On a New Characterization of Some Class Nonlinear Eigenvalue Problem

In this paper, we derive a new boundedness and compactness result for the Hardy operator in variable exponent Lebesgue spaces (VELS) Lp(.)(0, l). A maximally weak condition is assumed on the exponent function. The last time such a study was carried out was in [1–12]. For a study of the Dirichlet problem of some class nonlinear eigenvalue problem with nonstandard growth condition the obtained results are applied. Such equations arise in the studies of the socalled Winslow effect physical phenomena [13] in the smart materials. In this connection, we mention recent studies for the multidimensional cases with application of AmbrosettiRabinowitz’s Mountain Pass theorem approaches (see, e.g., [1, 14, 15]). Theorem A. Let q, p : (0, l) 󳨀→ (1,∞) be measurable functions with q(x) ≥ p(x) on (0, l). Assume p is monotonically increasing and the function x−1/p󸀠(x)+δ is almost decreasing on (0, l).. Then operator H boundedly acts the space Lp(0, l) into Lq(.),−1/p󸀠−1/q(.)(0, l). Moreover, the norm of mapping depends on p−, p+, δ, β. Theorem B. Let q, p : (0, l) 󳨀→ (1,∞) be measurable functions such that ∞ > q+ ≥ q(x) ≥ p(x) ≥ p− > 1 for all x ∈ (0, l). Assume that p is monotonically increasing and x−1/p󸀠+ε is almost decreasing. Then the identity operator maps boundedly the space W1. p(.)(0, l) into Lq(.),−1/p󸀠−1/q(.)(0, l). Moreover, the norm of mapping is estimated by a constant depending on p−, p+, q, ε, β.


Introduction
In this paper, we derive a new boundedness and compactness result for the Hardy operator in variable exponent Lebesgue spaces (VELS)  (.) (0, ).A maximally weak condition is assumed on the exponent function.The last time such a study was carried out was in [1][2][3][4][5][6][7][8][9][10][11][12].For a study of the Dirichlet problem of some class nonlinear eigenvalue problem with nonstandard growth condition the obtained results are applied.Such equations arise in the studies of the socalled Winslow effect physical phenomena [13] in the smart materials.In this connection, we mention recent studies for the multidimensional cases with application of Ambrosetti-Rabinowitz's Mountain Pass theorem approaches (see, e.g., [1,14,15]).
We need the following assertion.
In the light of the information given above, we can give proof of Theorem A.
Proof.In order to proof Theorem C, we may apply the approaches from [3][4][5].In this way, insert the operators As it was stated in [3],  3 is a limit of finite rank operators, while  2 is a finite rank operator.From the condition lim →0 () = 0 it follows that      −  2  −  3      (.) (0,) ≤      1      (.) (0,) as  → 0. To show the last estimation we shall use the arguments of Theorem A. Repeating all constructions there, we get the following estimates: Notice that we have used  −1/  + ≤  −1/  ()+ for any  ∈   (), where  belongs to the natural number.
To prove the PS condition, we must prove that such a sequence is compact; that is, it contains a subsequence {   } converging in  to a function ∈ .In order to show it, establish the boundedness of {  }.
Now, what remains is to find a point  0 ∈  where   ( 0 ) < 0. To show this, apply the fibering method; for  ∈  to be fixed and sufficiently large  > 1 it holds that Therefore,   − ≡ 0; using imbedding Theorem B we infer  − ≡ 0, which implies that  0 () > 0.
We have proved the existence of problem (27) for any  > 0.