On a New Characterization of Some Class Nonlinear Eigenvalue Problem

Abstract

A normal mode analysis of a vibrating mechanical or electrical system gives rise to an eigenvalue problem. Faber made a fairly complete study of the existence and asymptotic behavior of eigenvalues and eigenfunctions, Green’s function, and expansion properties. We will investigate a new characterization of some class nonlinear eigenvalue problem.

1. Introduction

In this paper, we derive a new boundedness and compactness result for the Hardy operator in variable exponent Lebesgue spaces (VELS) . 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 so-called 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]).

Theorem A. Let be measurable functions with on . Assume is monotonically increasing and the function is almost decreasing on . Then operator H boundedly acts the space into . Moreover, the norm of mapping depends on .

Theorem B. Let be measurable functions such that for all . Assume that is monotonically increasing and is almost decreasing. Then the identity operator maps boundedly the space into . Moreover, the norm of mapping is estimated by a constant depending on

Notice that Theorem B states the inequalityfor any absolutely continuous function with

In the given assertions, denotes the space of measurable functions with finite norm , while stands for the space of absolutely continuous functions y with and finite norm We say that the function is almost increasing (decreasing) if there exists a constant such that for any it holds .

We need the following assertion.

Lemma 1. Let be increasing for . Let , where is the natural number. Then it holdswhere .

We will be inspired by [6–8] while proving Lemma 1.

Proof. Let be a point with . Let and both lie in . Then using the almost decrease of it follows that Using it follows thatNow let ; then, using the increase of , also will be increasing. Since , it follows thatwhere .
Lemma 1 has been proved.

In the light of the information given above, we can give proof of Theorem A.

Proof of Theorem A. Let be a positive measurable function. It holds the identityAssume Using the triangle property of normswith (recall ).
Derive estimation for every summand in (8). In this purpose get estimation for the proper modularApplying the assumption on (decreasing of ), and using the expression for , we have Notice that we have used for any and by using the almost decrease of .
Therefore, from (8), using Hölder’s inequality, it follows thatApplying Lemma 1 and estimate (3) it follows from (11) thatSince It follows thatHence,Therefore, it has been proved that which implies Inserting (17) into (8), we get Theorem A has been proved.

Theorem C. Let be measurable functions such that .
Assume that the function increases on and is almost decreasing in Then operator H acts compactly the space into for any .

Proof. In order to proof Theorem C, we may apply the approaches from [3–5]. In this way, insert the operatorsAs it was stated in [3], is a limit of finite rank operators, while is a finite rank operator. From the condition it follows that oras 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 for any , where belongs to the natural number.
Therefore, using Hölder’s inequality, Applying Lemma 1 and the arguments above, we attain the estimatesTherefore, it has been shown that if . This implies Inserting these estimates over in the expression The last estimate is a needed estimation which completes the proof of Theorem C.
Consider the problemwhere

Theorem D. Let be measurable functions on such that is increasing and the function is almost decreasing on . Then for any fixed there exists a nontrivial solution of the problem (27).

Proof. To prove this assertion, we shall use the well-known Mountain Pass theorem approaches. Define the functional Define the space a closure of absolutely continuous functions on , such that with a normDefine also the space as a space of measurable functions with finite normApplying the standard approaches (see, e.g., [13]), it is not difficult to see that the functional Further, and By applying Theorem B, we get the implication Show that Palaisce-Smale (PS) condition is satisfied for the problem (27). Let be a sequence such that it is satisfied by the following two conditions:(1);(2) as 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 .
From (1) it follows that Then Using condition (2) for , it follows that ; that is,Inserting this into (27), it follows that From this, since , it follows thator Using Young’s inequality from here it follows that This completes the boundedness of in . Applying well-known fact, there exists a weak convergent subsequence in Denote it again as ∙It follows from the compact embedding
Theorem C that a strong convergence in holds; that is,Now, we are ready to show the strong convergence in . For this purpose, insert into the equalityThen From this, since in and using Holder’s inequality, it follows thatsince is bounded in .
Therefore, From this we infer that Since weakly in it holds thatThis ensures that We will apply the following two inequalities:for and for Applying (47), for the case , we get As to the case , we have the inequality Using Young’s inequality from here it follows that Therefore, where does not depend on . This and the above inequality and Young’s inequality giveThis inequality yields in .
Now, we are ready to apply the Mountain Pass theorem. If from preceding equality one getsTherefore, using assumption and Young’s inequality we have that is, in strongly.
The proof of PS property has been completed.

Now, apply the Mountain Pass theorem in order to show the existence of solution for the problem (27).

For we haveBy using Theorem A, it follows that Then (56) implies thatHence for it follows thatTherefore,If we choose , it follows thatChoose to apply the Mountain Pass theorem.

Now, what remains is to find a point where . To show this, apply the fibering method; for to be fixed and sufficiently large it holds thatInsert in order to get some point laid out of the ball in , such that . Applying Mountain Pass theorem, there exists a point with such that , where Therefore,

, . To show that is a positive solution of (27) insert in , Since the second integral is zero , we haveTherefore, ; using imbedding Theorem B we infer , which implies that .

We have proved the existence of problem (27) for any .

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The author declares that he has no conflicts of interest.

Acknowledgments

The author would like to thank the referee for the careful reading of the paper and the valuable suggestions.