Pohozaev-type inequalities and nonexistence results for non C 2 solutions of p(x) -laplacian equations
aa r X i v : . [ m a t h . A P ] M a r POHOZAEV-TYPE INEQUALITIES AND NONEXISTENCERESULTS FOR NON C SOLUTIONS OF p ( x ) -LAPLACIANEQUATIONS. GABRIEL L ´OPEZ
Abstract.
In this paper a Pohozaev type inequality is stated for variableexponent Sobolev spaces in order to prove non existence of nontrivial weaksolutions for a Dirichlet problem with non-standard growth. The obtainedresults generalize a previous work of M. ˆOtani. Introduction
Let Ω be a bounded domain in R N with smooth boundary ∂ Ω . The domain Ω issaid to be star shaped (respectively strictly star shaped ) if ( x · ν ( x )) > x · ν ( x )) > ρ >
0) holds for all x ∈ ∂ Ω with a suitable choice of the origin, where ν ( x ) = ( ν ( x ) , . . . , ν N ( x )) denotes the outward normal unit vector at x ∈ ∂ Ω . Consider the problem ( − ∆ p ( x ) u = f ( u ) , x ∈ Ω u ( x ) = 0 , x ∈ ∂ Ω . (1.1)In [3] in order to obtain some non existence results for Problem (1.1) with Ω starshaped some Pohozaev type identities are stated and applied to the case in which f does not depend of p ( x ) and u ∈ C (Ω) . Nevertheless, it is known [9] that for f ( u ) = | u | q − u, < q < ∞ , < p < ∞ , and p, q constants, nontrivial solutions of(1.1) does not belong to C (Ω) ∩ C (Ω) . The arguments in [9, Proposition 1.1] areeasily extended to the variable exponent case, so that in general, results in [3] cannot be applied when ∇ u ( x ) = 0 , not even for solutions in W ,p ( x ) (Ω) ∩ W ,p ( x ) (Ω) . In this way, solutions of the problem,( E ) ( − ∆ p ( x ) u = | u | q ( x ) − u, x ∈ Ω u ( x ) = 0 , x ∈ ∂ Ω , (1.2)where ∆ p ( x ) u = div( |∇ u | p ( x ) − ∇ u ) , in general do not belong to C (Ω) . Existence of solutions for problem ( E ) is studied in [6] and [12]. The authorsin [12] prove existence for the case in which the embbeding from W ,p ( · )0 (Ω) to L q ( · ) (Ω) is compact and moreover, they prove existence even for the case in whichthe embbeding from W ,p ( · )0 (Ω) to L q ( · ) (Ω) is not compact provided that certainfunctional inequality holds true.This paper is organized as follows. In section 2 some necessary background inVariable Exponent Sobolev Spaces is provided including some required CompactEmbedding results. In section 3, Theorem 3.2 we state and prove a Pohozaev-type Mathematics Subject Classification.
Key words and phrases.
Pohozaev-type inequality, p ( x )-Laplace operator, variable exponentSobolev spaces. inequality. In Section 4, as a consequence of the Pohozaev type inequality, we provesome nonexistence results of nontrivial weak solutions of problem (1.2).2. Variable exponent setting
We recall some definitions and basic properties of the variable exponent Lebesgue-Sobolev spaces L p ( · ) (Ω) and W ,p ( · )0 (Ω), where Ω is a bounded domain in R N .For any p ∈ C (Ω) we define p + = sup x ∈ Ω p ( x ) and p − = inf x ∈ Ω p ( x ) . The variable exponent Lebesgue space for measurable real-valued functions is de-fined as the set L p ( · ) (Ω) = (cid:26) u : Z Ω | u ( x ) | p ( x ) dx < ∞ (cid:27) , endowed with the Luxemburg norm k u k p ( · ) = 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 ≤ ) , which is a separable and reflexive Banach space if 1 < p − p + < ∞ . For basicproperties of the variable exponent Lebesgue spaces we refer to [2], [10].Let L p ′ ( · ) (Ω) be the conjugate space of L p ( · ) (Ω), obtained by conjugating theexponent pointwise that is, 1 /p ( x ) + 1 /p ′ ( x ) = 1, [10, Corollary 2.7]. For any u ∈ L p ( · ) (Ω) and v ∈ L p ′ ( · ) (Ω) the following H¨older type inequality is valid (cid:12)(cid:12)(cid:12)(cid:12)Z Ω uv dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) p − + 1 p ′− (cid:19) k u k p ( · ) k v k p ′ ( · ) . (2.1)An important role in manipulating the generalized Lebesgue-Sobolev spaces isplayed by the p ( · ) -modular of the L p ( · ) (Ω) space, which is the mapping ρ p ( · ) : L p ( · ) (Ω) → R defined by ρ p ( · ) ( u ) = Z Ω | u | p ( x ) dx. If ( u n ), u ∈ L p ( · ) (Ω) then the following relations hold k u k p ( · ) < > ⇔ ρ p ( · ) ( u ) < >
1) (2.2) k u k p ( · ) > ⇒ k u k p − p ( · ) ≤ ρ p ( · ) ( u ) ≤ k u k p + p ( · ) (2.3) k u k p ( · ) < ⇒ k u k p + p ( · ) ≤ ρ p ( · ) ( u ) ≤ k u k p − p ( · ) (2.4) k u n − u k p ( · ) → ⇔ ρ p ( · ) ( u n − u ) → , (2.5)since p + < ∞ . For a proof of these facts see [10].The set W ,p ( x )0 (Ω) is defined as the closure of C ∞ (Ω) under the norm k u k p ( x ) = k∇ u k p ( x ) . The space ( W ,p ( x )0 (Ω) , k · k p ( x ) ) is a separable and reflexive Banach space if 1
The bounded variable exponent p is said to be Log-H¨older continuous if there isa constant C > | p ( x ) − p ( y ) | C − log( | x − y | ) (2.6)for all x, y ∈ R N , such that | x − y | ≤ . A bounded exponent p is Log-H¨oldercontinuous in Ω if and only if there exists a constant C > | B | p − B − p + B ≤ C (2.7)for every ball B ⊂ Ω [2, Lemma 4.1.6, page 101]. Under the Log-H¨older conditionsmooth functions are dense in variable exponent Sobolev space [2, Proposition11.2.3, page 346].Finally, Compact Embedding results, as many other facts, are a very delicate andinteresting issue in variable exponent spaces. For instance in [12, prop 3.1] is shownthat for certain exponents with p ∗ ( x ) > q ( x ) > p ∗ ( x ) − ǫ (in our notation) with x in some subset of Ω the embedding from W ,p ( · )0 (Ω) to L q ( · ) (Ω) is not compact. Onthe other hand, surprisingly, if q ( x ) = p ∗ ( x ) at some point, it is known that theembedding is compact in R N see [2, Thm. 8.4.6] and references therein. In thispaper we will use Proposition 3.3 of [12] which in our notation can be stated as thefollowing Proposition. Proposition 2.1.
Let p ( · ) satisfying the log-H¨older condition on the open andbounded set Ω ⊂ R N . Suppose that ∂ Ω ∈ C or Ω satisfies the cone condition, and p + < N. Let q ( · ) be a variable exponent on Ω such that q − and ess inf x ∈ Ω ( p ∗ ( x ) − q ( x )) > . (2.8) Then W ,p ( · )0 (Ω) ֒ → ֒ → L q ( · ) (Ω) , i. e. W ,p ( · )0 (Ω) is compactly embedded in L q ( · ) (Ω) . In the next section we will require also the following Lemma.
Lemma 2.2.
Let < p ( x ) < q − < q ( x ) < q + < ∞ a.e. in Ω . Assume that k u n k r < C for r < ∞ and u n → u as n → ∞ in L p ( · ) (Ω) . Then u n → u as n → ∞ in L q ( · ) (Ω) . Proof.
Given (2.2) to (2.5) it is enough to show that ρ q ( · ) ( u n − u ) → n → ∞ . For some θ ∈ (0 ,
1) satisfying 1 /q − = θ/p − + (1 − θ ) /q + we have ρ q ( · ) ( u n − u ) = Z Ω | u n − u | q ( x ) dx Z Ω | u n − u | q − dx (cid:18)Z Ω | u n − u | p − dx (cid:19) θq − /p − (cid:18)Z Ω | u n − u | q + dx (cid:19) (1 − θ ) q − /q + C (cid:18)Z Ω | u n − u | p − dx (cid:19) θq − /p − → n → ∞ , (2.9)given Thm. 2.11 in [1], and since u n → u in L p − (Ω) . (cid:3) GABRIEL L ´OPEZ G Pohozaev-type inequalitiy
In this section we state a Pohozaev-type inequality for weak solutions u belongingto the class P defined as P = n u ∈ (cid:16) W ,p ( · )0 ∩ L q ( · ) (cid:17) (Ω) : x i | u | q ( x ) − u ∈ L p ′ ( · ) (Ω) , i = 1 , , . . . , N o (3.1)where p ′ ( x ) = p ( x ) / ( p ( x ) −
1) and p + < N. To this aim, we employ the techniquesintroduced by Hashimoto and tani in [9], [8], [13], but within the framework ofvariable exponent spaces, which, as the reader may notice, require much morecareful estimations than those in the constant case.Let g n ( · ) ∈ C ( R ) be the cutoff functions such that 0 g ′ n ( s ) , s ∈ R and g n ( s ) = ( s | s | n, ( n + 1)sign s | s | > n + 1 . (3.2)Let u be a weak solution of (1.2) and set u n = g n ( u ) then | u n | r − u n ∈ (cid:16) W ,p ( · )0 ∩ L ∞ (cid:17) (Ω)for r ∈ [2 , ∞ ) . Consider now the approximate problem( E ) n ( | w n | q ( x ) − w n − ∆ p ( x ) w n = 2 | u n | q ( x ) − u n , in Ω ,w n = 0 on ∂ Ω . (3.3)Since u n ∈ L ∞ (Ω) , there exists a sequence { v εn } ⊂ C ∞ (Ω) satisfying k v εn k L ∞ (Ω) C o , for all ε ∈ (0 , , (3.4) v εn → | u n | q ( x ) − u n , strongly in L r ( · ) (Ω) as ε → , for all r ∈ [1 , ∞ ) . (3.5)In turn, we require another approximate equation for ( E ) n given by( E ) εn ( | w εn | q ( x ) − w εn + A ε w εn = v εn in Ω w εn = 0 on ∂ Ω , (3.6)where A ε u ( x ) = − div (cid:8) ( |∇ u ( x ) | + ε ) ( p ( x ) − / ∇ u ( x ) (cid:9) and ε > . It is possible toshow that (3.3) and (3.6) have unique solutions and that (3.6) and (3.3) providegood approximations respectively for (3.3) and (1.2) according to
Lemma 3.1.
Let p ( · ) satisfying the log-H¨older condition on the open and boundedset Ω ⊂ R N . Suppose that ∂ Ω ∈ C or Ω satisfies the cone condition, and p + < N. Then the following statements hold true:(i) For each ε ∈ (0 , and n ∈ N , there exists a unique solution w εn ∈ C (Ω) of (3.6).(ii) For each n ∈ N there exists a unique solution w n ∈ C ,α (Ω) ∩ W ,p ( x )0 (Ω) of (3.3).(iii) w εn converges to w n as ε → in the following sense: Z Ω |∇ w εn | p ( x ) dx → Z Ω |∇ w n | p ( x ) dx as ε → , (3.7) w εn → w n strongly in L r ( x )(Ω) , (3.8) for r ( · ) such that < r − < r ( x ) < r + a.e. in Ω and p + < N. OHOZAEV TYPE INEQ. 5 (iv) w n converges to u as n → ∞ in the following sense: Z Ω |∇ w n | p ( x ) dx → Z Ω |∇ u | p ( x ) dx as n → ∞ (3.9) Z Ω | w n | q ( x ) dx → Z Ω | u | q ( x ) dx, as n → ∞ , (3.10) Proof. (i) Since u n ∈ L ∞ (Ω) , there exists a sequence { v εn } ⊂ C ∞ (Ω) satisfying k v εn k L ∞ (Ω) C o , for all ε ∈ (0 , , (3.11) v εn → | u n | q ( x ) − u n , strongly in L r (Ω) as ε → , for all r ∈ [1 , ∞ ) . (3.12)Given that v εn belongs to C (Ω) and since A ε u is elliptic, Theorem 15.10 in [15]guarantees the existence of a unique solution w εn ∈ C (Ω) of (3.6).(ii) Set F ( z ) = Z Ω |∇ z | p ( x ) p ( x ) dx + Z Ω | z | q ( x ) q ( x ) dx − Z Ω | u n | q ( x ) − u n zdx, so that F ( z ) is strictly convex, coercive and Fr´echet differentiable on (cid:16) W ,p ( x )0 ∩ L q ( x ) (cid:17) (Ω) . Now, if z n ⇀ z o weakly in (cid:16) W ,p ( x )0 ∩ L q ( x ) (cid:17) (Ω) , then since p ∈ P (Ω , µ ) (for defini-tions see [2]) the modulars R Ω |∇ z | p ( x ) /p ( x ) dx and R Ω | z | q ( x ) /q ( x ) dx are sequentiallyweakly lower semicontinuous [2, Thm. 3.2.9] and R Ω | u n | q ( x ) − u n zdx ∈ ( L q ( x ) (Ω)) ∗ we conclude lim inf n →∞ F ( z n ) > F ( z o ) . Since F is bounded below, there exists w n ∈ (cid:16) W ,p ( x )0 ∩ L q ( x ) (cid:17) (Ω) where F attains its minimum, and since F is Fr´echetdifferentiable h F ′ ( w n ) , φ i = 0 for all φ ∈ (cid:16) W ,p ( x )0 ∩ L q ( x ) (cid:17) (Ω) , i.e. w n solves (3 . F ( z ) . Multiplying (3.6) by | w n | r − w n ( r > ε -inequality with ε = 1 / , and considering that | u n | q ( x ) − u n belongs to L ∞ (Ω) we obtain Z Ω | w n | q ( x )+ r − dx + ( r − Z Ω | w n | p ( x ) | w n | r − dx Z Ω n + 1) q ( x ) − | w n | r − dx Z Ω | w n | q ( x )+ r − dx + 2 ( q + +2 r − / ( q − − ( n + 1) q + + r − | Ω | . (3.13)So, by [7, Thm. 1.3, p. 427] k w n k q ± + r − L q ( x )+ r − · ( q + +2 r − / ( q − − ( n + 1) q + + r − | Ω | , where q ± = ( q + if k w n k L q ( x )+ r − < ,q − if k w n k L q ( x )+ r − > . In this way we can obtain an a priori bound for k w n k L q ( x )+ r − independent of r. Letting r → ∞ we get an L ∞ -estimate for w n . Therefore using [4, Thm. 1.2, p.400] we conclude w n ∈ C ,α (Ω) . GABRIEL L ´OPEZ G (iii) With a similar argumentation as in (ii) we obtain k w εn k L ∞ (Ω) C n for all ε > . (3.14)Multiply (3.6) by w ǫn , to obtain Z Ω | w εn | q ( x ) dx + Z Ω ( |∇ w εn | + ε ) ( p ( x ) − / |∇ w εn | dx = Z Ω v εn w εn dx. On the other hand, note that Z Ω |∇ w εn | p ( x ) dx = Z Ω ( |∇ w εn | ) ( p ( x ) − / |∇ w εn | dx Z Ω ( |∇ w εn | + ε ) ( p ( x ) − / |∇ w εn | dx And hence Z Ω |∇ w εn | p ( x ) dx Z Ω v εn w εn dx. Now use Young’s inequality and the fact that q ( x ) , q ′ ( x ) > Z Ω |∇ w εn | p ( x ) dx Z Ω | v εn | q ′ ( x ) dx + Z Ω | w εn | q ( x ) dx. so by (3.14) and given that v n ∈ C ∞ (Ω) we deduce k∇ w εn k L p ( x ) (Ω) C n for all ε > . (3.15)Together (3.14), (3.15), and compactness Proposition 2.1 and Lemma 2.2 implythat there exists a sequence { w ε k n } such that for p + < Nw ε k n → w strongly in L r (Ω) , with 1 r − < r ( x ) < r + < ∞ (3.16) ∇ w ε k n ⇀ ∇ w weakly in L p ( x ) (Ω) , (3.17) Z Ω | w ε k n | q ( x ) − w ε k n v → Z Ω | w | q ( x ) − wv as ε k → , for all v ∈ W p ( x )0 (Ω) . (3.18)Weak convergence holds since L p ( x ) spaces are uniformly convex [2, Thm. 3.4.9],and hence reflexive.From this point we refer to [11] for all the notations and results concerning tosubdifferentials. Set φ ε ( z ) := Z Ω p ( x ) ( |∇ z | + ε ) p ( x ) / dx with D ( φ ε ) = W ,p ( x )0 (Ω) so that φ ε is a convex operator according to definition insection 1.3.3 p. 24 in [11]. Noting that φ ε is Fr´echet differentiable and that actually φ ′ ε ( z ) v = h A ε z, v i = Z Ω ( |∇ z | + ε ) p ( x ) / ∇ z · ∇ vdx. So according to [11] section 4.2.2, A ε ∈ ∂φ ε where ∂φ ε is the subdifferential of φ ε . Hence w εn satisfies φ ε ( v ) − φ ε ( w εn ) > Z Ω ( |∇ w εn | + ε ) p ( x ) / ∇ w εn · ∇ ( v − w εn ) dx, ∀ v ∈ W ,p ( x )0 (Ω) . Now, by (3.6) φ ε ( v ) − φ ε ( w εn ) > Z Ω ( −| w εn | q ( x ) − w εn + v εn ) · ( v − w εn ) dx. (3.19) OHOZAEV TYPE INEQ. 7
On the other hand, given strong convergence of w εn → w n as ε → v n → | u n | q ( x ) − u n in L (Ω) , we have that v εn w εn → | u n | q ( x ) − u n w n as ε → L (Ω) since Z Ω | v εn w εn − | u n | q ( x ) − u n w n | dx Z Ω | v εn || w εn − w n | dx + Z Ω | w n | (cid:12)(cid:12)(cid:12) v εn − | u n | q ( x ) − u n (cid:12)(cid:12)(cid:12) dx C o Z Ω | w εn − w n | dx + Z Ω | w n | (cid:12)(cid:12)(cid:12) v εn − | u n | q ( x ) − u n (cid:12)(cid:12)(cid:12) dx, (3.20)given that (3.11) holds. That the last integral goes to zero as ε → w n ∈ L r (Ω) , and (3.12).Given that φ ε ( v ) → φ ( v ) as ε → v ∈ W ,p ( x ) (Ω) andlim inf k →∞ φ ε k ( w ε k n ) > φ ε k ( w ) > φ ( w ) (3.21)since modulars are weakly lower semicontinuous [2, Thm. 2.2.8]. Taking limits as ε → φ ( v ) − φ ( w ) > Z Ω (cid:16) −| w | q ( x ) − w + 2 | u n | q ( x ) − u n (cid:17) · ( v − w ) dx, for all v ∈ W ,p ( x )0 (Ω) which imply, by subdifferential’s definition, that Z Ω div( |∇ w | p ( x ) − ∇ w ) · ∇ ϕ = Z Ω ( −| w | q ( x ) − w + 2 | u n | q ( x ) − u n ) · ϕ, (3.22)for all ϕ ∈ W ,p ( x )0 (Ω) . We conclude that w = w n , since the argument above doesnot depend on the choice of { ε k } . Multiply equation in (3.3) by w n and equation in (3.6) by w εn and integrate byparts to get Z Ω |∇ w n | p ( x ) dx = − Z Ω | w n | q ( x ) dx + 2 Z Ω | u n | q ( x ) − u n w n dx Z Ω ( |∇ w εn | + ε ) ( p ( x ) − / |∇ w εn | dx = − Z Ω | w εn | q ( x ) dx + Z Ω v εn w εn dx. So that (3.12) and (3.16) imply Z Ω ( |∇ w εn | + ε ) ( p ( x ) − / |∇ w εn | dx → Z Ω |∇ w n | p ( x ) dx as ε → . (3.23)Take v = w = w n in (3.19) and let ε → ε → φ ε ( w εn ) φ ( w n ) , Last inequality and (3.21) imply Z Ω ( |∇ w εn | + ε ) p ( x ) / dx → Z Ω |∇ w n | p ( x ) dx as ε → . (3.24)Moreover, since (3.17) holds thenlim inf ε Z Ω |∇ w εn | p ( x ) dx > Z Ω |∇ w n | p ( x ) GABRIEL L ´OPEZ G since modulars are weakly lower semicontinuous.On the other hand, since ( |∇ w εn | ) p ( x ) / ( |∇ w εn | + ε ) p ( x ) / we havelim sup ε Z Ω |∇ w εn | p ( x ) dx lim sup ε Z Ω ( |∇ w εn | + ε ) p ( x ) / dx Z Ω |∇ w n | p ( x ) dx Therefore we conclude (3.7).iv) We proceed first by noticing that | u n | q ( x ) − u n → | u | q ( x ) − u strongly in L q ′ ( x ) (Ω) as n → ∞ , (3.25)by the uniform convexity of L q ′ ( x ) (Ω) . Multiply (3.3) by w n and integrate by partsto obtain Z Ω | w n | q ( x ) dx + Z Ω |∇ w n | p ( x ) dx = 2 Z Ω | u n | q ( x ) − u n w n dx (3.26) k| u n | q ( x ) − k L q ′ ( x ) (Ω) k w n k L q ( x ) (Ω) , by H¨older’s inequality for variable exponent Sobolev spaces [2, lemma 2.6.5]. Now,using [7, Thm. 1.3] and (3.26) we get k w n k q ± L q ( x ) (Ω) + k∇ w n k p ± L p ( x ) (Ω) C k w n k L q ( x ) (Ω) , (3.27)where q ± = ( q + if k w n k L q ( x ) (Ω) < q − if k w n k L q ( x ) (Ω) > ,p ± = ( p + if k∇ w n k L q ( x ) (Ω) < p − if k∇ w n k L q ( x ) (Ω) > , The fact that p ± , q ± > k w n k q ± L q ( x ) (Ω) , k∇ w n k p ± L p ( x ) (Ω) C. We use againProposition 2.1 and Lemma 2.2 to obtain that, up to a subsequence { n k } , ∇ w n k ⇀ ∇ w weakly in L p ( x ) (Ω)(3.28) w n k ⇀ w weakly in L q ( x ) (Ω)(3.29) w n k → w strongly in L q ( x ) (Ω) for all q such that 1 q − < q ( x ) <, q + < ∞ Z Ω | w n k | q ( x ) − w n k · vdx → Z Ω | w | q ( x ) − w · vdx for all v ∈ L q ′ ( x ) (Ω) as k → ∞ . (3.30)Given that w n is solution of (3.3) subdifferential’s definition leads to Z Ω p ( x ) |∇ v | p ( x ) dx − Z Ω p ( x ) |∇ w n | p ( x ) dx = Z Ω p ( x ) |∇ v | p ( x ) dx − Z Ω p ( x ) |∇ w n | p ( x ) dx > Z Ω ( −| w n | q ( x ) − w n + 2 | u n | q ( x ) − u n )( v − w n ) dx (3.31) > Z Ω | w n | q ( x ) dx − Z Ω | w n | q ( x ) − w n vdx + 2 Z Ω | u n | q ( x ) − u n ( v − w n ) dx, for all v ∈ C ∞ (Ω) and for n such that supp v ⊂ Ω . Let n = n k → ∞ in (3.31) andrecall (3.25), (3.28), (3.29) and (3.30) to obtain Z Ω p ( x ) |∇ v | p ( x ) dx − Z Ω p ( x ) |∇ w | p ( x ) dx > Z Ω ( −| w | q ( x ) − w + 2 | u | q ( x ) − u )( v − w ) dx, (3.32) OHOZAEV TYPE INEQ. 9 for all v ∈ C ∞ (Ω) . Now put v = w + tz with z ∈ C ∞ o (Ω) and let t → + , t → − in (3.32) and use the definition of Fr´echet derivative to see that w satisfies Z Ω |∇ w | p ( x ) − ∇ w · ∇ z + Z Ω | w | q ( x ) − wzdx = 2 Z Ω | u | q ( x ) − uzdx for all z ∈ C ∞ o (Ω) . Hence | w | q ( x ) − w − ∆ p ( x ) w = | u | q ( x ) − u − ∆ p ( x ) u in the sense of distributions. That w = u follows from well known inequality | a − b | p C p (cid:8) ( | a | p − a − | b | p − b ) · ( a − b ) (cid:9) s/ ( | a | p + | b | p ) − s/ which holds for all a, b ∈ R N where s = p if p ∈ (1 ,
2) and s = 2 if p > , and C p > a, b. Since the above argument does not depend on thechoice of subsequences, (3.28), (3.29) and (3.30) hold for n k = n. Taking into account (3.25), (3.26), (3.28) and (3.29) we get2 Z Ω | u | q ( x ) dx = Z Ω | u | q ( x ) dx + Z Ω |∇ u | p ( x ) dx lim inf n →∞ (cid:18)Z Ω | w n | q ( x ) dx + Z Ω |∇ w n | p ( x ) dx (cid:19) = lim n →∞ (cid:18)Z Ω | w n | q ( x ) dx + Z Ω |∇ w n | p ( x ) dx (cid:19) Z Ω | u | q ( x ) dx. Consequentlylim n →∞ (cid:18)Z Ω | w n | q ( x ) dx + Z Ω |∇ w n | p ( x ) dx (cid:19) = Z Ω | u | q ( x ) dx + Z Ω |∇ u | p ( x ) dx Further, notice that Z Ω | u | q ( x ) dx lim inf n →∞ Z Ω | w n | q ( x ) dx lim sup n →∞ Z Ω | w n | q ( x ) dx = lim sup n →∞ (cid:18)Z Ω | w n | q ( x ) dx + Z Ω |∇ w n | p ( x ) p ( x ) dx − Z Ω |∇ w n | p ( x ) p ( x ) dx (cid:19) lim sup n →∞ (cid:18)Z Ω | w n | q ( x ) dx + Z Ω |∇ w n | p ( x ) p ( x ) dx (cid:19) − lim inf n →∞ Z Ω |∇ w n | p ( x ) p ( x ) dx Z Ω | u | q ( x ) dx. Therefore lim n →∞ Z Ω | w n | q ( x ) dx = Z Ω | u | p ( x ) dx and lim n →∞ Z Ω |∇ w n | p ( x ) dx = Z Ω |∇ u | p ( x ) dx. (cid:3) In order to obtain a Pohozaev type inequality we introduce the function F ( x, u, s ) := | u ( x ) | q ( x ) q ( x ) + ( | s | + ε ) p ( x ) / p ( x ) − v εn ( x ) u ( x ) (3.33)where s = ( s , . . . , s N ) , which will be used in the context of a Pucci-Serrin formula[14]. Theorem 3.2 (Pohozaev type inequality) . Let u be a weak solution of (1.2) be-longing to P . Then u satisfies − Z Ω Nq ( x ) | u | q ( x ) dx + Z Ω N − p ( x ) p ( x ) |∇ u | p ( x ) dx + Z Ω x · ∇ p ( x ) |∇ u | p ( x ) p ( x ) log (cid:16) e − |∇ u | p ( x ) (cid:17) dx − Z Ω x · ∇ q ( x ) | u | q ( x ) q ( x ) log (cid:16) e − | u | q ( x ) (cid:17) dx + R ≤ , (3.34) where R = p † − p + lim sup n →∞ lim sup ε → Z ∂ Ω (cid:0) |∇ w εn | + ε (cid:1) p ( x ) / ( x · ν ( x )) dS, p † = min x ∈ Ω { , p ( x ) } , and w εn is the solution of (3.6) uniquely determined by u. Proof.
In (3.33) denote by F s ( x, u, s ) = ( ∂ s F , . . . , ∂ s N F ) , so that ∂ s i F ( x, u, s ) = ( | s | + ε ) p ( x ) / − s i . hence we denote ∂ s i F ( x, u, ∇ u ) = ( |∇ u | + ε ) p ( x ) / − ∂ i u. and F s ( x, u, ∇ u ) = ( |∇ u | + ε ) ( p ( x ) − / ∇ u. So that div F ( x, u, ∇ u ) = − A ε u, where, we recall, A ε is defined after (3.6). Finally, we denote ∇F ( x, u, ∇ u ) = ( ∂ x F , . . . , ∂ x N F )= ( ∂ F , . . . , ∂ N F )with ∂ i F = ∂ i (cid:18) | u ( x ) | q ( x ) q ( x ) + ( | s | + ε ) p ( x ) / p ( x ) − v εn ( x ) u ( x ) (cid:19) = | u | q ( x ) ( q ( x )) (cid:0) log | u | q ( x ) − (cid:1) ∂ i q ( x ) + | u | q ( x ) − u∂ i u + ( |∇ u | + ε ) p ( x ) / p ( x )) (cid:0) log( |∇ u | + ε ) p ( x ) − (cid:1) ∂ i p ( x )+( |∇ u | + ε ) p ( x ) / − ∂ i ( |∇ u | ) − (cid:2) ( ∂ i v εn ) u + v εn ∂ i u (cid:3) OHOZAEV TYPE INEQ. 11
We will make use the Pucci-Serrin formula [14, Prop. 1, p. 683] in the form Z ∂ Ω h F ( x, , ∇ u ) −∇ u ·F s ( x, , ∇ u ) i ( h · ν ) dS = Z Ω h F ( x, u, ∇ u ) div h + h · ∇F ( x, u, ∇ u ) − ( h · ∇ u ) div F s ( x, u, ∇ u ) − F s ( x, u, ∇ u ) · ∇ ( h · ∇ u ) − au div F s ( x, u, ∇ u ) − ∇ ( au ) · F s ( x, u, ∇ u ) i dx (3.35)Taking a constant, h = x = ( x , . . . , x n ) , u = w εn equation (3.35) becomes Z ∂ Ω ( |∇ w εn | + ε ) p ( x ) / p ( x ) ( x · ν ) dS − Z ∂ Ω ( |∇ w εn | + ε ) p ( x ) / − |∇ w εn | ( x · ν ) dS == Z Ω N (cid:18) | w εn | q ( x ) q ( x ) + ( |∇ w εn | + ε ) p ( x ) / p ( x ) − v εn w εn (cid:19) dx + Z Ω ( x ·∇ q ( x )) | w εn | q ( x ) ( q ( x )) (cid:0) log | w εn | q ( x ) − (cid:1) dx + Z Ω ( x ·∇ p ( x )) ( |∇ w εn | + ε ) p ( x ) / ( p ( x )) (cid:0) log( |∇ w εn | + ε ) p ( x ) / − (cid:1) dx − Z Ω w εn ( x ·∇ v εn ) dx − Z Ω ( |∇ w εn | + ε ) ( p ( x ) − / |∇ w εn | dx + Z Ω aw εn A ε w εn dx − Z Ω ( ∇ ( aw εn ) ·∇ w εn )( |∇ w εn | + ε ) ( p ( x ) − / dx. (3.36)For the surface integrals in (3.36) adding and subtracting the integral ε R ∂ Ω ( |∇ w εn | + ε ) p ( x ) / − ( x · ν ) dS we have Z ∂ Ω ( |∇ w εn | + ε ) p ( x ) / p ( x ) ( x · ν ) dS − Z ∂ Ω ( |∇ w εn | + ε ) p ( x ) / − |∇ w εn | ( x · ν ) dS == Z ∂ Ω (cid:18) p ( x ) − (cid:19) ( |∇ w εn | + ε ) p ( x ) / ( x · ν ) dS + ε Z ∂ Ω ( |∇ w εn | + ε ) p ( x ) / − ( x · ν ) dS (3.37)On the other hand, since ( x · ν ( x )) > x ∈ ∂ Ω , then ε Z ∂ Ω ( |∇ w εn | + ε ) p ( x ) / − ( x · ν ) dS R ∂ Ω ε p ( x ) / ( x · ν ( x )) dS, if 1 < p ( x ) , R ∂ Ω p ( x ) − p ( x ) ( |∇ w εn | + ε ) p ( x ) / ( x · ν ) dS ++ R ∂ Ω 2 p ( x ) ε p ( x ) / ( x · ν ( x )) dS, if 2 < p ( x ) . (3.38)Now we analyze what happen with each term in (3.36) as ε → . We begin withthe last term and we continue the analysis going down to up into the equation:(1) − R Ω ( ∇ ( aw εn ) ·∇ w εn )( |∇ w εn | + ε ) ( p ( x ) − / dx → − a R Ω |∇ w n | p ( x ) dx by (3.23).(2) R Ω aw εn A ε w εn dx → a (cid:0)R Ω | u n | q ( x ) − u n w n dx − R Ω | w n | q ( x ) dx (cid:1) by (3.6) and(3.20).(3) − R Ω ( |∇ w εn | + ε ) ( p ( x ) − / |∇ w εn | dx → − R Ω |∇ w n | p ( x ) dx by (3.23).(4) For the term − R Ω w εn ( x · ∇ v εn ) dx we make the following estimations − Z Ω w εn ( x · ∇ v εn ) dx = − Z Ω x · ∇ ( w εn v εn ) dx + Z Ω v εn x · ∇ w εn dx. (3.39)Note that R Ω v εn x · ∇ w εn dx → R Ω | u n | q ( x ) − u n x · ∇ w n dx as ε → , by asimilar proof as in (3.20). On the other hand, calculating the first term in the right hand side of(3.39), − Z Ω x · ∇ ( w εn v εn ) dx = Z Ω v εn w εn div x dx − Z ∂ Ω v εn w εn ( x · ν ) dS = N Z Ω v εn w εn dx. (3.40)(5) We claim that Z Ω ( x · ∇ q ( x )) | w εn | q ( x ) ( q ( x )) (cid:0) log | w εn | q ( x ) − (cid:1) dx → Z Ω ( x · ∇ q ( x )) | w n | q ( x ) ( q ( x )) (cid:0) log | w n | q ( x ) − (cid:1) dx (3.41)and Z Ω ( x · ∇ p ( x )) ( |∇ w εn | + ε ) p ( x ) / ( p ( x )) (cid:0) log( |∇ w εn | + ε ) p ( x ) / − (cid:1) dx → Z Ω ( x · ∇ p ( x )) |∇ w n | p ( x ) ( p ( x )) (cid:0) log |∇ w n | p ( x ) − (cid:1) dx (3.42)for η > . Fix I := Z Ω ( x · ∇ q ( x )) | w εn | q ( x ) ( q ( x )) log | w εn | q ( x ) dx and I := Z Ω ( x · ∇ p ( x )) ( |∇ w εn | + ε ) p ( x ) / ( p ( x )) log( |∇ w εn | + ε ) p ( x ) / dx. In order to prove (3.41) and (3.42), we estimate I by distinguishing thecases | w εn | ≤ , and | w εn | >
1. Notice that the relationssup ≤ t ≤ t η | log t | < ∞ (3.43)sup t> t − η log t < ∞ (3.44)hold for η > := { x ∈ Ω : | w εn ( x ) | ≤ } and Ω := { x ∈ Ω : | w εn ( x ) | > } . We canchoose k ∈ N such that p ( x ) − /k ≥ p − . Since w εn ∈ L p − (Ω) and in Ω , | w εn ( x ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12) ( x · ∇ q ( x )) | w εn | q ( x ) ( q ( x )) log | w εn | q ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | w εn ( x ) | p ( x ) − /m ≤ C | w εn ( x ) | p − , (3.45)for m > k. For Ω we can choose k ′ such that p ( x ) + 1 /k ′ ≤ ( p ( x )) ∗ = N p ( x ) / ( N − p ( x )) . So (cid:12)(cid:12)(cid:12)(cid:12) ( x · ∇ q ( x )) | w εn | q ( x ) ( q ( x )) log | w εn | q ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | w εn ( x ) | p ( x )+1 /m ≤ C | w εn ( x ) | ( p ( x )) ∗ , (3.46) OHOZAEV TYPE INEQ. 13 for m > k ′ , and x ∈ Ω . Therefore (3.45), (3.46), and the convergence of w εn in Lemma 3.1 imply that there exists h ( x ) ∈ L (Ω) such that (cid:12)(cid:12)(cid:12)(cid:12) ( x · ∇ q ( x )) | w εn | q ( x ) ( q ( x )) log | w εn | q ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ h ( x ) (3.47)On the other hand, given the convergence Lemma 3.1, assertion (3.8) andthe continuity of the log function, we may conclude that( x · ∇ q ( x )) | w εn | q ( x ) ( q ( x )) log | w εn | q ( x ) → ( x · ∇ q ( x )) | w n | q ( x ) ( q ( x )) log | w n | q ( x ) (3.48) a.e. in Ω as ε → . With (3.47), (3.48), and the Lebesgue convergenceTheorem the claim of point (5) follows.(6) Finally, Z Ω N (cid:18) | w εn | q ( x ) q ( x ) + ( |∇ w εn | + ε ) p ( x ) / p ( x ) (cid:19) dx → Z Ω N (cid:18) | w n | q ( x ) q ( x ) + |∇ w n | p ( x ) p ( x ) (cid:19) dx as ε → N Z Ω | w n | q ( x ) q ( x ) dx + Z Ω N − p ( x ) p ( x ) |∇ w n | p ( x ) dx + Z Ω x ·∇ p ( x ) |∇ w n | p ( x ) p ( x ) (cid:16) log |∇ w n | p ( x ) − (cid:17) dx + Z Ω x · ∇ q ( x ) | w n | q ( x ) q ( x ) (cid:16) log | w n | q ( x ) − (cid:17) dx + 2 Z Ω | u n | q ( x ) − u n x · ∇ w n dx + a (cid:18)Z Ω | u n | q ( x ) − u n w n dx − Z Ω | w n | q ( x ) dx − Z Ω |∇ w n | p ( x ) dx (cid:19) + R n ≤ , (3.49)where R n = p † − p + lim sup ε → R ∂ Ω (cid:0) |∇ w εn | + ε (cid:1) p ( x ) / ( x · ν ( x )) dS, and p † = min x ∈ Ω { , p ( x ) } . Now let n → ∞ in (3.49) and take into account (3.9), (3.10) to obtain N Z Ω | u | q ( x ) q ( x ) dx + Z Ω N − p ( x ) p ( x ) |∇ u | p ( x ) dx + Z Ω x ·∇ p ( x ) |∇ u | p ( x ) p ( x ) (cid:16) log |∇ u | p ( x ) − (cid:17) dx + Z Ω x · ∇ q ( x ) | u | q ( x ) q ( x ) (cid:16) log | u | q ( x ) − (cid:17) dx + 2 Z Ω | u | q ( x ) − u ( x · ∇ u ) dx + a (cid:18)Z Ω | u | q ( x ) dx − Z Ω |∇ u | p ( x ) dx (cid:19) + R ≤ , (3.50)where R = p † − p + lim sup n →∞ lim sup ε → Z ∂ Ω (cid:0) |∇ w εn | + ε (cid:1) p ( x ) / ( x · ν ( x )) dS. Further, notice that since u is a weak solution of (1.2), Z Ω | u | q ( x ) dx − Z Ω |∇ u | p ( x ) dx = 0 . (3.51)In fact, multiplying (1.2) by ϕ ∈ W ,p ( · )0 (Ω) , and integrating by parts, we have Z Ω |∇ u | p ( x ) − ∇ udx = Z Ω | u | q ( x ) − uϕdx. Taking ϕ = u we get (3.51) as wanted. On the other hand, Z Ω x · ∇| u | q ( x ) q ( x ) dx = Z Ω | u | q ( x ) − u ( x · ∇ u ) dx + Z Ω q ( x ) | u | q ( x ) log | u | q ( x ) ( x · ∇ q ( x )) dx, (3.52)so that Z Ω x · ∇| u | q ( x ) q ( x ) dx = − Z Ω div (cid:18) xq ( x ) (cid:19) | u | q ( x ) dx + Z ∂ Ω | u | q ( x ) ∂∂ν (cid:18) xq ( x ) (cid:19) dS − N Z Ω | u | q ( x ) q ( x ) dx + Z Ω | u | q ( x ) x · ∇ q ( x ) q ( x ) dx. (3.53)Hence from (3.52), (3.53) Z Ω | u | q ( x ) − u ( x · ∇ u ) dx = − N Z Ω | u | q ( x ) q ( x ) dx + Z Ω | u | q ( x ) x · ∇ q ( x ) q ( x ) (cid:16) − log | u | q ( x ) (cid:17) dx (3.54)We derive inequality (3.34) by substituting (3.51) and (3.54) in (3.50) . (cid:3) Nonexistence of Nontrivial Solutions
Now we can state a Non Existence Theorem which is a generalization to variableexponent Sobolev spaces of Theorem III, p. 142 in [13]. The proofs are similar tothose in [13], but are included here for the reader’s convenience.
Theorem 4.1.
Consider the Problem (1.2), where Ω ⊂ R N is a bounded domainof Class C , p ( · ) is a log-H¨older exponent with < p − p ( x ) p + < N. Let P beas defined in (3.1). Then we have:i) If Ω is star-shaped and q − > ( p + ) ∗ then Problem (1.2) has not a nontrivialweak solution belonging to P ∩ E where E = u : Z Ω log (cid:0) |∇ u | p ( x ) e − (cid:1) x ·∇ pp |∇ u | p ( x ) (cid:0) | u | q ( x ) e − (cid:1) x ·∇ qq | u | q ( x ) dx > . ii) If Ω is strictly star-shaped and q − = ( p + ) ∗ then Problem (1.2) has not anontrivial weak solution of definite sign belonging to P ∩ E . Proof. i) If Ω is star-shaped, R > (cid:18) N − p + p + − Nq − (cid:19) Z Ω | u | q ( x ) dx . So u ≡ . ii) If Ω is strictly star-shaped, R = 0 in (3.34), so0 = R > ρ lim sup n →∞ lim sup ε → Z ∂ Ω (cid:0) |∇ w εn | + ε (cid:1) p ( x ) / dS. Since ρ > n →∞ lim sup ε → Z ∂ Ω (cid:0) |∇ w εn | + ε (cid:1) p ( x ) / dS. OHOZAEV TYPE INEQ. 15
Multiplying the PDE in (3.6) by v ( x ) ≡ , integrating by parts, and taking lim supas ε → n → ∞ we obtain (cid:12)(cid:12)(cid:12)(cid:12)Z Ω | u | q ( x ) − udx (cid:12)(cid:12)(cid:12)(cid:12) C lim sup n →∞ lim sup ε → Z ∂ Ω (cid:0) |∇ w εn | + ε (cid:1) p ( x ) / dS = 0 , C > . Therefore R Ω | u | q ( x ) − udx = 0 . (cid:3) References [1] Adams R. A., Fournier J. J. F,
Sobolev Spaces, 2nd edition,
Academic Press 2003.[2] Diening L., Harjulehto P., H¨ast¨o P., Ruˇziˇcka M.,
Lebesgue and Sobolev Spaces with VariableExponents,
Lecture Notes in Mathematics 2017, Springer-Verlag Berlin Heidelberg 2011.[3] Dinca G., Isia F.,
Generalizad Pohozaev and Pucci-Serrin identities and non existence resultsfor p ( x ) -laplacian type equations, Rendiconti del circolo Matematico di Palermo 59, 1-46(2010).[4] Fan X.,
Global C ,α regularity for variable exponent elliptic equations in divergence form, Journal of Differential Equations 235 (2007) 397-417.[5] Fan X., Shen J. and Zhao D.,
Sobolev embedding theorems for spaces W k,p ( x ) (Ω) . J. Math.Anal. Appl. 262 (2001), 749-760.[6] Fan X., Zhang.,
Existence of solutions for p(x)-Laplacian Dirichlet problem,
Nonlinear Anal-ysis 52 (2003) 1843-1852.[7] Fan X., Zhao D.,
On the spaces L p ( x ) (Ω) and W m,p ( x ) (Ω) , Journal of Mathematical Analysisand Applications 263, 424-446 (2001).[8] Hashimoto T.,
Pohozaev-type inequalities for weak solutions of elliptic equations,
KyotoUniversity Research Information Repository, 951, 1996, 126-135.[9] Hasimoto T. and tani M.,
Nonexistence of Weak Solutions of Nonlinear Elliptic Equationsin Exterior Domains,
Houston Journal of Mathematics, Vol.23, No. 2, 1997, 267-290.[10] Kov´aˇcik O. and R´akosnik J.,
On spaces L p ( x ) and W ,p ( x ) . Czech. Math. J. 41(1991), 592-618.[11] Kusraev A. G. and Kutateladze S. S.
Subdifferentials: Theory and Applications,
KluwerAcademic Publishers 1995.[12] Mizuta Y., Ohno T., Shimomura T., Shioji N.,
Compact Embeddings for Sobolev Spaces ofvariable Expopnent and existence of solutions for Nonlinear Elliptic Problems involving thep(x)-laplacian and its Critical Expopnent,
Annales Academiae Scienciarum Fennicae Mathe-matica Vol. 35, 2010, 115-130.[13] tani M.,
Existence and Nonexistence of Nontrivial Solutions of Some Nonlinear DegenerateElliptic Equations,
Journal of Functional Analysis, 76, 140-159 (1988).[14] Pucci P., Serrin J.,
A General Variational Identity,
Indiana University Mathematics Journal,Vol. 35, No. 3 (1986), 681-703.[15] Gilbarg D., Trudinger N. S.,
Elliptic Partial Differential Equations of Second Order,
Springer-Verlag Berlin Heidelberg 2001.
Gabriel L´opez G.Universidad Aut´onoma Metropolitana , M´exico D.F, M´exico
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