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Nonsmooth critical point theorems and its applications to quasilinear schrödinger equations
Acta Mathematica Scientia 2016,36B(1):73–86 http://actams.wipm.ac.cn
NONSMOOTH CRITICAL POINT THEOREMS AND ITS APPLICATIONS TO QUASILINEAR ¨ SCHRODINGER EQUATIONS∗
o±!)
Zhouxin LI (
School of Mathematics and Statistics, Central South University, Changsha 400083, China E-mail : [email protected]
!U)
Yaotian SHEN (
Department of Mathematics, South China University of Technology, Guangzhou 510640, China E-mail : [email protected] Abstract In this paper, the existence and nonexistence of solutions to a class of quasilinear elliptic equations with nonsmooth functionals are discussed, and the results obtained are applied to quasilinear Schr¨ odinger equations with negative parameter which arose from the study of self-channeling of high-power ultrashort laser in matter. Key words
nonsmooth critical point theorems; quasilinear elliptic equations; Schr¨ odinger equation
2010 MR Subject Classification
1
35J20; 35J65; 35J10
Introduction and Main Results
In this paper, we consider the existence of nontrivial solutions for the following class of quasilinear elliptic equations with natural growth 1 −Dj (aij (x, u)Di u) + ∂s aij (x, u)Di uDj u + c(x)u = |u|q−2 u, 2
x ∈ RN ,
(1.1)
where N ≥ 3, q > 2, c(x) ∈ C 1 (RN ), aij (x, s) are Carath´eodory functions, ∂s aij denotes the derivatives of aij with respect to s. The repeated indices indicate the summation from 1 to N . As an example, in this paper, we also consider a special case of equation (1.1). In the study of self-channeling of high-power ultrashort laser in matter [1], the following quasilinear Schr¨odinger equation was considered i∂t z = −∆z + W (x)z − l(|z|2 )z − κ∆h(|z|2 )h′ (|z|2 )z,
x ∈ RN ,
(1.2)
where W (x) is a given potential, κ is a real constant. Let h(s) = (1 + s)1/2 and l(s) = s(q−2)/2 . If we consider the standing waves of (1.2) of the form z(x, t) = exp(−iEt)u(x), we observe that ∗ Received
December 5, 2014; revised May 13, 2015. The first author was supported by NSF of China (11201488) and Hunan Provincial Natural Science Foundation of China (14JJ4002). The second author was supported by NSF of China (11371146).
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z(x, t) satisfies (1.2) if and only if u(x) solves the following equation κu −∆u + c(x)u − ∆(1 + |u|2 )1/2 = |u|q−2 u, 2(1 + |u|2 )1/2 Equation (1.3) is a special case of (1.1) with aij (x, u) = 1 +
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x ∈ RN .
(1.3)
κ|u|2 δij . 2(1 + |u|2 )
We make the following assumptions on aij (x, s). (a1 ) aij ≡ aji , and aij , ∂s aij ∈ L∞ (RN × R, R); (a2 ) there exist β ≥ α > 0 such that for (x, s, ξ) ∈ (RN × R × RN ), α|ξ|2 ≤ aij (x, s)ξi ξj ≤ β|ξ|2 ; (a3 )
there exist µ ∈ (0, 1), ν ∈ (0, q − 2) such that for (x, s, ξ) ∈ (RN × R × RN ), −2µaij (x, s)ξi ξj ≤ s∂s aij (x, s)ξi ξj ≤ νaij (x, s)ξi ξj .
As to c(x), we assume that (c) c(x) ∈ C(RN ), c(x) ≥ 0 for x ∈ RN and lim c(x) → +∞. |x|→∞
We define the Hilbert space Z 1 N X := u ∈ H (R ) :
c(x)u dx < +∞ 2
RN
endowed with the inner product hu, vi = RN (∇u∇v+c(x)uv) and with the norm kuk2X = hu, ui. Under condition (c), the imbedding from X into Lp (RN ) is continuous for 1 ≤ p ≤ 2∗ , and is compact for 1 ≤ p < 2∗ , see [2]. These imbedding results hold if (c) is replaced by one of the following conditions: N (c1 ) c(x) ∈ L∞ loc (R ), ess inf c(x) > 0, and for any M > 0 and any r > 0 there holds R
meas({x ∈ Br (y) : c(x) ≤ M }) → 0 as |y| → ∞, where Br (y) = {x ∈ RN : |x − y| < r} (see [3]). (c2 ) c(x) ∈ C(RN ), c(x) ≥ c0 > 0 for x ∈ RN , and [c(x)]−1 ∈ L1 (RN ) (see [4]). The corresponding functional of (1.1) is Z Z Z 1 1 2 I(u) = aij (x, u)Di uDj u + c(x)u − G(u), 2 RN 2 RN RN Rs where G(s) = 0 g(t)dt and g(s) = |s|q−2 s. Under assumptions (a1 )–(a3 ), I is well defined on X and is continuous on X, but it is not differentiable on X, it is differentiable only in ϕ ∈ X ∩ L∞ (RN ). For u ∈ X and for any ϕ ∈ X ∩ L∞ (RN ), we have Z Z 1 ′ hI (u), ϕi = aij (x, u)Di uDj ϕ + ϕ∂s aij (x, u)Di uDj u N N R R 2 Z Z + c(x)uϕ − g(u)ϕ. RN
RN
In order to study the existence of solutions to (1.1), we employ the nonsmooth critical point theorem in [5–7]. We say that u ∈ X is a weak solution to (1.1) if the weak slope (see the definition in Section 2) of I at u is zero. On a bounded domain Ω ∈ RN , (1.1) was extensively studied, one can refer to [8–15] and some references therein. Moreover, in [16–19], (1.1) is studied when the principal part
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aij (x, u)Di uDj u is unbounded in L1 (Ω). In [16, 17] the existence of solutions were obtained by method of approximation, in [19] the existence of infinitely many solutions were obtained by employing nonsmooth critical point theory for lower semicontinuous functionals, and in [18] the existence of solutions were proved by using a change of variable to reformulate the corresponding functional of the equation. On unbounded domain, the problems that the principal part of (1.1) is bounded was studied also and some existence results were obtained in [2, 20–22]. Recently, to find solutions of (1.1) on unbounded domain with principal part unbounded attracted increasing interest, see for examples [23, 24]. Gluing techniques were used in [23] for periodic problems. In [24] a new method of change of variable, which can be applied to a more general class of quasilinear elliptic problems, was introduced. In most of problems, (1.1) with unbounded principal part can be reformulated to equations with bounded principal part. In most of the studies, the natural growth term of equation (1.1) was assumed to be nonnegative, that is, for u ∈ X, u∂s aij (x, u)Di uDj u ≥ 0. This is called the sign condition of the equation. In this paper, we allow the natural growth term to be negative, see (a3 ). As for as we know, (a3 ) was first introduced in [12] to study (1.1) on bounded domain Ω when q = 2, and then was extended to the case 2 ≤ q < 2∗ in [13]. There are many equations like (1.3) with κ < 0, see also Example 4.1 in [25], that do not satisfy the sign condition. Parameter κ plays an important role in (1.3) especially when it is negative. Assumption (a3 ) make it possible for us to study the existence of solutions to (1.3) when κ is negative. In this paper, we prove that Theorem 1.1 Assume that (a1 )–(a3 ), (c) hold and q ∈ (2, 2∗ ), then problem (1.1) has a nontrivial weak solution in X. In the case that (a3 ) does not satisfied, that is, when s∂s aij (x, s)ξi ξj < −2µaij (x, s)ξi ξj ,
(1.4)
we prove that Theorem 1.2 Assume that aij are x-independent and that (a1 )–(a2 ), (c) hold. Furthermore, assume that q ∈ (2, 2∗ ), c(x) ∈ C 1 (RN ), x · ∇c(x) ≥ −(1 − 2q )N c(x) and (1.4) holds. Then problem (1.1) has no nontrivial weak solution in X ∩ C 1 (RN ). Remark 1.3 (i) Theorems 1.1 and 1.2 imply that −2µaij (x, s)ξi ξj is the lower bound to obtain solutions for (1.1). (ii) According to the proof of Theorem 1.2, the best constant for the lower bound is µ = 1 − q/2∗ . As for problem (1.3), we have Theorem 1.4 Assume that (c) holds and q ∈ (2, 2∗ ), then problem (1.3) has a nontrivial weak solution in X provided that one of the following conditions holds: (i) 4 ≤ q and 0 ≤ κ < +∞; (ii) 2 < q < 4 and 0 ≤ κ < κ∗ ;
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(iii) −κ∗ ≤ κ < 0, where κ∗ = 16(q−2) (4−q)2 > 0, κ∗ =
8µ (1+µ)2
> 0, µ = 1 −
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q 2∗ .
Remark 1.5 (i) Part (i) in Theorem 1.4 holds only in the case N = 3. Indeed, to ensure q ≥ 4 holds, we require that 2∗ > 4, and this holds when N ≤ 3. (ii) κ∗ is increasing in q ∈ (2, 4) and κ∗ → 0 as q → 2, κ∗ → +∞ as q → 4. ∗ (iii) κ∗ is decreasing in q ∈ (2, 2∗ ) and κ∗ → (N16N +2)2 as q → 2, κ∗ → 0 as q → 2 . Corollary 1.6 Assume that (c) holds, q ∈ (2, 2∗ ), κ < −κ∗ and c(x) ∈ C 1 (RN ), x · ∇c(x) ≥ −(1 − 2q )N c(x), then problem (1.3) has no nontrivial solution in X ∩ C 1 (RN ). Corollary 1.7 Assume that (c) holds, q = 2∗ , κ < 0 and c(x) ∈ C 1 (RN ), x · ∇c(x) > −2c(x), then problem (1.3) has no nontrivial solution in X ∩ C 1 (RN ). This paper is organized as the following. In Section 2, we present the nonsmooth variational framework and analyse the compactness of (PS) sequence. In Section 3, we prove the main theorems.
2
Compactness of (PS)c Sequence
We recall from [5–7] the variational framework of nonsmooth critical point theory. Let X be a complete metric space and f : X → R be a continuous function and u ∈ X. We denote by |df |(u) the supremum of the σ’s in [0, +∞) such that there exist δ > 0 and a continuous map η : B(u; δ) × [0, δ] → X such that d(η(v, t), v) ≤ t,
f (η(v, t)) ≤ f (v) − σt.
The extended real number |df |(u) is called the weak slope of f at u. We say that a point u ∈ X is a critical point if |df |(u) = 0 and that a real number c ∈ R is a critical value of f if there exists a critical point u ∈ X of f with f (u) = c. Let c ∈ R. We say that f satisfies the (nonsmooth) Palais-Smale condition at level c ((PC)c in short), if every sequence {un } in X with |df |(un ) → 0 and f (un ) → c contains a subsequence converging in X. Under the above conceptions, the classical mountain pass theorem can be extended to nonsmooth case. Theorem 2.1 Let X be a Banach space endowed with the norm k · k, f : X → R a continuous function. Suppose that there exist w ∈ X, η > f (0) and r > 0 such that f (u) > η,
∀u ∈ X, kuk = r,
f (w) < η,
kwk > r.
Setting Γ = {γ : [0, 1] → X, γ is continuous and γ(0) = 0, γ(1) = w}. Suppose that f satisfies (PS)c condition at the level c = inf sup f (γ(t)) < +∞. γ∈Γ t∈[0,1]
Then, there exists a nontrivial critical point u of f such that f (u) = c. In order to prove that the functional I satisfies the (PS)c condition, we introduce the following Concrete-Palais-Smale condition like [5].
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Definition 2.2 We say that a sequence (un ) ⊂ X is a Concrete-Palais-Smale sequence of I at level c ((CPS)c in short) if there exist (yn ) ⊂ X ∗ with yn → 0 such that I(un ) → c and Z Z 1 aij (x, un )Di un Dj φ + φ∂s aij (x, un )Di un Dj un 2 N N R R Z Z + c(x)un φ − g(un )φ = hyn , φi, ∀φ ∈ C0∞ (RN ). (2.1) RN
RN
We say that I satisfies the (CPS)c condition if every (CPS)c sequence is strongly compact in X.
Note that (2.1) holds for test function φ ∈ X ∩ L∞ (RN ) by approximation. In general, the (CPS)c condition is not equal to the (PS)c condition. The relationship between them can be elucidated by the following lemma. Lemma 2.3 For every u ∈ X, |dI|(u) ≥
sup
hI ′ (u), φi. In particular, if
φ∈C0∞ (RN ),kφkX =1
|dI|(u) < +∞, then |dI|(u) ≥ kI ′ (u)kX ∗ , where k · kX ∗ denote the norm of X ∗ . Proof
The proof is similar to that for [5, Theorem 2.1.3].
Let (un ) ⊂ X be a (PS)c sequence of I, Lemma 2.3 means that it is also a (CPS)c sequence of I, and thus admits a convergent subsequence, thus I satisfies the (PS)c condition. In view of this, we should only prove that I satisfies the (CPS)c condition. We devote the rest of this section to study the compactness of the (CPS)c sequence of I. Proposition 2.4 Assume that (a1 )–(a3 ), (c) hold, then every (CPS)c sequence {un } of I is bounded in X. Proof
and
For every n ∈ N, we denote N Ω+ n := x ∈ R : un ∂s aij (x, un )Di un Dj un ≥ 0 N Ω− n := x ∈ R : un ∂s aij (x, un )Di un Dj un < 0 .
From (a2 )–(a3 ), we obtain that
|s∂s aij (x, s)ξi ξj | ≤ (ν + 2µ)aij (x, s)ξi ξj ≤ (ν + 2µ)β|ξ|2 . Thus un ∂s aij (x, un )Di un Dj un ∈ L1 (RN ). Then, by approximation, we can take φ = un as test functions in (2.1) and get Z Z Z hyn , un i = aij (x, un )Di un Dj un + c(x)u2n − g(un )un RN RN RN Z 1 un ∂s aij (x, un )Di un Dj un . (2.2) + RN 2 It is obvious that Z Z Z un ∂s aij (x, un )Di un Dj un un ∂s aij (x, un )Di un Dj un = + RN Ω+ Ω− n Z n ≤ un ∂s aij (x, un )Di un Dj un . Ω+ n
Thus by (a3 ), we get Z
RN
aij (x, un )Di un Dj un +
1 2
Z
RN
un ∂s aij (x, un )Di un Dj un
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Z Z ν +2 aij (x, un )Di un Dj un + aij (x, un )Di un Dj un 2 Ω+ Ω− n n Z ν +2 ≤ aij (x, un )Di un Dj un . 2 RN
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≤
(2.3)
Taking (a2 ) and (2.3) into considering, the calculation of qI(un ) − hyn , un i gives q−ν −2 q−2 min α, kun k2X 2 2 Z Z q−ν −2 q−2 ≤ aij (x, un )Di un Dj un + c(x)u2n 2 2 N N R R ≤ qI(un ) − hyn , un i ≤ c + o(1)kun kX . This implies the conclusion of the proposition.
Now we prove the almost everywhere convergence of gradients of a bounded (CPS)c sequence of I. We denote Du = (D1 u, · · · , DN u). Proposition 2.5 Assume that (a1 )–(a3 ), (c) hold, then for every bounded (CPS)c sequence {un } of I, there exists u ∈ X such that Dun (x) → Du(x) a.e. in RN . Proof The proof is similar to that in [26, Theorem 2.1]. Note that the imbedding from X into Lp (RN ) is continuous for 1 ≤ p ≤ 2∗ , is compact for 1 ≤ p < 2∗ , and that {un } is bounded in X, there exists u ∈ X such that un ⇀ u in X, un → u in Lr (RN ), 1 ≤ r < 2∗ and un → u a.e. in RN . Now let K, K ′ be two compact subsets of RN such that K ′ ⊂⊂ K ⊂ RN . Let φK ∈ [0, 1] be a smooth function on RN such that φK = 1 on K ′ and φK = 0 on RN \ K. For k ∈ R, let Tk be a truncated function with Tk (s) = s for |s| ≤ k and Tk (s) = ks/|s| for |s| > k. We take φ = φK Tk (un − u) ∈ X as test functions in (2.1). Then by (a2 ) we deduce that Z 0 ≤ α φK |DTk (un − u)|2 N Z R ≤ φK aij (x, un )Di (un − u)Dj Tk (un − u) RN Z 1 = − φK ∂s aij (x, un )Di un Dj un Tk (un − u) 2 N ZR Z − φK aij (x, un )Di uDj Tk (un − u) − Tk (un − u)aij (x, un )Di un Dj φK RN RN Z Z − φK c(x)un Tk (un − u) + φK g(un )Tk (un − u) + hyn , φK Tk (un − u)i RN
RN
:= I + II + III + IV + V + VI.
By (a1 )–(a3 ) and note that q ∈ (2, 2∗ ), we can estimate that I ≤ Ck for some constant C > 0 and that II + III + IV + V + VI → 0 as n → ∞. In fact, since un is bounded in X, by (a1 ), we have ∂s aij (x, un )Di un Dj un is bounded in L1 (RN ), thus Z I ≤ k φK |∂s aij (x, un )Di un Dj un | ≤ Ck. RN
On the other hand, note that un ⇀ u in X, and un → u a.e. in RN , we have Z II = N φK [aij (x, un ) − aij (x, u)]Di uDj Tk (un − u) R Z + φK aij (x, u)Di uDj Tk (un − u) RN
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≤
N Z X
RN
i,j=1
Z ≤C
φK |Dj Tk (un − u)|2
1/2
·
φK [aij (x, un ) − aij (x, u)]2 |Di u|2
RN N X
i,j=1
Z
79
1/2
+ o(1)
φK [aij (x, un ) − aij (x, u)]2 |Di u|2
RN
1/2
+ o(1) → 0.
The proof of III → 0 is similar. Finally, since un → u in L2 (K) and Lq (K), we have 1/2 1/2 Z Z 2 2 IV ≤ → 0, φK c(x)|Tk (un − u)| φK c(x)|un | RN
RN
and similarly,
V ≤
Z
φK |un |q
RN
′
1/q′ Z
φK |Tk (un − u)|q RN
1/q
→ 0,
where q ′ = q/(q − 1). VI → 0 obviously. In conclusion, we have Z φK |DTk (un − u)|2 ≤ Ck.
(2.4)
RN
Define Ak,n := {x ∈ K : |un (x) − u(x)| > k}, Bk,n = K \ Ak,n . Then |Ak,n | → 0 as n → ∞ since un (x) → u(x) a.e. in K. Let en (x) := aij (x, un )Di (un − u)Dj (un − u). Then for 0 < s < 1, by H¨ older inequality, we have Z s Z Z Z Z esn = esn + esn ≤ en |Ak,n |1−s + K
Ak,n
Bk,n
Ak,n
Bk,n
en
s
|Bk,n |1−s .
Combining this and (2.4), we obtain, as k → 0, Z lim sup esn ≤ (Ck)s |K|1−s → 0, n→∞
K
which implies that en (x) → 0 a.e. in K. Thus we conclude that Dun → Du in L2 (K), which implies also that Dun (x) → Du(x) a.e. in K. Now covering RN with a sequence of compact sets with finite measure, we thus deduce that Dun (x) → Du(x) a.e. in RN . Proposition 2.6 Assume that (a1 )–(a3 ), (c) hold, then every bounded (CPS)c sequence {un } of I converges strongly in X to u ∈ X. Proof By Proposition 2.5, we have Dun (x) → Du(x) a.e. in RN , this make it possible for us to use special test functions and employ Fatou’s lemma to prove, like [11, Lemma 2.3], that hI ′ (u), φi = 0,
∀φ ∈ X ∩ L∞ (RN ).
(2.5)
We present the proof of (2.5) in the end of this section for the sake of completeness. Especially, we have hI ′ (u), ui = 0 by approximation. Then by a standard argument, we can prove that un converges strongly to u in X.
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In fact, first, from (2.2), we have Z h i (1 − µ)aij (x, un )Di un Dj un + c(x)u2n lim sup N n→∞ R Z g(un )un = lim sup hyn , un i + n→∞ RN Z h i 1 − un ∂s aij (x, un )Di un Dj un + µaij (x, un )Di un Dj un . RN 2
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(2.6)
Note that we have hyn , un i → 0, and that the imbedding X ֒→ Lq (RN ) is compact, we have Z Z lim g(un )un = g(u)u. n→∞
RN
RN
Moreover, (a3 ) implies that
1 un ∂s aij (x, un )Di un Dj un + µaij (x, un )Di un Dj un ≥ 0. 2 So that we can employ Fatou’s lemma to deduce from (2.6) and hI ′ (u), ui = 0 that Z h i lim sup (1 − µ)aij (x, un )Di un Dj un + c(x)u2n n→∞ RN Z Z h i 1 u∂s aij (x, u)Di uDj u + µaij (x, u)Di uDj u ≤ g(u)u − N RN 2 ZR h i = (1 − µ)aij (x, u)Di uDj u + c(x)u2 .
(2.7)
(2.8)
RN
Second, by Fatou’s lemma,
Z
c(x)u2 ≤ lim inf n→∞
RN
Z
aij (x, u)Di uDj u ≤ lim inf n→∞
RN
Z
Z
RN
c(x)u2n ,
aij (x, un )Di un Dj un .
RN
Since c(x)u2n and aij (x, un )Di un Dj un are nonnegative, we conclude that Z Z c(x)u2 = lim inf c(x)u2n , n→∞
RN
Z
aij (x, u)Di uDj u = lim inf n→∞
RN
Adding lim sup n→∞
R
RN
−c(x)u2n = lim
n→∞
Likewise, we have
Z
R
RN
RN
Z
aij (x, un )Di un Dj un .
RN
−c(x)u2 and (2.8), we get Z
aij (x, un )Di un Dj un =
RN
aij (x, u)Di uDj u.
(2.9)
RN
lim
n→∞
Z
RN
c(x)u2n
=
Z
c(x)u2 .
(2.10)
RN
Third, since aij ∈ L∞ (RN × R), by Lebesgue’s dominated convergence theorem, we have Z Z lim aij (x, un )Di uDj u = aij (x, u)Di uDj u. (2.11) n→∞
RN
RN
We claim that
lim
n→∞
Z
RN
aij (x, un )Di un Dj u =
Z
RN
aij (x, u)Di uDj u.
(2.12)
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In fact, by Lebesgue’s dominated convergence theorem, for 1 ≤ i, j ≤ N , we have aij (x, un )Dj u → aij (x, u)Dj u strongly in L2 (RN ). Thus H¨ older’s inequality gives Z lim aij (x, un ) − aij (x, u) Dj uDi un = 0. n→∞ N R
Since by the weak convergence of un ⇀ u in X, Z Z lim aij (x, u)Di un Dj u = n→∞
RN
aij (x, u)Di uDj u,
RN
thus we conclude that (2.12) hold. Finally, by the weak convergence of un ⇀ u in X, we have Z Z lim c(x)un u = c(x)u2 . n→∞
RN
(2.13)
RN
Through the above analysis, we conclude from (2.9)–(2.13) that Z Z α |D(un − u)|2 + c(x)(un − u)2 RN RN Z Z ≤ aij (x, un )Di (un − u)Dj (un − u) + c(x)(un − u)2 RN RN Z Z ≤ aij (x, un )Di un Dj un + aij (x, un )Di uDj u RN RN Z Z −2 aij (x, un )Di un Dj u + c(x)(un − u)2 → 0. RN
RN
This implies the strong convergence of {un } and completes the proof.
Remark 2.7 Assume that (a1 )–(a3 ) and (c) hold, {un } is a bounded (CPS)c sequence of I converges weakly to u ∈ X. Then by an argument similar to the proof of (2.9) and (2.10), we can prove that Z Z lim un ∂s aij (x, un )Di un Dj un = u∂s aij (x, u)Di uDj u. n→∞
RN
RN
In fact, on one hand, the convergence of g(un )un → g(u)u in L1 (RN ), yn → 0 in X ∗ , and (2.9)–(2.10) imply that Z Z lim sup un ∂s aij (x, un )Di un Dj un ≤ u∂s aij (x, u)Di uDj u. n→∞
RN
RN
On the other hand, (2.7) ensures that we can employ Fatou’s lemma and (2.9) to deduce that Z Z u∂s aij (x, u)Di uDj u ≤ lim inf un ∂s aij (x, un )Di un Dj un . n→∞
RN
RN
Now we prove equality (2.5). The crucial point is to dominate the integral Z φ∂s aij (x, un )Di un Dj un .
(2.14)
Ω− n
This can be done by modifying the test function φ = ϕ exp{−M u+ n } in [5], where ϕ ∈ X ∩ L∞ (RN ) and M > 0 sufficiently large. Proof of (2.5) Let H ∈ [0, 1] be a smooth cutoff function on R with H = 1 on [− 12 , 21 ], H = 0 outside [−1, 1], and |H ′ | ≤ C for some constant C > 0. Define s+ = max{s, 0}, s− =
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max{−s, 0} and let 0 ≤ ϕ ∈ C0∞ (RN ). According to (a1 )–(a2 ), there exists M > 0 such that |∂s aij (x, s)ξi ξj | ≤ 2M aij (x, s)ξi ξj . For k > 0 large, we take φ = ϕH(un /k) exp{−M T2k [(un + k)+ ]} in (2.1), where T2k is defined similar to Tk in the proof of Proposition 2.5. Then Z 0= aij (x, un )Di un Dj ϕ · H(un /k) exp{−M T2k [(un + k)+ ]} RN Z 1 + aij (x, un )Di un Dj un · ϕH ′ (un /k) exp{−M T2k [(un + k)+ ]}} RN k Z − M aij (x, un )Di un Dj un · ϕH(un /k) exp{−M T2k [(un + k)+ ]} N ZR 1 + ∂s aij (x, un )Di un Dj un · ϕH(un /k) exp{−M T2k [(un + k)+ ]} RN 2 Z + c(x)un ϕH(un /k) exp{−M T2k [(un + k)+ ]} RN Z − g(un )ϕH(un /k) exp{−M T2k [(un + k)+ ]} RN
−hyn , ϕH(un /k) exp{−M T2k [(un + k)+ ]}i.
(2.15)
Note that 1 ∂s aij (x, un )Di un Dj un · ϕH(un /k) exp{−M T2k [(un + k)+ ]} 2 −M aij (x, un )Di un Dj un · ϕH(un /k) exp{−M T2k [(un + k)+ ]} ≤ 0, and by (a2 ), Z
RN
C 1 ′ + aij (x, un )Di un Dj un · ϕH (un /k) exp{−M T2k [(un + k) ]} ≤ , k k
by Fatou’s lemma, we can take lim sup in (2.15) and get Z C aij (x, u)Di uDj ϕ · H(u/k) exp{−M T2k [(u + k)+ ]} − ≤ k RN Z − M aij (x, u)Di uDj u · ϕH(u/k) exp{−M T2k [(u + k)+ ]} RN Z 1 + ∂s aij (x, u)Di uDj u · ϕH(u/k) exp{−M T2k [(u + k)+ ]} 2 N ZR + c(x)uϕH(u/k) exp{−M T2k [(u + k)+ ]} RN Z − g(u)ϕH(u/k) exp{−M T2k [(u + k)+ ]}.
(2.16)
RN
Choose ϕ exp{M T2k [(u + k)+ ]} instead of ϕ in (2.16), and then let k → ∞, we get Z Z Z Z 1 0≤ aij (x, u)Di uDj ϕ + ϕ∂s aij (x, u)Di uDj u + c(x)uϕ − g(u)ϕ. RN RN 2 RN RN
Now we take φ = ϕH(un /k) exp{−M T2k [(un − k)− ]} in (2.1) to obtain the opposite inequality. Thus we have hI ′ (u), ϕi = 0 holds for ϕ ≥ 0 and holds for ϕ ≤ 0 similarly. Then it holds also for general ϕ ∈ X ∩ L∞ (RN ) by approximation. Remark 2.8 The main idea of the proof of (2.5) is to truncate the part in (2.14) that is out of control. Note that Dj Tk (un ) ≡ 0 on the set Ak,n (RN ) := {x ∈ RN : |un (x) − u(x)| > k}. Another method to prove (2.5) is to add additional integral of the principal part to dominate
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(2.14). Let 0 ≤ ϕ ∈ C0∞ (RN ) and let φ = ϕ exp{−M T2k [(u + k)+ ]} for k > 0. According to the proof of Proposition 3.3 in [13], we have lim kun − Tk (un )kX ≤ εk , where 0 < εk → 0 as n→∞
k → ∞. Then by (a2 )–(a3 ), we have Z 1 ∂s aij (x, un )Di un Dj un · ϕ exp{−M T2k [(u + k)+ ]} RN 2 Z − M aij (x, un )Di un Dj Tk (un ) · ϕ exp{−M T2k [(u + k)+ ]} N Z R 1 = ∂s aij (x, un )Di un Dj un · ϕ exp{−M T2k [(u + k)+ ]} RN 2 Z − M aij (x, un )Di un Dj un · ϕ exp{−M T2k [(u + k)+ ]} RN Z + M aij (x, un )Di (un − Tk (un ))Dj (un − Tk (un )) · ϕ exp{−M T2k [(u + k)+ ]} RN
≤ Cεk .
Thus we can take φ as test function in (2.1) and obtain (2.5) similarly.
3
Proof of Main Results
In this section, we prove the existence and nonexistence of weak solutions of (1.1). Proof of Theorem 1.1 Theorem 2.1 applied. First, Proposition 2.6 shows that I satisfies (PS)c condition. Secondly, we prove that I has the geometrical conditions stated in Theorem 2.1. In fact, we have I(0) = 0. For every u ∈ X, by (a2 ) and Sobolev’s inequality, we have 1 I(u) ≥ min (α, 1)kuk2X − λCkukqX , 2 where C > 0 is a constant. Thus there exist r > 0, δ > 0 such that I(u) ≥ δ for kukX = r. Now let ϕ0 ∈ X be such that ϕ0 (x) > 0 and kϕ0 kX = 1 and let t ∈ R+ , then by (a2 ) again, t2 max (β, 1)kϕ0 k2X − λtq kϕ0 kqq < 0 2 for t large. According to Theorem 2.1, I admit a critical point u ∈ X. I(tϕ0 ) ≤
Remark 3.1 (i) Denote u+ = max{u, 0}. Replacing |u|q−2 u by |u|q−2 u+ in eq. (1.1), the solutions to the equation can be proved to be nonnegative. (ii) Note that the measure of the set Ak = {x ∈ RN : |u(x)| > k} is finite for k > 0, the solution to (1.1) is in fact belongs to X ∩ L∞ (RN ). Indeed, if 2 ≤ q < 2∗ , it can be deduced as in [12]. If q = 2∗ , the same device as the proof for Theorem 4.1 in [18] can be applied. In order to prove the nonexistence result Theorem 1.2, we empoy Pohozaev’s identity in [27] (see also [28]). Let Ω ⊂ RN be open and F (x, u, r) : Ω × R × RN be a functional of class C 1 , we consider the following equation div{Fr (x, u, Du)} = Fu (x, u, Du),
(3.1)
here we write Du = (∂u/∂x1 , · · · , ∂u/∂xN ), Fu = ∂F/∂u, Fxi = ∂F/∂xi and Fri = ∂F/∂ri , r = (r1 , · · · , rN ). Assume that F (x, 0, 0) = 0. Let a(x) ∈ C 1 (Ω) ∩ C(Ω), h(x) ∈ C 1 (Ω, RN )
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and u ∈ C 1 (Ω) ∩ C(Ω) be a solution to problem (3.1), then we have the following Pohozaev’s identity. Z F (x, 0, Du) − Di uFri (x, 0, Du) (h · ν)ds Z∂Ω Z Z = F (x, u, Du)divh + hi Fxi (x, u, Du) − Dj uDi hj + uDi a(x) Fri (x, u, Du) Ω Ω Ω Z − a(x) Di uFri (x, u, Du) + uFu (x, u, Du) . (3.2) Ω
Proof of Theorem 1.2 to (1.1). Let
Assume on the contrary that u ∈ X ∩C 1 (RN ) is a weak solution
1 1 aij (u)Di uDj u + c(x)u2 − G(u), 2 2 and let a be independent on x, h = x in (3.2), we get Z Z N − 2 a −a aij (u)Di uDj u − ua′ (u)Di uDj u 0= 2 2 RN ij RN Z N Z N Z 1 + −a c(x)u2 + (x · ∇c(x))u2 − −a |u|q . 2 2 q N N N R R R F (x, u, Du) =
Set a = N/q. Note that by assumption, we have (1 − 2q )N c(x) + (x · ∇c(x)) ≥ 0, thus Z Z q 1 1− ∗ aij (u)Di uDj u + ua′ij (u)Di uDj u ≥ 0. 2 2 N N R R Therefore if u satisfies (1.4), then (3.3) implies that u ≡ 0. This completes the proof.
(3.3)
Remark 3.2 The parameter µ in (a3 ) is, in fact, should not greater than 1 − 2q∗ according to Theorem 1.2. Moreover, as q tends to 2∗ , µ tends to 0. This is partially consistent with [10, Theorem 5.1], where the following result was proved: if bij (s), ∂s bij (s) ∈ L∞ (R), bij (s)ξi ξj ≥ γ|ξ|2 , γ ∈ (0, 1] and s∂s bij (s)ξi ξj < 0, ∀s 6= 0, then the equation ∗ 1 −Dj (bij (u)Di u) + ∂s bij (u)Di uDj u = |u|2 −2 u 2 has no nontrivial solution in H01 (Ω) ∩ L∞ (Ω), where Ω ⊂ RN (N ≥ 3) is bounded and starshaped.
Now we are ready to prove Theorem 1.4. We prove that problem (1.3) has nontrivial solutions even if κ is less than zero. Proof of Theorem 1.4 By direct computation, we have u −∆u − κ ∆(1 + |u|2 )1/2 2(1 + |u|2 )1/2 κ|u|2 κu = − 1+ ∆u − |∇u|2 2(1 + |u|2 ) 2(1 + |u|2 )2 and
−div Thus we get
κ|u|2 κ|u|2 κu 1+ ∇u = − 1 + ∆u − |∇u|2 . 2(1 + |u|2 ) 2(1 + |u|2 ) (1 + |u|2 )2 u −∆u − κ ∆(1 + |u|2 )1/2 2(1 + |u|2 )1/2 κ|u|2 κu = −div 1 + ∇u + |∇u|2 , 2(1 + |u|2 ) 2(1 + |u|2 )2
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κ|u|2 that is, we have aij (x, u) = 1 + 2(1+|u| 2 ) δij . Since (a1 ) and (a2 ) are satisfied obviously, we only need to prove that (a3 ) holds. Case κ ≥ 0. We prove that κ|u|2 κ|u|2 2 0 ≤ u∂s aij (x, u)Di uDj u = |∇u| ≤ ν 1 + |∇u|2 , (1 + |u|2 )2 2(1 + |u|2 )
that is,
ν+
(ν − 2)κ|u|2 + νκ|u|4 ≥ 0. 2(1 + |u|2 )2
16ν This inequality holds either (i) ν ≥ 2 and κ ≥ 0 or (ii) 0 < ν < 2 and 0 ≤ κ ≤ (2−ν) 2 . Since 16ν is increasing in ν ∈ (0, 2) and tends to +∞ as ν tends to 2, we thus draw a conclusion (2−ν)2
that the inequality holds either (i) q ≥ 4 and κ ≥ 0 or (ii) 2 < q < 4 and 0 ≤ κ < 16(q−2) (4−q)2 . Case κ < 0. We prove that κ|u|2 κ|u|2 2 0 > u∂s aij (x, u)Di uDj u = |∇u| ≥ −2µ 1 + |∇u|2 , (1 + |u|2 )2 2(1 + |u|2 ) that is,
2µ + κ
(µ + 1)|u|2 + µ|u|4 ≥ 0. (1 + |u|2 )2
q 8µ This inequality holds when κ ≥ − (1+µ) 2 > −2, here we take µ = (1 − 2∗ ) ∈ (0, 1). In this case, we need to ensure that problem (1.3) is still elliptic, that is, for such κ, there exists γ > 0 such that κ|u|2 aij (x, u)Di uDj u = 1 + |∇u|2 ≥ γ|∇u|2 . 2(1 + |u|2 ) κ|u|2 Since 1 + 2(1+|u| is decreasing in |u|, the above inequality is equivalent to 1 + κ2 ≥ γ, i.e., 2) κ ≥ 2(γ − 1). Since κ > −2, we obtain the existence of such γ.
Proof of Corollary 1.7 Under the assumptions of Corollary 1.7, assume that there is still a solution 0 6= u ∈ X ∩ C 1 (RN ) to (1.3). Then Pohozaev’s inequality, with a = N/2∗ and h = x in (3.2), implies that Z Z Z N 1 ′ 2 ua (x · ∇c(x))u2 ≥ 0, (u)D uD u = c(x)u + i j 22∗ RN ij 2 N N R R where a′ij (u) =
κ|u|2 (1+|u|2 )2 δij .
But this is impossible unless u ≡ 0.
Acknowledgements The first author thanks the staff of Chern Institute of Mathematical, Nankai University and Professor Zhiqiang Wang for their hospitality. Part of this work was done when the first author was visiting there. References [1] Borovskii A V, Galkin A L. Dynamical modulation of an ultrashort high-intensity laser pulse in matter. JETP, 1993, 77: 562–573 [2] Aouaoui S. Multiplicity of solutions for quasilinear elliptic equations in RN . J Math Anal Appl, 2010, 370: 639–648 [3] Bartsch T, Pankov A, Wang Z Q. Nonlinear Schr¨ odinger equations with steep potential well. Comm Contemp Math, 2001, 4: 549–569 [4] do O J M, Severo U. Quasilinear Schr¨ odinger equations involving concave and convex nonlinearities. Comm Pure Appl Anal, 2009, 8: 621–644
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[5] Canino A, Degiovanni M. Nonsmooth critical point theory and quasilinear elliptic equations//Granas A, Frigon M, Sabidussi G, eds. Topological Methods in Differential Equations and Inclusions, Montreal, 1994, NATO ASI Series. Dordrecht: Kluwer Academic Publishers, 1995: 1–50 [6] Corvellec J N, Degiovanni M, Marzocchi M. Deformation properties for continuous functionals and critical point theory. Topol Methods Nonl Anal, 1993, 1: 151–171 [7] Degiovanni M, Marzocchi M. A critical point theory for nonsmooth functional. Ann Mat Pura Appl, 1994, 167(4): 73–100 [8] Arcoya D, Boccardo L. Critical points for multiple integrals of the calculus of variations. Arch Rational Meth Anal, 1996, 134: 249–274 [9] Arcoya D, Boccardo L, Orsina L. Existence of critical points for some noncoercive functionals. Ann I H Poincar´ e-AN, 2001, 18(4): 437–457 [10] Arioli G, Gazzola F. Quasilinear elliptic equations at critical growth. NoDEA Nonl Eiffer Equ Appl, 1998, 5: 83–97 [11] Canino A. Multiplicity of solutions for quasilinear elliptic equations. Topol Methods Nonl Anal, 1995, 6: 357–370 [12] Shen Y. Nontrivial solution for a class of quasilinear equation with natural growth. Acta Math Sinica, Chinese Series, 2003, 46: 683–690 [13] Li Z, Shen Y, Yao Y. Nontrivial solutions for quasilinear elliptic equations with natural growth. Acta Math Sinica, Chinese Series, 2009, 52: 785–798 [14] Shen Y, Li Z, Wang Y. Sign-changing critical points for noncoercive functionals. Topol Methods Nonl Anal, 2014, 43: 373–384 [15] Squassina M. Existence, multiplicity, perturbation, and concentration results for a class of quasi-linear elliptic problems. Electronic Journal of Differential Equations, Monograph 7, 2006 [16] Abdellaoui B, Boccardo L, Peral I, et al. Quasilinear elliptic equations with natural growth. Differ Integr Equ, 2007, 20: 1005–1020 [17] Boccardo L, Spagnolo S. Positive solutions for some quasilinear elliptic equations with natural growths. Rend Mat Acc Lincei, 2000, 11: 31–39 [18] Li Z. Existence of nontrivial solutions for quasilinear elliptic equations at critical growth. Appl Math Comput, 2011, 218: 76–87 [19] Pellacci B, Squassina M. Unbounded critical points for a class of lower semicontinuous functionals. J Differ Equ, 2004, 201: 25–62 [20] Arioli G, Gazzola F. On a quasilinear elliptic differential equation in unbounded domains. Rend Istit Mat Univ, Trieste, 1998, 30: 113–128 [21] Conti M, Gazzola F. Positive entire solutions of quasilinear elliptic problems via nonsmooth critical point theory. Topol Methods Nonl Anal, 1996, 8: 275–294 [22] Gazzola F. Positive solutions of critical quasilinear elliptic problems in general domains. Abstr Appl Anal, 1998, 3(1/2): 65–84 [23] Liu J Q, Wang Z Q, Guo Y X. Multibump solutions for quasilinear elliptic equations. J Funct Anal, 2012, 262: 4040–4102 [24] Shen Y, Wang Y. Soliton solutions for generalized quasilinear Schr´ odinger. Nonl Anal TMA, 2013, 80: 194–201 [25] Li Z, Shen Y, Zhang Y. An application of nonsmooth critical point theory. Topol Methods Nonl Anal, 2010, 35: 203–219 [26] Boccardo L, Murat F. Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonl Anal TMA, 1992, 19: 581–597 [27] Pucci P, Serrin J. A general variational identity. Indiana Univ Math J, 1986, 35(3): 681–703 [28] Degiovanni M, Musesti A, Squassina M. On the regularity of solutions in the Pucci-Serrin identity. Calc Var Partial Differential Equations, 2003, 18: 317–334
NONSMOOTH CRITICAL POINT THEOREMS AND ITS APPLICATIONS TO QUASILINEAR ¨ SCHRODINGER EQUATIONS∗
o±!)
Zhouxin LI (
School of Mathematics and Statistics, Central South University, Changsha 400083, China E-mail : [email protected]
!U)
Yaotian SHEN (
Department of Mathematics, South China University of Technology, Guangzhou 510640, China E-mail : [email protected] Abstract In this paper, the existence and nonexistence of solutions to a class of quasilinear elliptic equations with nonsmooth functionals are discussed, and the results obtained are applied to quasilinear Schr¨ odinger equations with negative parameter which arose from the study of self-channeling of high-power ultrashort laser in matter. Key words
nonsmooth critical point theorems; quasilinear elliptic equations; Schr¨ odinger equation
2010 MR Subject Classification
1
35J20; 35J65; 35J10
Introduction and Main Results
In this paper, we consider the existence of nontrivial solutions for the following class of quasilinear elliptic equations with natural growth 1 −Dj (aij (x, u)Di u) + ∂s aij (x, u)Di uDj u + c(x)u = |u|q−2 u, 2
x ∈ RN ,
(1.1)
where N ≥ 3, q > 2, c(x) ∈ C 1 (RN ), aij (x, s) are Carath´eodory functions, ∂s aij denotes the derivatives of aij with respect to s. The repeated indices indicate the summation from 1 to N . As an example, in this paper, we also consider a special case of equation (1.1). In the study of self-channeling of high-power ultrashort laser in matter [1], the following quasilinear Schr¨odinger equation was considered i∂t z = −∆z + W (x)z − l(|z|2 )z − κ∆h(|z|2 )h′ (|z|2 )z,
x ∈ RN ,
(1.2)
where W (x) is a given potential, κ is a real constant. Let h(s) = (1 + s)1/2 and l(s) = s(q−2)/2 . If we consider the standing waves of (1.2) of the form z(x, t) = exp(−iEt)u(x), we observe that ∗ Received
December 5, 2014; revised May 13, 2015. The first author was supported by NSF of China (11201488) and Hunan Provincial Natural Science Foundation of China (14JJ4002). The second author was supported by NSF of China (11371146).
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z(x, t) satisfies (1.2) if and only if u(x) solves the following equation κu −∆u + c(x)u − ∆(1 + |u|2 )1/2 = |u|q−2 u, 2(1 + |u|2 )1/2 Equation (1.3) is a special case of (1.1) with aij (x, u) = 1 +
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x ∈ RN .
(1.3)
κ|u|2 δij . 2(1 + |u|2 )
We make the following assumptions on aij (x, s). (a1 ) aij ≡ aji , and aij , ∂s aij ∈ L∞ (RN × R, R); (a2 ) there exist β ≥ α > 0 such that for (x, s, ξ) ∈ (RN × R × RN ), α|ξ|2 ≤ aij (x, s)ξi ξj ≤ β|ξ|2 ; (a3 )
there exist µ ∈ (0, 1), ν ∈ (0, q − 2) such that for (x, s, ξ) ∈ (RN × R × RN ), −2µaij (x, s)ξi ξj ≤ s∂s aij (x, s)ξi ξj ≤ νaij (x, s)ξi ξj .
As to c(x), we assume that (c) c(x) ∈ C(RN ), c(x) ≥ 0 for x ∈ RN and lim c(x) → +∞. |x|→∞
We define the Hilbert space Z 1 N X := u ∈ H (R ) :
c(x)u dx < +∞ 2
RN
endowed with the inner product hu, vi = RN (∇u∇v+c(x)uv) and with the norm kuk2X = hu, ui. Under condition (c), the imbedding from X into Lp (RN ) is continuous for 1 ≤ p ≤ 2∗ , and is compact for 1 ≤ p < 2∗ , see [2]. These imbedding results hold if (c) is replaced by one of the following conditions: N (c1 ) c(x) ∈ L∞ loc (R ), ess inf c(x) > 0, and for any M > 0 and any r > 0 there holds R
meas({x ∈ Br (y) : c(x) ≤ M }) → 0 as |y| → ∞, where Br (y) = {x ∈ RN : |x − y| < r} (see [3]). (c2 ) c(x) ∈ C(RN ), c(x) ≥ c0 > 0 for x ∈ RN , and [c(x)]−1 ∈ L1 (RN ) (see [4]). The corresponding functional of (1.1) is Z Z Z 1 1 2 I(u) = aij (x, u)Di uDj u + c(x)u − G(u), 2 RN 2 RN RN Rs where G(s) = 0 g(t)dt and g(s) = |s|q−2 s. Under assumptions (a1 )–(a3 ), I is well defined on X and is continuous on X, but it is not differentiable on X, it is differentiable only in ϕ ∈ X ∩ L∞ (RN ). For u ∈ X and for any ϕ ∈ X ∩ L∞ (RN ), we have Z Z 1 ′ hI (u), ϕi = aij (x, u)Di uDj ϕ + ϕ∂s aij (x, u)Di uDj u N N R R 2 Z Z + c(x)uϕ − g(u)ϕ. RN
RN
In order to study the existence of solutions to (1.1), we employ the nonsmooth critical point theorem in [5–7]. We say that u ∈ X is a weak solution to (1.1) if the weak slope (see the definition in Section 2) of I at u is zero. On a bounded domain Ω ∈ RN , (1.1) was extensively studied, one can refer to [8–15] and some references therein. Moreover, in [16–19], (1.1) is studied when the principal part
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aij (x, u)Di uDj u is unbounded in L1 (Ω). In [16, 17] the existence of solutions were obtained by method of approximation, in [19] the existence of infinitely many solutions were obtained by employing nonsmooth critical point theory for lower semicontinuous functionals, and in [18] the existence of solutions were proved by using a change of variable to reformulate the corresponding functional of the equation. On unbounded domain, the problems that the principal part of (1.1) is bounded was studied also and some existence results were obtained in [2, 20–22]. Recently, to find solutions of (1.1) on unbounded domain with principal part unbounded attracted increasing interest, see for examples [23, 24]. Gluing techniques were used in [23] for periodic problems. In [24] a new method of change of variable, which can be applied to a more general class of quasilinear elliptic problems, was introduced. In most of problems, (1.1) with unbounded principal part can be reformulated to equations with bounded principal part. In most of the studies, the natural growth term of equation (1.1) was assumed to be nonnegative, that is, for u ∈ X, u∂s aij (x, u)Di uDj u ≥ 0. This is called the sign condition of the equation. In this paper, we allow the natural growth term to be negative, see (a3 ). As for as we know, (a3 ) was first introduced in [12] to study (1.1) on bounded domain Ω when q = 2, and then was extended to the case 2 ≤ q < 2∗ in [13]. There are many equations like (1.3) with κ < 0, see also Example 4.1 in [25], that do not satisfy the sign condition. Parameter κ plays an important role in (1.3) especially when it is negative. Assumption (a3 ) make it possible for us to study the existence of solutions to (1.3) when κ is negative. In this paper, we prove that Theorem 1.1 Assume that (a1 )–(a3 ), (c) hold and q ∈ (2, 2∗ ), then problem (1.1) has a nontrivial weak solution in X. In the case that (a3 ) does not satisfied, that is, when s∂s aij (x, s)ξi ξj < −2µaij (x, s)ξi ξj ,
(1.4)
we prove that Theorem 1.2 Assume that aij are x-independent and that (a1 )–(a2 ), (c) hold. Furthermore, assume that q ∈ (2, 2∗ ), c(x) ∈ C 1 (RN ), x · ∇c(x) ≥ −(1 − 2q )N c(x) and (1.4) holds. Then problem (1.1) has no nontrivial weak solution in X ∩ C 1 (RN ). Remark 1.3 (i) Theorems 1.1 and 1.2 imply that −2µaij (x, s)ξi ξj is the lower bound to obtain solutions for (1.1). (ii) According to the proof of Theorem 1.2, the best constant for the lower bound is µ = 1 − q/2∗ . As for problem (1.3), we have Theorem 1.4 Assume that (c) holds and q ∈ (2, 2∗ ), then problem (1.3) has a nontrivial weak solution in X provided that one of the following conditions holds: (i) 4 ≤ q and 0 ≤ κ < +∞; (ii) 2 < q < 4 and 0 ≤ κ < κ∗ ;
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(iii) −κ∗ ≤ κ < 0, where κ∗ = 16(q−2) (4−q)2 > 0, κ∗ =
8µ (1+µ)2
> 0, µ = 1 −
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q 2∗ .
Remark 1.5 (i) Part (i) in Theorem 1.4 holds only in the case N = 3. Indeed, to ensure q ≥ 4 holds, we require that 2∗ > 4, and this holds when N ≤ 3. (ii) κ∗ is increasing in q ∈ (2, 4) and κ∗ → 0 as q → 2, κ∗ → +∞ as q → 4. ∗ (iii) κ∗ is decreasing in q ∈ (2, 2∗ ) and κ∗ → (N16N +2)2 as q → 2, κ∗ → 0 as q → 2 . Corollary 1.6 Assume that (c) holds, q ∈ (2, 2∗ ), κ < −κ∗ and c(x) ∈ C 1 (RN ), x · ∇c(x) ≥ −(1 − 2q )N c(x), then problem (1.3) has no nontrivial solution in X ∩ C 1 (RN ). Corollary 1.7 Assume that (c) holds, q = 2∗ , κ < 0 and c(x) ∈ C 1 (RN ), x · ∇c(x) > −2c(x), then problem (1.3) has no nontrivial solution in X ∩ C 1 (RN ). This paper is organized as the following. In Section 2, we present the nonsmooth variational framework and analyse the compactness of (PS) sequence. In Section 3, we prove the main theorems.
2
Compactness of (PS)c Sequence
We recall from [5–7] the variational framework of nonsmooth critical point theory. Let X be a complete metric space and f : X → R be a continuous function and u ∈ X. We denote by |df |(u) the supremum of the σ’s in [0, +∞) such that there exist δ > 0 and a continuous map η : B(u; δ) × [0, δ] → X such that d(η(v, t), v) ≤ t,
f (η(v, t)) ≤ f (v) − σt.
The extended real number |df |(u) is called the weak slope of f at u. We say that a point u ∈ X is a critical point if |df |(u) = 0 and that a real number c ∈ R is a critical value of f if there exists a critical point u ∈ X of f with f (u) = c. Let c ∈ R. We say that f satisfies the (nonsmooth) Palais-Smale condition at level c ((PC)c in short), if every sequence {un } in X with |df |(un ) → 0 and f (un ) → c contains a subsequence converging in X. Under the above conceptions, the classical mountain pass theorem can be extended to nonsmooth case. Theorem 2.1 Let X be a Banach space endowed with the norm k · k, f : X → R a continuous function. Suppose that there exist w ∈ X, η > f (0) and r > 0 such that f (u) > η,
∀u ∈ X, kuk = r,
f (w) < η,
kwk > r.
Setting Γ = {γ : [0, 1] → X, γ is continuous and γ(0) = 0, γ(1) = w}. Suppose that f satisfies (PS)c condition at the level c = inf sup f (γ(t)) < +∞. γ∈Γ t∈[0,1]
Then, there exists a nontrivial critical point u of f such that f (u) = c. In order to prove that the functional I satisfies the (PS)c condition, we introduce the following Concrete-Palais-Smale condition like [5].
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Definition 2.2 We say that a sequence (un ) ⊂ X is a Concrete-Palais-Smale sequence of I at level c ((CPS)c in short) if there exist (yn ) ⊂ X ∗ with yn → 0 such that I(un ) → c and Z Z 1 aij (x, un )Di un Dj φ + φ∂s aij (x, un )Di un Dj un 2 N N R R Z Z + c(x)un φ − g(un )φ = hyn , φi, ∀φ ∈ C0∞ (RN ). (2.1) RN
RN
We say that I satisfies the (CPS)c condition if every (CPS)c sequence is strongly compact in X.
Note that (2.1) holds for test function φ ∈ X ∩ L∞ (RN ) by approximation. In general, the (CPS)c condition is not equal to the (PS)c condition. The relationship between them can be elucidated by the following lemma. Lemma 2.3 For every u ∈ X, |dI|(u) ≥
sup
hI ′ (u), φi. In particular, if
φ∈C0∞ (RN ),kφkX =1
|dI|(u) < +∞, then |dI|(u) ≥ kI ′ (u)kX ∗ , where k · kX ∗ denote the norm of X ∗ . Proof
The proof is similar to that for [5, Theorem 2.1.3].
Let (un ) ⊂ X be a (PS)c sequence of I, Lemma 2.3 means that it is also a (CPS)c sequence of I, and thus admits a convergent subsequence, thus I satisfies the (PS)c condition. In view of this, we should only prove that I satisfies the (CPS)c condition. We devote the rest of this section to study the compactness of the (CPS)c sequence of I. Proposition 2.4 Assume that (a1 )–(a3 ), (c) hold, then every (CPS)c sequence {un } of I is bounded in X. Proof
and
For every n ∈ N, we denote N Ω+ n := x ∈ R : un ∂s aij (x, un )Di un Dj un ≥ 0 N Ω− n := x ∈ R : un ∂s aij (x, un )Di un Dj un < 0 .
From (a2 )–(a3 ), we obtain that
|s∂s aij (x, s)ξi ξj | ≤ (ν + 2µ)aij (x, s)ξi ξj ≤ (ν + 2µ)β|ξ|2 . Thus un ∂s aij (x, un )Di un Dj un ∈ L1 (RN ). Then, by approximation, we can take φ = un as test functions in (2.1) and get Z Z Z hyn , un i = aij (x, un )Di un Dj un + c(x)u2n − g(un )un RN RN RN Z 1 un ∂s aij (x, un )Di un Dj un . (2.2) + RN 2 It is obvious that Z Z Z un ∂s aij (x, un )Di un Dj un un ∂s aij (x, un )Di un Dj un = + RN Ω+ Ω− n Z n ≤ un ∂s aij (x, un )Di un Dj un . Ω+ n
Thus by (a3 ), we get Z
RN
aij (x, un )Di un Dj un +
1 2
Z
RN
un ∂s aij (x, un )Di un Dj un
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Z Z ν +2 aij (x, un )Di un Dj un + aij (x, un )Di un Dj un 2 Ω+ Ω− n n Z ν +2 ≤ aij (x, un )Di un Dj un . 2 RN
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≤
(2.3)
Taking (a2 ) and (2.3) into considering, the calculation of qI(un ) − hyn , un i gives q−ν −2 q−2 min α, kun k2X 2 2 Z Z q−ν −2 q−2 ≤ aij (x, un )Di un Dj un + c(x)u2n 2 2 N N R R ≤ qI(un ) − hyn , un i ≤ c + o(1)kun kX . This implies the conclusion of the proposition.
Now we prove the almost everywhere convergence of gradients of a bounded (CPS)c sequence of I. We denote Du = (D1 u, · · · , DN u). Proposition 2.5 Assume that (a1 )–(a3 ), (c) hold, then for every bounded (CPS)c sequence {un } of I, there exists u ∈ X such that Dun (x) → Du(x) a.e. in RN . Proof The proof is similar to that in [26, Theorem 2.1]. Note that the imbedding from X into Lp (RN ) is continuous for 1 ≤ p ≤ 2∗ , is compact for 1 ≤ p < 2∗ , and that {un } is bounded in X, there exists u ∈ X such that un ⇀ u in X, un → u in Lr (RN ), 1 ≤ r < 2∗ and un → u a.e. in RN . Now let K, K ′ be two compact subsets of RN such that K ′ ⊂⊂ K ⊂ RN . Let φK ∈ [0, 1] be a smooth function on RN such that φK = 1 on K ′ and φK = 0 on RN \ K. For k ∈ R, let Tk be a truncated function with Tk (s) = s for |s| ≤ k and Tk (s) = ks/|s| for |s| > k. We take φ = φK Tk (un − u) ∈ X as test functions in (2.1). Then by (a2 ) we deduce that Z 0 ≤ α φK |DTk (un − u)|2 N Z R ≤ φK aij (x, un )Di (un − u)Dj Tk (un − u) RN Z 1 = − φK ∂s aij (x, un )Di un Dj un Tk (un − u) 2 N ZR Z − φK aij (x, un )Di uDj Tk (un − u) − Tk (un − u)aij (x, un )Di un Dj φK RN RN Z Z − φK c(x)un Tk (un − u) + φK g(un )Tk (un − u) + hyn , φK Tk (un − u)i RN
RN
:= I + II + III + IV + V + VI.
By (a1 )–(a3 ) and note that q ∈ (2, 2∗ ), we can estimate that I ≤ Ck for some constant C > 0 and that II + III + IV + V + VI → 0 as n → ∞. In fact, since un is bounded in X, by (a1 ), we have ∂s aij (x, un )Di un Dj un is bounded in L1 (RN ), thus Z I ≤ k φK |∂s aij (x, un )Di un Dj un | ≤ Ck. RN
On the other hand, note that un ⇀ u in X, and un → u a.e. in RN , we have Z II = N φK [aij (x, un ) − aij (x, u)]Di uDj Tk (un − u) R Z + φK aij (x, u)Di uDj Tk (un − u) RN
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≤
N Z X
RN
i,j=1
Z ≤C
φK |Dj Tk (un − u)|2
1/2
·
φK [aij (x, un ) − aij (x, u)]2 |Di u|2
RN N X
i,j=1
Z
79
1/2
+ o(1)
φK [aij (x, un ) − aij (x, u)]2 |Di u|2
RN
1/2
+ o(1) → 0.
The proof of III → 0 is similar. Finally, since un → u in L2 (K) and Lq (K), we have 1/2 1/2 Z Z 2 2 IV ≤ → 0, φK c(x)|Tk (un − u)| φK c(x)|un | RN
RN
and similarly,
V ≤
Z
φK |un |q
RN
′
1/q′ Z
φK |Tk (un − u)|q RN
1/q
→ 0,
where q ′ = q/(q − 1). VI → 0 obviously. In conclusion, we have Z φK |DTk (un − u)|2 ≤ Ck.
(2.4)
RN
Define Ak,n := {x ∈ K : |un (x) − u(x)| > k}, Bk,n = K \ Ak,n . Then |Ak,n | → 0 as n → ∞ since un (x) → u(x) a.e. in K. Let en (x) := aij (x, un )Di (un − u)Dj (un − u). Then for 0 < s < 1, by H¨ older inequality, we have Z s Z Z Z Z esn = esn + esn ≤ en |Ak,n |1−s + K
Ak,n
Bk,n
Ak,n
Bk,n
en
s
|Bk,n |1−s .
Combining this and (2.4), we obtain, as k → 0, Z lim sup esn ≤ (Ck)s |K|1−s → 0, n→∞
K
which implies that en (x) → 0 a.e. in K. Thus we conclude that Dun → Du in L2 (K), which implies also that Dun (x) → Du(x) a.e. in K. Now covering RN with a sequence of compact sets with finite measure, we thus deduce that Dun (x) → Du(x) a.e. in RN . Proposition 2.6 Assume that (a1 )–(a3 ), (c) hold, then every bounded (CPS)c sequence {un } of I converges strongly in X to u ∈ X. Proof By Proposition 2.5, we have Dun (x) → Du(x) a.e. in RN , this make it possible for us to use special test functions and employ Fatou’s lemma to prove, like [11, Lemma 2.3], that hI ′ (u), φi = 0,
∀φ ∈ X ∩ L∞ (RN ).
(2.5)
We present the proof of (2.5) in the end of this section for the sake of completeness. Especially, we have hI ′ (u), ui = 0 by approximation. Then by a standard argument, we can prove that un converges strongly to u in X.
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In fact, first, from (2.2), we have Z h i (1 − µ)aij (x, un )Di un Dj un + c(x)u2n lim sup N n→∞ R Z g(un )un = lim sup hyn , un i + n→∞ RN Z h i 1 − un ∂s aij (x, un )Di un Dj un + µaij (x, un )Di un Dj un . RN 2
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(2.6)
Note that we have hyn , un i → 0, and that the imbedding X ֒→ Lq (RN ) is compact, we have Z Z lim g(un )un = g(u)u. n→∞
RN
RN
Moreover, (a3 ) implies that
1 un ∂s aij (x, un )Di un Dj un + µaij (x, un )Di un Dj un ≥ 0. 2 So that we can employ Fatou’s lemma to deduce from (2.6) and hI ′ (u), ui = 0 that Z h i lim sup (1 − µ)aij (x, un )Di un Dj un + c(x)u2n n→∞ RN Z Z h i 1 u∂s aij (x, u)Di uDj u + µaij (x, u)Di uDj u ≤ g(u)u − N RN 2 ZR h i = (1 − µ)aij (x, u)Di uDj u + c(x)u2 .
(2.7)
(2.8)
RN
Second, by Fatou’s lemma,
Z
c(x)u2 ≤ lim inf n→∞
RN
Z
aij (x, u)Di uDj u ≤ lim inf n→∞
RN
Z
Z
RN
c(x)u2n ,
aij (x, un )Di un Dj un .
RN
Since c(x)u2n and aij (x, un )Di un Dj un are nonnegative, we conclude that Z Z c(x)u2 = lim inf c(x)u2n , n→∞
RN
Z
aij (x, u)Di uDj u = lim inf n→∞
RN
Adding lim sup n→∞
R
RN
−c(x)u2n = lim
n→∞
Likewise, we have
Z
R
RN
RN
Z
aij (x, un )Di un Dj un .
RN
−c(x)u2 and (2.8), we get Z
aij (x, un )Di un Dj un =
RN
aij (x, u)Di uDj u.
(2.9)
RN
lim
n→∞
Z
RN
c(x)u2n
=
Z
c(x)u2 .
(2.10)
RN
Third, since aij ∈ L∞ (RN × R), by Lebesgue’s dominated convergence theorem, we have Z Z lim aij (x, un )Di uDj u = aij (x, u)Di uDj u. (2.11) n→∞
RN
RN
We claim that
lim
n→∞
Z
RN
aij (x, un )Di un Dj u =
Z
RN
aij (x, u)Di uDj u.
(2.12)
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In fact, by Lebesgue’s dominated convergence theorem, for 1 ≤ i, j ≤ N , we have aij (x, un )Dj u → aij (x, u)Dj u strongly in L2 (RN ). Thus H¨ older’s inequality gives Z lim aij (x, un ) − aij (x, u) Dj uDi un = 0. n→∞ N R
Since by the weak convergence of un ⇀ u in X, Z Z lim aij (x, u)Di un Dj u = n→∞
RN
aij (x, u)Di uDj u,
RN
thus we conclude that (2.12) hold. Finally, by the weak convergence of un ⇀ u in X, we have Z Z lim c(x)un u = c(x)u2 . n→∞
RN
(2.13)
RN
Through the above analysis, we conclude from (2.9)–(2.13) that Z Z α |D(un − u)|2 + c(x)(un − u)2 RN RN Z Z ≤ aij (x, un )Di (un − u)Dj (un − u) + c(x)(un − u)2 RN RN Z Z ≤ aij (x, un )Di un Dj un + aij (x, un )Di uDj u RN RN Z Z −2 aij (x, un )Di un Dj u + c(x)(un − u)2 → 0. RN
RN
This implies the strong convergence of {un } and completes the proof.
Remark 2.7 Assume that (a1 )–(a3 ) and (c) hold, {un } is a bounded (CPS)c sequence of I converges weakly to u ∈ X. Then by an argument similar to the proof of (2.9) and (2.10), we can prove that Z Z lim un ∂s aij (x, un )Di un Dj un = u∂s aij (x, u)Di uDj u. n→∞
RN
RN
In fact, on one hand, the convergence of g(un )un → g(u)u in L1 (RN ), yn → 0 in X ∗ , and (2.9)–(2.10) imply that Z Z lim sup un ∂s aij (x, un )Di un Dj un ≤ u∂s aij (x, u)Di uDj u. n→∞
RN
RN
On the other hand, (2.7) ensures that we can employ Fatou’s lemma and (2.9) to deduce that Z Z u∂s aij (x, u)Di uDj u ≤ lim inf un ∂s aij (x, un )Di un Dj un . n→∞
RN
RN
Now we prove equality (2.5). The crucial point is to dominate the integral Z φ∂s aij (x, un )Di un Dj un .
(2.14)
Ω− n
This can be done by modifying the test function φ = ϕ exp{−M u+ n } in [5], where ϕ ∈ X ∩ L∞ (RN ) and M > 0 sufficiently large. Proof of (2.5) Let H ∈ [0, 1] be a smooth cutoff function on R with H = 1 on [− 12 , 21 ], H = 0 outside [−1, 1], and |H ′ | ≤ C for some constant C > 0. Define s+ = max{s, 0}, s− =
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max{−s, 0} and let 0 ≤ ϕ ∈ C0∞ (RN ). According to (a1 )–(a2 ), there exists M > 0 such that |∂s aij (x, s)ξi ξj | ≤ 2M aij (x, s)ξi ξj . For k > 0 large, we take φ = ϕH(un /k) exp{−M T2k [(un + k)+ ]} in (2.1), where T2k is defined similar to Tk in the proof of Proposition 2.5. Then Z 0= aij (x, un )Di un Dj ϕ · H(un /k) exp{−M T2k [(un + k)+ ]} RN Z 1 + aij (x, un )Di un Dj un · ϕH ′ (un /k) exp{−M T2k [(un + k)+ ]}} RN k Z − M aij (x, un )Di un Dj un · ϕH(un /k) exp{−M T2k [(un + k)+ ]} N ZR 1 + ∂s aij (x, un )Di un Dj un · ϕH(un /k) exp{−M T2k [(un + k)+ ]} RN 2 Z + c(x)un ϕH(un /k) exp{−M T2k [(un + k)+ ]} RN Z − g(un )ϕH(un /k) exp{−M T2k [(un + k)+ ]} RN
−hyn , ϕH(un /k) exp{−M T2k [(un + k)+ ]}i.
(2.15)
Note that 1 ∂s aij (x, un )Di un Dj un · ϕH(un /k) exp{−M T2k [(un + k)+ ]} 2 −M aij (x, un )Di un Dj un · ϕH(un /k) exp{−M T2k [(un + k)+ ]} ≤ 0, and by (a2 ), Z
RN
C 1 ′ + aij (x, un )Di un Dj un · ϕH (un /k) exp{−M T2k [(un + k) ]} ≤ , k k
by Fatou’s lemma, we can take lim sup in (2.15) and get Z C aij (x, u)Di uDj ϕ · H(u/k) exp{−M T2k [(u + k)+ ]} − ≤ k RN Z − M aij (x, u)Di uDj u · ϕH(u/k) exp{−M T2k [(u + k)+ ]} RN Z 1 + ∂s aij (x, u)Di uDj u · ϕH(u/k) exp{−M T2k [(u + k)+ ]} 2 N ZR + c(x)uϕH(u/k) exp{−M T2k [(u + k)+ ]} RN Z − g(u)ϕH(u/k) exp{−M T2k [(u + k)+ ]}.
(2.16)
RN
Choose ϕ exp{M T2k [(u + k)+ ]} instead of ϕ in (2.16), and then let k → ∞, we get Z Z Z Z 1 0≤ aij (x, u)Di uDj ϕ + ϕ∂s aij (x, u)Di uDj u + c(x)uϕ − g(u)ϕ. RN RN 2 RN RN
Now we take φ = ϕH(un /k) exp{−M T2k [(un − k)− ]} in (2.1) to obtain the opposite inequality. Thus we have hI ′ (u), ϕi = 0 holds for ϕ ≥ 0 and holds for ϕ ≤ 0 similarly. Then it holds also for general ϕ ∈ X ∩ L∞ (RN ) by approximation. Remark 2.8 The main idea of the proof of (2.5) is to truncate the part in (2.14) that is out of control. Note that Dj Tk (un ) ≡ 0 on the set Ak,n (RN ) := {x ∈ RN : |un (x) − u(x)| > k}. Another method to prove (2.5) is to add additional integral of the principal part to dominate
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(2.14). Let 0 ≤ ϕ ∈ C0∞ (RN ) and let φ = ϕ exp{−M T2k [(u + k)+ ]} for k > 0. According to the proof of Proposition 3.3 in [13], we have lim kun − Tk (un )kX ≤ εk , where 0 < εk → 0 as n→∞
k → ∞. Then by (a2 )–(a3 ), we have Z 1 ∂s aij (x, un )Di un Dj un · ϕ exp{−M T2k [(u + k)+ ]} RN 2 Z − M aij (x, un )Di un Dj Tk (un ) · ϕ exp{−M T2k [(u + k)+ ]} N Z R 1 = ∂s aij (x, un )Di un Dj un · ϕ exp{−M T2k [(u + k)+ ]} RN 2 Z − M aij (x, un )Di un Dj un · ϕ exp{−M T2k [(u + k)+ ]} RN Z + M aij (x, un )Di (un − Tk (un ))Dj (un − Tk (un )) · ϕ exp{−M T2k [(u + k)+ ]} RN
≤ Cεk .
Thus we can take φ as test function in (2.1) and obtain (2.5) similarly.
3
Proof of Main Results
In this section, we prove the existence and nonexistence of weak solutions of (1.1). Proof of Theorem 1.1 Theorem 2.1 applied. First, Proposition 2.6 shows that I satisfies (PS)c condition. Secondly, we prove that I has the geometrical conditions stated in Theorem 2.1. In fact, we have I(0) = 0. For every u ∈ X, by (a2 ) and Sobolev’s inequality, we have 1 I(u) ≥ min (α, 1)kuk2X − λCkukqX , 2 where C > 0 is a constant. Thus there exist r > 0, δ > 0 such that I(u) ≥ δ for kukX = r. Now let ϕ0 ∈ X be such that ϕ0 (x) > 0 and kϕ0 kX = 1 and let t ∈ R+ , then by (a2 ) again, t2 max (β, 1)kϕ0 k2X − λtq kϕ0 kqq < 0 2 for t large. According to Theorem 2.1, I admit a critical point u ∈ X. I(tϕ0 ) ≤
Remark 3.1 (i) Denote u+ = max{u, 0}. Replacing |u|q−2 u by |u|q−2 u+ in eq. (1.1), the solutions to the equation can be proved to be nonnegative. (ii) Note that the measure of the set Ak = {x ∈ RN : |u(x)| > k} is finite for k > 0, the solution to (1.1) is in fact belongs to X ∩ L∞ (RN ). Indeed, if 2 ≤ q < 2∗ , it can be deduced as in [12]. If q = 2∗ , the same device as the proof for Theorem 4.1 in [18] can be applied. In order to prove the nonexistence result Theorem 1.2, we empoy Pohozaev’s identity in [27] (see also [28]). Let Ω ⊂ RN be open and F (x, u, r) : Ω × R × RN be a functional of class C 1 , we consider the following equation div{Fr (x, u, Du)} = Fu (x, u, Du),
(3.1)
here we write Du = (∂u/∂x1 , · · · , ∂u/∂xN ), Fu = ∂F/∂u, Fxi = ∂F/∂xi and Fri = ∂F/∂ri , r = (r1 , · · · , rN ). Assume that F (x, 0, 0) = 0. Let a(x) ∈ C 1 (Ω) ∩ C(Ω), h(x) ∈ C 1 (Ω, RN )
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and u ∈ C 1 (Ω) ∩ C(Ω) be a solution to problem (3.1), then we have the following Pohozaev’s identity. Z F (x, 0, Du) − Di uFri (x, 0, Du) (h · ν)ds Z∂Ω Z Z = F (x, u, Du)divh + hi Fxi (x, u, Du) − Dj uDi hj + uDi a(x) Fri (x, u, Du) Ω Ω Ω Z − a(x) Di uFri (x, u, Du) + uFu (x, u, Du) . (3.2) Ω
Proof of Theorem 1.2 to (1.1). Let
Assume on the contrary that u ∈ X ∩C 1 (RN ) is a weak solution
1 1 aij (u)Di uDj u + c(x)u2 − G(u), 2 2 and let a be independent on x, h = x in (3.2), we get Z Z N − 2 a −a aij (u)Di uDj u − ua′ (u)Di uDj u 0= 2 2 RN ij RN Z N Z N Z 1 + −a c(x)u2 + (x · ∇c(x))u2 − −a |u|q . 2 2 q N N N R R R F (x, u, Du) =
Set a = N/q. Note that by assumption, we have (1 − 2q )N c(x) + (x · ∇c(x)) ≥ 0, thus Z Z q 1 1− ∗ aij (u)Di uDj u + ua′ij (u)Di uDj u ≥ 0. 2 2 N N R R Therefore if u satisfies (1.4), then (3.3) implies that u ≡ 0. This completes the proof.
(3.3)
Remark 3.2 The parameter µ in (a3 ) is, in fact, should not greater than 1 − 2q∗ according to Theorem 1.2. Moreover, as q tends to 2∗ , µ tends to 0. This is partially consistent with [10, Theorem 5.1], where the following result was proved: if bij (s), ∂s bij (s) ∈ L∞ (R), bij (s)ξi ξj ≥ γ|ξ|2 , γ ∈ (0, 1] and s∂s bij (s)ξi ξj < 0, ∀s 6= 0, then the equation ∗ 1 −Dj (bij (u)Di u) + ∂s bij (u)Di uDj u = |u|2 −2 u 2 has no nontrivial solution in H01 (Ω) ∩ L∞ (Ω), where Ω ⊂ RN (N ≥ 3) is bounded and starshaped.
Now we are ready to prove Theorem 1.4. We prove that problem (1.3) has nontrivial solutions even if κ is less than zero. Proof of Theorem 1.4 By direct computation, we have u −∆u − κ ∆(1 + |u|2 )1/2 2(1 + |u|2 )1/2 κ|u|2 κu = − 1+ ∆u − |∇u|2 2(1 + |u|2 ) 2(1 + |u|2 )2 and
−div Thus we get
κ|u|2 κ|u|2 κu 1+ ∇u = − 1 + ∆u − |∇u|2 . 2(1 + |u|2 ) 2(1 + |u|2 ) (1 + |u|2 )2 u −∆u − κ ∆(1 + |u|2 )1/2 2(1 + |u|2 )1/2 κ|u|2 κu = −div 1 + ∇u + |∇u|2 , 2(1 + |u|2 ) 2(1 + |u|2 )2
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κ|u|2 that is, we have aij (x, u) = 1 + 2(1+|u| 2 ) δij . Since (a1 ) and (a2 ) are satisfied obviously, we only need to prove that (a3 ) holds. Case κ ≥ 0. We prove that κ|u|2 κ|u|2 2 0 ≤ u∂s aij (x, u)Di uDj u = |∇u| ≤ ν 1 + |∇u|2 , (1 + |u|2 )2 2(1 + |u|2 )
that is,
ν+
(ν − 2)κ|u|2 + νκ|u|4 ≥ 0. 2(1 + |u|2 )2
16ν This inequality holds either (i) ν ≥ 2 and κ ≥ 0 or (ii) 0 < ν < 2 and 0 ≤ κ ≤ (2−ν) 2 . Since 16ν is increasing in ν ∈ (0, 2) and tends to +∞ as ν tends to 2, we thus draw a conclusion (2−ν)2
that the inequality holds either (i) q ≥ 4 and κ ≥ 0 or (ii) 2 < q < 4 and 0 ≤ κ < 16(q−2) (4−q)2 . Case κ < 0. We prove that κ|u|2 κ|u|2 2 0 > u∂s aij (x, u)Di uDj u = |∇u| ≥ −2µ 1 + |∇u|2 , (1 + |u|2 )2 2(1 + |u|2 ) that is,
2µ + κ
(µ + 1)|u|2 + µ|u|4 ≥ 0. (1 + |u|2 )2
q 8µ This inequality holds when κ ≥ − (1+µ) 2 > −2, here we take µ = (1 − 2∗ ) ∈ (0, 1). In this case, we need to ensure that problem (1.3) is still elliptic, that is, for such κ, there exists γ > 0 such that κ|u|2 aij (x, u)Di uDj u = 1 + |∇u|2 ≥ γ|∇u|2 . 2(1 + |u|2 ) κ|u|2 Since 1 + 2(1+|u| is decreasing in |u|, the above inequality is equivalent to 1 + κ2 ≥ γ, i.e., 2) κ ≥ 2(γ − 1). Since κ > −2, we obtain the existence of such γ.
Proof of Corollary 1.7 Under the assumptions of Corollary 1.7, assume that there is still a solution 0 6= u ∈ X ∩ C 1 (RN ) to (1.3). Then Pohozaev’s inequality, with a = N/2∗ and h = x in (3.2), implies that Z Z Z N 1 ′ 2 ua (x · ∇c(x))u2 ≥ 0, (u)D uD u = c(x)u + i j 22∗ RN ij 2 N N R R where a′ij (u) =
κ|u|2 (1+|u|2 )2 δij .
But this is impossible unless u ≡ 0.
Acknowledgements The first author thanks the staff of Chern Institute of Mathematical, Nankai University and Professor Zhiqiang Wang for their hospitality. Part of this work was done when the first author was visiting there. References [1] Borovskii A V, Galkin A L. Dynamical modulation of an ultrashort high-intensity laser pulse in matter. JETP, 1993, 77: 562–573 [2] Aouaoui S. Multiplicity of solutions for quasilinear elliptic equations in RN . J Math Anal Appl, 2010, 370: 639–648 [3] Bartsch T, Pankov A, Wang Z Q. Nonlinear Schr¨ odinger equations with steep potential well. Comm Contemp Math, 2001, 4: 549–569 [4] do O J M, Severo U. Quasilinear Schr¨ odinger equations involving concave and convex nonlinearities. Comm Pure Appl Anal, 2009, 8: 621–644
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