On weighted eigenvalue problems and applications

Abstract

The main objective of this thesis is to _nd a large class of weight functions that admits a positive principal eigenvalue for the weighted eigenvalue problems for the Laplacian and the p-Laplacian. More speci_cally, for a connected domain in RN with N _ 2; we study su_cient conditions for a function g 2 L1 loc() to admit a pair (_; u); with _ 2 R+ and u gt 0 a.e. such that u is a weak solution of the following problem: _ _pu = _gjujp_2u; in ; (1) where 1 lt p lt N and _pu := div(jrujp_2ru) is the p-Laplace operator. Such a _; if exists, is called a principal eigenvalue of (1). In the literature, most of the su_cient conditions for the existence of a positive principal eigenvalue demand that the weight function g or its positive part g+ to be in L p (): However, in the _eld of applications one may need to consider weights that are not belonging to any of the Lebesgue spaces. We look for a weak solution of (1) in D1;p 0 (); where newlineD1;p 0 () := completion of C1c ()with respect to kr_kp norm : Now the existence of a positive principal eigenvalue for (1) is closely related with the existence of a minimizer for the functional J(u)= Z jrujp on the level set Mp=nu 2 D1;2 newline0 (): R gjujp=1o: If the map G; G(u) = R g+jujp; is compact, then a direct variational method ensures the existence of a minvii imizer for J on Mp: If g+ is in LN p (), the dual of Lp_ p (); then the map G is compact. This is mainly a consequence of three facts (i) the continuous embedding of D1;p 0 () into Lp_() (ii) the compactness of the embedding of D1;p 0 () into Lp loc() (iii) the density of C1c () in LN p (): The main novelty of our results is that we allow weights that are not in any of the Lebesgue spaces, but only in certain weak Lebesgue spaces. For this we make use of the _nest embedding of D1;p 0 () into the Lorentz space L(p_; p) = (u measurable : Z jj 0 [t1 p_ u_(t)]p dt t lt 1); where u_ denotes the one dimensional decreasing rearrangement of u: The Lorentz space L(p_; p) is a Banach space with a suitable norm and it is a proper subspace.