Remarks on eigenvalue problems involving the p(x)-Laplacian

Abstract

This paper deals with the eigenvalue problem involving the p(x)-Laplacian of the formis the Sobolev critical exponent. It is shown that for every t > 0, the problem has at least one sequence of solutions {(u n,t , λ n,t )} such that Ω 1 p(x) |∇u n,t | p(x) = t and λ n,t → ∞ as n → ∞. The principal eigenvalues for the problem in several important cases are discussed especially. The similarities and the differences in the eigenvalue problem between the variable exponent case and the constant exponent case are exposed. Some known results on the eigenvalue problem are extended.

Anisotropic (p, q)-equations with gradient dependent reaction

“…From (17) it follows that u 0 ∈ S + v = ∅. Then theorem 4.1 of Fan and Zhao [19] (see also Gasiński and Papageorgiou [25,proposition 3.1] and Papageorgiou, Rȃdulescu and Zhang [38, Proposition A1]), we have that u 0 ∈ L ∞ (Ω).…”

“…Remark 1. As we already mentioned earlier in this section, the hypotheses on the exponent p(•), imply that the eigenvalue problem (3) has a principal eigenvalue λ1 > 0 and an associated positive eigenfunction û1 ∈ int C + (see Fan, Zhang and Zhao [18], Fan [17], and Byun and Ko [9]).…”

“…For the anisotropic eigenvalue problem, in contrast to the isotropic one, we can have inf L = 0 (see Fan, Zhang and Zhao [18, theorem 3.1]). If we can find η ∈ R N (N > 1) such that for all z ∈ Ω, the function t → ϑ(t) = p(z + tη) is monotone on T z = {t ∈ R : z + tη ∈ Ω} and p ∈ C 1 (Ω), then problem (3) has a principal eigenvalue λ1 > 0 with corresponding positive eigenfunction û1 ∈ int C + (see Fan, Zhang and Zhao [18, theorem 3.3]) (see also Byun and Ko [9] and Fan [17]). We have…”

A nonlinear eigenvalue problem with <inline-formula><tex-math id="M1">\begin{document}$ p(x) $\end{document}</tex-math></inline-formula>-growth and generalized Robin boundary value condition

“…Nonlinear eigenvalue problems for p(x)-Laplacian operator similar to eigenvalue problems for p-Laplacian with Dirichlet, Neumann and Steklov boundary condition have been investigated previously, see, e.g., [8,9,10,19].…”

“…In the following we present some sufficient conditions for problem (P 1 ) in order to obtain that inf Λ P1 is either zero or positive. Let us to refer [8,9], where related arguments are provided. We start with the following preliminary result.…”