radial, positive solution u R ∈ W 1,p 0 (B R ) in the ball B R centered at the origin. Such a solution is also bounded, and belongs to the class C 1,α (B R ) [10,11,23,29].…”