Low perturbations for a class of nonuniformly elliptic problems
aa r X i v : . [ m a t h . A P ] J un LOW PERTURBATIONS FOR A CLASS OFNONUNIFORMLY ELLIPTIC PROBLEMS
ANOUAR BAHROUNI AND DUˇSAN D. REPOVˇS
Abstract.
We introduce and study a new functional which was motivated by our paper onthe Caffarelli-Kohn-Nirenberg inequality with variable exponent (Bahrouni, R˘adulescu & Repovˇs,Nonlinearity 31 (2018), 1518-1534). We also study the eigenvalue problem for equations involvingthis new functional. Introduction
The Caffarelli-Kohn-Nirenberg inequality plays an important role in studying various problems ofmathematical physics, spectral theory, analysis of linear and nonlinear PDEs, harmonic analysis, andstochastic analysis. We refer to Chaudhuri & Ramaswamy [2], Baroni, Colombo & Mingione [4],Colasuonno & Pucci [7], and Colombo & Mingione [8] for relevant applications of the Caffarelli-Kohn-Nirenberg inequality.Let Ω ⊂ R N ( N ≥
2) be a bounded domain with smooth boundary. The following Caffarelli-Kohn-Nirenberg inequality (see Caffarelli, Kohn & Nirenberg [5]) establishes that given p ∈ (1 , N )and real numbers a, b , and q such that −∞ < a < N − pp , a ≤ b ≤ a + 1 , q = N pN − p (1 + a − b ) , there is a positive constant C a,b such that for every u ∈ C c (Ω),(1) (cid:18)Z Ω | x | − bq | u | q dx (cid:19) p/q ≤ C a,b Z Ω | x | − ap |∇ u | p dx . This inequality has been extensively studied (see, e.g. Abdellaoui & Peral [1], Chaudhuri &Ramaswamy [2], Bahrouni, R˘adulescu & Repovˇs [3], Catrina & Wang [6], and Mih˘ailescu, R˘adulescu& Stancu [11], and the references therein).In particular, Bahrouni, R˘adulescu & Repovˇs [3] gives a new version of the Caffarelli-Kohn-Nirenberg inequality with variable exponent. The next theorem is proved under the following as-sumptions: let Ω ⊂ R N ( N ≥
2) be a bounded domain with smooth boundary and suppose that thefollowing hypotheses are satisfied(A) a : Ω → R is a function of class C and there exist x ∈ Ω, r >
0, and s ∈ (1 , + ∞ ) such that:(1) | a ( x ) | 6 = 0 , for every x ∈ Ω \ { x } ;(2) | a ( x ) | ≥ | x − x | s , for every x ∈ B ( x , r );(P) p : Ω → R is a function of class C and 2 < p ( x ) < N for every x ∈ Ω . Theorem 1.1. (see Bahrouni, R˘adulescu & Repovˇs [3])
Suppose that hypotheses ( A ) and ( P ) aresatisfied. Let Ω ⊂ R N ( N ≥ ) be a bounded domain with smooth boundary. Then there exists a Key words and phrases.
Caffarelli-Kohn-Nirenberg inequality, eigenvalue problem, critical point theorem, gener-alized Lebesgue-Sobolev space, Luxemburg norm. : Primary 35J60, Secondary 35J91, 58E30. positive constant β such that Z Ω | a ( x ) | p ( x ) | u ( x ) | p ( x ) dx ≤ β Z Ω | a ( x ) | p ( x ) − ||∇ a ( x ) || u ( x ) | p ( x ) dx + β (cid:18)Z Ω | a ( x ) | p ( x ) |∇ u ( x ) | p ( x ) dx + Z Ω | a ( x ) | p ( x ) |∇ p ( x ) || u ( x ) | p ( x )+1 dx (cid:19) + β Z Ω | a ( x ) | p ( x ) − |∇ p ( x ) || u ( x ) | p ( x ) − dx. for every u ∈ C c (Ω) . Motivated by Bahrouni, R˘adulescu & Repovˇs [3], we introduce and study in the present paper anew functional T : E → R via the Caffarelli-Kohn-Nirenberg inequality, in the framework of variableexponents. More precisely, we study the eigenvalue problem in which functional T is present. Ourmain result is Theorem 4.2 and we prove it in Section 5. Function spaces with variable exponent
We recall some necessary properties of variable exponent spaces. We refer to Hajek, Santalucia,Vanderwerff & Zizler [10], Musielak [12], Papageorgiou, R˘adulescu & Repovˇs [13], R˘adulescu [15],R˘adulescu [16], and R˘adulescu & Repovˇs [17], and the references therein.Consider the set C + (Ω) = { p ∈ C (Ω) | p ( x ) > x ∈ Ω } . For any p ∈ C + (Ω), let p + = sup x ∈ Ω p ( x ) and p − = inf x ∈ Ω p ( x ) , and define the variable exponent Lebesgue space as follows L p ( x ) (Ω) = (cid:26) u | u is measurable real-valued function such that Z Ω | u ( x ) | p ( x ) dx < ∞ (cid:27) , with the Luxemburg norm | u | p ( x ) = inf ( µ > | Z Ω (cid:12)(cid:12)(cid:12)(cid:12) u ( x ) µ (cid:12)(cid:12)(cid:12)(cid:12) p ( x ) dx ≤ ) . We recall that the variable exponent Lebesgue spaces are separable and reflexive Banach spaces ifand only if 1 < p − ≤ p + < ∞ , and continuous functions with compact support are dense in L p ( x ) (Ω)if p + < ∞ .Let L q ( x ) (Ω) denote the conjugate space of L p ( x ) (Ω), where 1 /p ( x ) + 1 /q ( x ) = 1. If u ∈ L p ( x ) (Ω)and v ∈ L q ( x ) (Ω) then the following H¨older-type inequality holds:(2) (cid:12)(cid:12)(cid:12)(cid:12)Z Ω uv dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) p − + 1 q − (cid:19) | u | p ( x ) | v | q ( x ) . An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the p ( . ) − modular of the L p ( x ) (Ω) space, which is the mapping ρ : L p ( x ) (Ω) → R defined by ρ ( u ) = Z Ω | u | p ( x ) dx. Proposition 2.1. (see R˘adulescu & Repovˇs [17])
The following properties hold ( i ) | u | p ( x ) < resp ., = 1; > ⇔ ρ ( u ) < resp ., = 1; > ; ( ii ) | u | p ( x ) > ⇒ | u | p − p ( x ) ≤ ρ ( u ) ≤ | u | p + p ( x ) ; and ( iii ) | u | p ( x ) < ⇒ | u | p + p ( x ) ≤ ρ ( u ) ≤ | u | p − p ( x ) . ONUNIFORMLY ELLIPTIC PROBLEMS 3
Proposition 2.2. (see R˘adulescu & Repovˇs [17]) If u, u n ∈ L p ( x ) (Ω) and n ∈ N , then the followingstatements are equivalent (1) lim n → + ∞ | u n − u | p ( x ) = 0 . (2) lim n → + ∞ ρ ( u n − u ) = 0 . (3) u n → u in measure in Ω and lim n → + ∞ ρ ( u n ) = ρ ( u ) . We define the variable exponent Sobolev space by W ,p ( x ) (Ω) = { u ∈ L p ( x ) (Ω) | |∇ u | ∈ L p ( x ) (Ω) } . On W ,p ( x ) (Ω) we consider the following norm k u k p ( x ) = | u | p ( x ) + |∇ u | p ( x ) . Then W ,p ( x ) (Ω) is a reflexive separable Banach space. Functional T We shall introduce a new functional T : E → R motivated by the Caffarelli-Kohn-Nirenberginequality obtained in Bahrouni, R˘adulescu & Repovˇs [3].We denote by E the closure of C c (Ω) under the norm k u k = || B ( x ) | p ( x ) ∇ u ( x ) | p ( x ) + | A ( x ) p ( x ) u ( x ) | p ( x ) + || D ( x ) | p ( x )+1 u ( x ) | p ( x )+1 + || C ( x ) | p ( x ) − u ( x ) | p ( x ) − , where the potentials A , B , C , and D are defined by(3) A ( x ) = | a ( x ) | p ( x ) − |∇ a ( x ) | B ( x ) = | a ( x ) | p ( x ) C ( x ) = | a ( x ) | p ( x ) − |∇ p ( x ) | D ( x ) = B ( x ) |∇ p ( x ) | . We now define T : E → R as follows T ( u ) = Z Ω B ( x ) p ( x ) |∇ u ( x ) | p ( x ) dx + Z Ω A ( x ) p ( x ) | u ( x ) | p ( x ) dx + Z Ω D ( x ) p ( x ) + 1 | u ( x ) | p ( x )+1 dx + Z Ω C ( x ) p ( x ) − | u ( x ) | p ( x ) − dx. The following properties of T will be useful in the sequel. Lemma 3.1.
Suppose that hypotheses ( A ) and ( P ) are satisfied. Then the functional T is well-defined on E . Moreover, T ∈ C ( E , R ) with the derivative given by h L ( u ) , v i = h T ′ ( u ) , v i = Z Ω B ( x ) |∇ u ( x ) | p ( x ) − ∇ u ( x ) ∇ v ( x ) dx + Z Ω A ( x ) | u ( x ) | p ( x ) − u ( x ) v ( x ) dx + Z Ω D ( x ) | u ( x ) | p ( x ) − u ( x ) v ( x ) dx + Z Ω C ( x ) | u ( x ) | p ( x ) − u ( x ) v ( x ) dx, for every u, v ∈ E .Proof. The proof is standard, see R˘adulescu & Repovˇs [17]. (cid:3)
Lemma 3.2.
Suppose that hypotheses ( A ) and ( P ) are satisfied. Then the following properties hold ( i ) L : E → E ∗ is a continuous, bounded and strictly monotone operator; ( ii ) L is a mapping of type ( S + ) , i.e. if u n ⇀ u in E and lim sup n → + ∞ h L ( u n ) − L ( u ) , u n − u i ≤ , then u n → u in E . A. BAHROUNI AND D.D. REPOVˇS
Proof. (i) Evidently, L is a bounded operator. Recall the following Simon inequalities (see Si-mon [18]):(4) | x − y | p ≤ c p (cid:16) | x | p − x − | y | p − y (cid:17) . ( x − y ) for p ≥ | x − y | p ≤ C p h(cid:16) | x | p − x − | y | p − y (cid:17) . ( x − y ) i p ( | x | p + | y | p ) − p for 1 < p < , for every x, y ∈ R N , where c p = ( 12 ) − p and C p = 1 p − . Using inequalities (4) and recalling that 2 < p − , we can prove that L is a strictly monotoneoperator.(ii) The proof is identical to the proof of Theorem 3 . (cid:3) Main theorem
We recall our Compactness Lemma:
Lemma 4.1. (see Bahrouni, R˘adulescu & Repovˇs [3])
Suppose that hypotheses ( A ) and ( P ) aresatisfied and that p − > s . Then E is compactly embeddable in L q (Ω) for each q ∈ (1 , Np − N + sp + ) .Moreover, the same conclusion holds if we replace L q (Ω) by L q ( x ) (Ω) , provided that q + < Np − N + sp + . We are concerned with the following nonhomogeneous problem(5) − div ( B ( x ) |∇ u | p ( x ) − ∇ u ) + ( A ( x ) | u | p ( x ) − + C ( x ) | u | p ( x ) − ) u =( λ | u | q ( x ) − − D ( x ) | u | p ( x ) − ) u in Ω ,u = 0 on ∂ Ω , where λ > q is continuous on Ω. We assume that q satisfies the followingbasic inequalities( Q ) 1 < min x ∈ Ω q ( x ) < min x ∈ Ω ( p ( x ) − < max x ∈ Ω q ( x ) < N p − N + sp + . We can now state the main result of this paper.
Theorem 4.2.
Suppose that all hypotheses of Lemma 4.1 are satisfied and that inequalities ( Q ) hold. Then there exists λ > such that every λ ∈ (0 , λ ) is an eigenvalue for problem (5) . In order to prove Theorem 4.2 (this will be done in the Section 5), we shall need some preliminaryresults. We begin by defining the functional I λ : E → R ,I λ ( u ) = Z Ω B ( x ) p ( x ) |∇ u ( x ) | p ( x ) dx + Z Ω A ( x ) p ( x ) | u ( x ) | p ( x ) dx + Z Ω C ( x ) p ( x ) − | u ( x ) | p ( x ) − dx + Z Ω D ( x ) p ( x ) + 1 | u ( x ) | p ( x )+1 dx − λ Z Ω | u ( x ) | q ( x ) q ( x ) dx. Standard argument shows that I λ ∈ C ( E , R ) and h I ′ λ ( u ) , v i = Z Ω B ( x ) |∇ u ( x ) | p ( x ) − ∇ u ( x ) ∇ v ( x ) dx + Z Ω A ( x ) | u ( x ) | p ( x ) − u ( x ) v ( x ) dx + Z Ω D ( x ) | u ( x ) | p ( x ) − u ( x ) v ( x ) dx + Z Ω C ( x ) | u ( x ) | p ( x ) − u ( x ) v ( x ) dx − λ Z Ω | u ( x ) | q ( x ) − u ( x ) v ( x ) , for every u, v ∈ E .Thus the weak solutions of problem (5) coincide with the critical points of I λ . ONUNIFORMLY ELLIPTIC PROBLEMS 5
Lemma 4.3.
Suppose that all hypotheses of Theorem 4.2 are satisfied. Then there exists λ > such that for any λ ∈ (0 , λ ) there exist ρ, α > such that I λ ( u ) ≥ α for any u ∈ E with k u k = ρ. Proof.
By Lemma 4.1, there exists β > | u | r ( x ) ≤ β k u k , for every u ∈ E and r + ∈ (1 , N p − N + sp + ) . We fix ρ ∈ (0 , min(1 , β )). Invoking Proposition 2.1, for every u ∈ E with k u k = ρ , we can get | u | q ( x ) < . Combining the above relations and Proposition 2.1, for any u ∈ E with k u k = ρ , we can thendeduce that(6) I λ ( u ) ≥ p + (cid:18)Z Ω B ( x ) |∇ u ( x ) | p ( x ) dx + Z Ω A ( x ) | u ( x ) | p ( x ) dx (cid:19) + 1 p + + 1 Z Ω D ( x ) | u ( x ) | p ( x )+1 dx + 1 p + − Z Ω C ( x ) | u ( x ) | p ( x ) − dx − λq − Z Ω | u ( x ) | q ( x ) dx ≥ p + ( p + + 1) k u k p + +1 − λ β q − q − k u k q − ≥ p + ( p + + 1) ρ p + +1 − λ β q − q − ρ q − = ρ q − ( p + ( p + +1) ρ p + +1 − q − − λ β q − q − ) . Put λ = ρ p ++1 − q − p + (2 p + +2) q − β q − . It now follows from (6) that for any λ ∈ (0 , λ ), I λ ( u ) ≥ α with k u k = ρ, and α = ρ p ++1 p + (2 p + +2) > . This completes the proof of Lemma 4.3. (cid:3)
Lemma 4.4.
Suppose that all hypotheses of Theorem 4.2 are satisfied. Then there exists ϕ ∈ E such that ϕ > and I λ ( tϕ ) < , for small enough t .Proof. By virtue of hypotheses ( P ) and ( Q ), there exist ǫ > ⊂ Ω such that(7) q ( x ) < q − + ǫ < p − − , for every x ∈ Ω . Let ϕ ∈ C ∞ (Ω) such that Ω ⊂ supp( ϕ ), ϕ = 1 for every x ∈ Ω and 0 ≤ ϕ ≤ t ∈ (0 , I λ ( tϕ ) = Z Ω t p ( x ) B ( x ) p ( x ) |∇ ϕ ( x ) | p ( x ) dx + Z Ω t p ( x ) A ( x ) p ( x ) | ϕ ( x ) | p ( x ) dx + Z Ω t p ( x ) − C ( x ) p ( x ) − | ϕ | p ( x ) − dx + Z Ω t p ( x )+1 D ( x ) p ( x ) + 1 | ϕ ( x ) | p ( x )+1 dx − λ Z Ω t q ( x ) | ϕ ( x ) | q ( x ) q ( x ) dx ≤ t p − − p − − Z Ω B ( x ) p ( x ) |∇ ϕ ( x ) | p ( x ) dx + Z Ω A ( x ) p ( x ) | ϕ ( x ) | p ( x ) dx + Z Ω C ( x ) p ( x ) − | ϕ | p ( x ) − dx + Z Ω D ( x ) p ( x ) + 1 | ϕ ( x ) | p ( x )+1 dx ) − λt q − + ǫ Z Ω | ϕ ( x ) | q ( x ) q ( x ) dx. (8)Combining (7) and (8), we finally arrive at the desired conclusion.This completes the proof of Lemma 4.4. (cid:3) A. BAHROUNI AND D.D. REPOVˇS Proof of Theorem 4.2
In the last section we shall prove the main theorem of this paper.Let λ be defined as in Lemma 4.3 and choose any λ ∈ (0 , λ ).Again, invoking Lemma 4.3, we can deduce that(9) inf u ∈ ∂B (0 ,ρ ) I λ ( u ) > . On the other hand, by Lemma 4.4, there exists ϕ ∈ E such that I λ ( tϕ ) < > . Moreover, by Proposition 2.1, when k u k < ρ , we have I λ ( u ) ≥ p + ( p + + 1) k u k p + +1 − c k u k q − , where c is a positive constant. It follows that −∞ < m = inf u ∈ B (0 ,ρ ) I λ ( u ) < . Applying Ekeland’s variational principle to the functional I λ : B (0 , ρ ) → R , we can find a (PS) sequence ( u n ) ∈ B (0 , ρ ), that is, I λ ( u n ) → m and I ′ λ ( u n ) → . It is clear that ( u n ) is bounded in E . Thus there exists u ∈ E such that, up to a subsequence,( u n ) ⇀ u in E . Using Theorem 4.1, we see that ( u n ) strongly converges to u in L q ( x ) (Ω).So, by the H¨older inequality and Proposition 2.2, we can obtain the followinglim n → + ∞ Z Ω | u n | q ( x ) − u n ( u n − u ) dx = lim n → + ∞ Z Ω | u | q ( x ) − u ( u n − u ) dx = 0 . On the other hand, since ( u n ) is a (PS) sequence, we can also infer thatlim n → + ∞ h I ′ λ ( u n ) − I ′ λ ( u ) , u n − u i = 0 . Combining the above pieces of information with Lemma 3.2, we can now conclude that u n → u in E . Therefore I λ ( u ) = m < I ′ λ ( u ) = 0 . We have thus shown that u is a nontrivial weak solution for problem (5) and that every λ ∈ (0 , λ )is an eigenvalue of problem (5).This completes the proof of Theorem 4.2. (cid:3) Acknowledgements
The second author was supported by the Slovenian Research Agency grants P1-0292, J1-7025, J1-8131, N1-0064, N1-0083, and N1-0114. We thank the referee for comments and suggestions.
ONUNIFORMLY ELLIPTIC PROBLEMS 7
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Mathematics Department, University of Monastir, Faculty of Sciences, 5019 Monastir,Tunisia
E-mail address : [email protected] (D.D. Repovˇs) Faculty of Education and Faculty of Mathematics and Physics, University of Ljubljana,& Institute of Mathematics, Physics and Mechanics, 1000 Ljubljana, Slovenia
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