Nonlinear Kato Class and Unique Continuation of Eigenfunctions for p-Laplacian Operator

René Erlín Castillo and Julio C. Ramos Fernández 1 Departamento de Matemáticas, Universidad Nacional de Colombia, Apartado, 360354 Bogotá, Colombia 2Departamento de Matemáticas, Universidad de Oriente, Cumaná 6101 Estado Sucre, Venezuela Correspondence should be addressed to René Erĺın Castillo; recastillo@unal.edu.co Received 25 April 2013; Revised 27 August 2013; Accepted 28 August 2013 Academic Editor: Janusz Matkowski Copyright © 2013 R. E. Castillo and J. C. Ramos Fernández.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Journal of Function Spaces and Applications in our case where 1 <  < , it is written as where and  ∈  ∞ 0 (Ω), with Ω bounded domain in R  .We are concerned with the following problem: and the weight function  is assumed to be not equivalent to zero and lies in M (R  ) in the case  < .Specifically, we are interested in studying a family of functions which enjoys the strong unique continuation property, that is, functions besides the possible zero functions which have zero of infinite order.Definition 1.We say that a function  ∈   loc (Ω) vanishes of infinite order at point  0 if for any natural number  there exists a constant   , such that ∫ ( 0 ,) | ()|   ≤     , (7) for all  ∈ N and for small positive number .Here,  ( 0 , ) = { ∈ R  :      −  0     < } .
Definition 2. We say that (6) has a strong unique continuation property if and only if any solution  of (6) in Ω is identically zero in Ω provided that  vanishes of infinite order at a point in Ω.
There is an extensive literature on unique continuation.We refer to the work of Zamboni on unique continuation for nonnegative solutions of quasilinear elliptic equation [3], also the work of Jerison-Kenig on the unique continuation for Schrödinger operators [4].The same work is done by Chiarenza and Frasca, but for linear elliptic operator in the case where  ∈  /2 when  > 2 [5].

Definitions and Notation
In this section, we gather definitions and notations that will be used throughout the paper.We also include several simple lemmas.By  1 loc (R  ), we will denote the space of functions which are locally integrable on R  , and by
Definition 4. We say that a function We are now ready to formulate some simple properties of the classes   and M .
Lemma 5 (see [3], page 152).For 1 <  < , one has From Lemma 5 we conclude that both   (R  ) and M (R  ) are generalizations of   .Remark 6.The following example shows that   is properly contained in   (R  ) for  > 2. It is known that the function This can be shown by splitting the domain of integration in the interior integral into the following three parts: After routine calculations, we can see that is majorized by || −1 .Finally, we have This shows that (13) holds.Thus,  ∈ ⋂ >2   .
Definition 7. The distribution function   of a measurable function  is given by where  denotes the Lebesgue measure on R  .The distribution function   provides information about the size of  but not about the behavior of  itself near any given point.For instance, a function on R  and each of its translates have the same distribution function.It follows from Definition 7 that   is a decreasing function of  (not strictly necessary).
Definition 8. Let  be a measurable function in R  .The decreasing rearrangement of  is the function  defined on [0, ∞) by We use here the convention that inf 0 = ∞.
(ii) For  > 2, the expression (26) satisfies the following inequality: for all  and  in M (R  ).
If  is a neighborhood of 0 from (27), we have then, M (R  ) is a topological vector space.
For more details on nonlinear Kato class, we refer the readers to [9].

Some Useful Inequalities
For the sake of completeness and convenience of the reader, we include the proof of the next result which is due to Schechter [3].
Proof.For any  ∈  ∞ 0 (R  ) supported in ( 0 , ), using the well-known inequality On the other hand, using Hölder's inequality one more time, we have The next corollary is an easy consequence of the previous theorem.It can be obtained via a standard partition of unity.
Corollary 16.Let  ∈ M (R  ) and let Ω be a bounded subset of R  , supp  ⊆ Ω.Then, for any  > 0 there exists a positive constant  depending on , such that for all  ∈  ∞ 0 (Ω).
Proof.Let  > 0. Let  be a positive number that will be chosen later.Let {   },  = 1, 2, . . ., (), be a finite partition of the unity of Ω, such that supp  ⊆ (  , ) with   ∈ Ω.We apply Theorem 15 to the functions   and we get (44) Finally, to obtain the result, it is sufficient to choose  such that Φ  (2) = .After that, we note that () ≈  − and the corollary follows.
and this shows that (7) holds, which means that  has a zero of infinite order in -mean at  0 .