Concentration phenomena in nonlinear eigenvalue problems with variable exponents and sign-changing potential

Abstract

In this paper we establish the concentration of the spectrum in an unbounded interval for a class of eigenvalue problems involving variable growth conditions and a sign-changing potential. We also study the optimization problem for the particular eigenvalue given by the infimum of the associated Rayleigh quotient when the variable potential lies in a bounded, closed and convex subset of a certain variable exponent Lebesgue space.

Some remarks on an eigenvalue problem for an anisotropic elliptic equation with indefinite weight

“…Our goal is to complement and extend the previous ones in [13] and [10] in the sense that Mihailescu et al [13] considered problem (1) in the special case V ≡ 1 while Kefi [10] considered p(x)-Laplacian problem (2) only in the sublinear case. We also find that our results are better than those presented in [15,16] since we consider the problem in the anisotropic case. By considering two different situations concerning the growth rates involved in the problem, we prove the existence of a continuous family of eigenvalues by using variational methods.…”

“…Now, we are in the position to prove that λ * > 0. Assume by contradiction that λ * = 0, from (16) we get λ * = 0. Then, there exists a sequence {u n } ⊂ W + \{0} such that…”

Multiple solutions for eigenvalue problems involving an indefinite potential and with (p 1(x), p 2(x)) balanced growth

“…In this paper we extend the results obtained by M. Mihȃilescu and V. Rȃdulescu in [15] in the framework of the new operators introduced by I. H. Kim and Y. H. Kim. We study the presence of two operators with variable growth and the influence of an indefinite sign-changing potential on their spectral properties.…”

Multiple Solutions for a Class of (P1(x), P2(x))-Laplacian Problems With Neumann Boundary Conditions

“…In that context, we refer to the book of Musielak [17] and the papers of Fan et al [9], Mihȃilescu and Rȃdulescu [16]. Set…”

“…By the presence of two variable exponents p 1 (x) and p 2 (x), problem (3) are more complicated. Some extensions of [15] can be found in [7,16,20,23].…”