Acta Mathematica Scientia 2017,37B(6):1870–1880 http://actams.wipm.ac.cn
EXISTENCE OF NONTRIVIAL SOLUTIONS FOR ¨ GENERALIZED QUASILINEAR SCHRODINGER EQUATIONS WITH CRITICAL OR SUPERCRITICAL GROWTHS∗
o )
Quanqing LI (
Department of Mathematics, Honghe University, Mengzi 661100, China E-mail :
[email protected]
Çm)
Xian WU (
†
Department of Mathematics, Yunnan Normal University, Kunming 650092, China E-mail :
[email protected] Abstract In this paper, we study the following generalized quasilinear Schr¨ odinger equations with critical or supercritical growths −div(g 2 (u)∇u) + g(u)g ′ (u)|∇u|2 + V (x)u = f (x, u) + λ|u|p−2 u, x ∈ RN , where λ > 0, N ≥ 3, g : R → R+ is a C 1 even function, g(0) = 1, g ′ (s) ≥ 0 for all s ≥ 0, ′ ∗ lim |s|g(s) α−1 := β > 0 for some α ≥ 1 and (α − 1)g(s) > g (s)s for all s > 0 and p ≥ α2 . |s|→+∞
Under some suitable conditions, we prove that the equation has a nontrivial solution for small λ > 0 using a change of variables and variational method. Key words
quasilinear Schr¨ odinger equations; critical or supercritical growths; variational methods
2010 MR Subject Classification
1
35J20; 35J70; 35P05; 35P30; 34B15; 58E05; 47H04
Introduction and Preliminaries
Consider the following generalized quasilinear Schr¨odinger equations with critical or supercritical growths −div(g 2 (u)∇u) + g(u)g ′ (u)|∇u|2 + V (x)u = f (x, u) + λ|u|p−2 u, x ∈ RN ,
(1.1)
where λ > 0, N ≥ 3, g : R → R+ is a C 1 even function, g(0) = 1, g ′ (s) ≥ 0 for all s ≥ 0, ′ ∗ lim |s|g(s) α−1 := β > 0 for some α ≥ 1 and (α − 1)g(s) > g (s)s for all s > 0 and p ≥ α2 .
|s|→+∞
∗ Received
February 29, 2016; revised April 29, 2017. This work was supported in part by the National Natural Science Foundation of China (11501403; 11461023) and the Shanxi Province Science Foundation for Youths under grant 2013021001-3. † Corresponding author: Xian WU.
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The equation is related to the existence of solitary wave solutions for quasilinear Schr¨odinger equations izt = −∆z + W (x)z − k(x, |z|)z − ∆l(|z|2 )l′ (|z|2 )z, (1.2) where z : R × RN → C, W : RN → R is a given potential, l : R → R and k : RN × R → R are suitable functions. The form of (1.2) was derived as models of several physical phenomena corresponding to various types of l(s). For instance, the case l(s) = s models the time evolution of the condensate wave function in super-fluid film [13, 16], and is called the 1 superfluid film equation in fluid mechanics by Kurihara [13]. In the case l(s) = (1+s) 2 , problems (1.2) models the self-channeling of a high-power ultra short laser in matter, the propagation of a high-irradiance laser in a plasma creates an optical index depending nonlinearly on the light intensity and this leads to interesting new nonlinear wave equations, see [2, 6, 7, 24]. For more physical motivations and more references dealing with applications, we can refer to [1, 3, 10, 11, 15, 21, 23, 25] and references therein. Set z(t, x) = exp(−iEt)u(x), where E ∈ R and u is a real function. Then (1.2) can be reduced to the corresponding equation of elliptic type (see [4]) −∆u + V (x)u − ∆l(u2 )l′ (u2 )u = f (x, u), x ∈ RN , where f (x, u) = k(x, |u|)u. If we take g 2 (u) = 1 +
(1.3)
[(l(u2 ))′ ]2 , 2
then (1.1) turns into (1.3) (see [27]). Moreover, problem (1.1) also arises in biological models and propagation of laser beams when g(u) is a positive constant (see [12, 14]). In (1.1), if we set g 2 (u) = 1 + 2u2 , then we get the superfluid film equation in plasma physics −∆u + V (x)u − ∆(u2 )u = f (x, u) + λ|u|p−2 u, x ∈ RN ; if we set g 2 (u) = 1 +
2
u 2(1+u2 ) ,
(1.4)
then we get the equation 1
−∆u + V (x)u − [∆(1 + u2 ) 2 ]
u 1
2(1 + u2 ) 2
= f (x, u) + λ|u|p−2 u, x ∈ RN ,
(1.5)
which models the self-channeling of a high-power ultrashort laser in matter. In the past, the research on the existence of solitary wave solutions of Schr¨odinger equation (1.2) is for some given special function l(s). In this paper, we will use a unified new variable replacement to study equation (1.2), which constructed by Shen and Wang in [27]. Here we point out that the authors in [27] studied the existence of nontrivial solutions for the following generalized quasilinear Schr¨ odinger equations with subcritical growth −div(g 2 (u)∇u) + g(u)g ′ (u)|∇u|2 + V (x)u = h(u), x ∈ RN , where |h(s)| ≤ C(1 + g(s)|G(s)|p−1 ) for all s ∈ R and 2 < p < 2∗ ; in [8], Deng and Peng and Yan proved that the existence of positive solutions for generalized quasilinear Schr¨odinger equations with critical growth ∗
−div(g 2 (u)∇u) + g(u)g ′ (u)|∇u|2 + V (x)u = h(x, u) + g(u)|G(u)|2 −2 G(u), x ∈ RN , Ru where G(u) := 0 g(s)ds. However, the assumptions in [8] imposed on g(s) and h(x, s) depend mutually on each other. From the above statements we know that what they say the critical
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exponent is 2∗ . But, as observed in [19], the number 22∗ behaves like a critical exponent for (1.4). Consequently, equation (1.1) maybe possess the other critical exponent. Recently, by adding a proper condition, Deng et al. in [9] found that α2∗ is the critical exponent for problem (1.1) if lim sg(s) α−1 := β > 0 for some α ≥ 1, and studied the existence of positive solutions for s→+∞
the equation (1.1) with λ = 1 and p = α2∗ . Moreover, they pointed out that if p ≥ α2∗ and x · ∇V (x) ≥ 0 in RN , then equation (1.1) with λ = 1 and f (x, u) ≡ 0 in RN × R has no positive solution. Inspired by them, in this paper, we study the existence of nontrivial solution for the equation (1.1) with λ > 0 and p ≥ α2∗ . The energy functional associated with (1.1) is Z Z Z Z 1 1 λ Iλ (u) = g 2 (u)|∇u|2 dx + V (x)u2 dx − F (x, u)dx − |u|p dx, 2 RN 2 RN p N N R R Ru 1 N where F (x, u) := 0 f (x, t)dt. However, Iλ is not well defined in H (R ) because of the term R g 2 (u)|∇u|2 dx. To overcome this difficulty, we make a change of variable constructed by RN Ru Shen and Wang in [27]: v := G(u) := 0 g(t)dt. Then we obtain Z Z 1 1 Jλ (v) = |∇v|2 dx + V (x)|G−1 (v)|2 dx 2 RN 2 RN Z Z λ −1 − F (x, G (v))dx − |G−1 (v)|p dx. p RN RN We say u is a solution of (1.1) if Z hIλ′ (u), ϕi = [g 2 (u)∇u∇ϕ + g(u)g ′ (u)|∇u|2 ϕ + V (x)uϕ − f (x, u)ϕ − λ|u|p−2 uϕ]dx RN
=0
(1.6)
1 ψ. By [27] we know that (1.6) is equivalent to for all ϕ ∈ C0∞ (RN ). Let ϕ = g(u) Z G−1 (v) f (x, G−1 (v)) |G−1 (v)|p−2 G−1 (v) ′ hJλ (v), ψi = ∇v∇ψ + V (x) ψ− ψ−λ ψ dx g(G−1 (v)) g(G−1 (v)) g(G−1 (v)) RN =0
for all ψ ∈ C0∞ (RN ). Therefore, in order to find the nontrivial solution of (1.1), it suffices to study the existence of the nontrivial solutions of the following equations −∆v + V (x)
G−1 (v) f (x, G−1 (v)) |G−1 (v)|p−2 G−1 (v) − − λ = 0. g(G−1 (v)) g(G−1 (v)) g(G−1 (v))
In order to reduce the statements for main results, we list the assumptions as follows. (V) +∞ > V∞ := lim V (x) ≥ V (x) ≥ V0 := inf V (x) > 0 for all x ∈ RN . |x|→∞
x∈RN
(f1 ) f ∈ C(R × R, R) and there exists 2α < q < α2∗ such that N
|f (x, t)| ≤ C(1 + |t|q−1 ) for all (x, t) ∈ RN × R. (f2 ) f (x, t) = o(|t|) uniformly in x ∈ RN as |t| → 0. Rt (f3 ) f (x, t)t ≥ qF (x, t) := q 0 f (x, s)ds ≥ 0 for all (x, t) ∈ RN × R. Set E = H 1 (RN ). Our main result is the following.
(1.7)
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Theorem 1.1 Suppose that (V) and (f1 )–(f3 ) are satisfied. Then there exists some λ0 > 0 such that for λ ∈ (0, λ0 ], equation (1.1) admits a nontrivial solution. Remark 1.1 If we take g 2 (u) = 1 + 2κu2 with κ > 0, then α = 2 and Theorem 1.1 implies that there exists some λ0 > 0 such that for λ ∈ (0, λ0 ], the quasilinear Schr¨odinger equation −∆u + V (x)u − κ∆(u2 )u = f (x, u) + λ|u|p−2 u has a nontrivial solution. At this point, if we again take f (x, u) = |u|q−2 u with 4 < q < 22∗ , then for λ ∈ (0, λ0 ], the quasilinear Schr¨ odinger equation −∆u + V (x)u − κ∆(u2 )u = |u|q−2 u + λ|u|p−2 u has a nontrivial solution. Hence our result generalizes Theorem 1.1 in [28]. Remark 1.2 In [9], Deng et al. studied the existence of nontrivial solutions for critical problem ∗ −div(g 2 (u)∇u) + g(u)g ′ (u)|∇u|2 + V (x)u = h(x, u) + |u|α2 −2 u, where h satisfies the following assumptions. (h1 ) h(x, t) ≥ 0 is differentiable in t ∈ [0, +∞) for all x ∈ RN and continuous in x ∈ RN for all t ∈ [0, +∞). Moreover, we extend h(x, t) ≡ 0 for all t ∈ (−∞, 0), x ∈ RN . h(x,t) (h2 ) lim |t|h(x,t) = 0 uniformly in x ∈ RN . α2∗ −1 = 0 and lim t t→+∞
t→0+
(h3 ) There exists µ ∈ (2, 2∗ ) such that for any t > 0, there holds h′t (x, t)t ≥ (αµ − 1)h(x, t). ¯ (h4 ) lim h(x, t) = h(t) uniformly on any compact subset of [0, ∞) and there exists a |x|→+∞
constant ν > 2 such that for any ε > 0 we can find Cε > 0 satisfying ¯ ≥ −e−ν|x| (εtα + Cε tαp )tα−1 h(x, t) − h(t) for all x ∈ RN , t ≥ 0, where p ∈ (1, 2∗ − 1). Clearly, our theorem is different from the main result in [9]. Moreover, we have considered the supercritical case. 2 Remark 1.3 If we take g 2 (u) = 1 + κ (r+1) |u|2r with κ > 0 and r > 0, then α = r + 1 2 and equation (1.1) can be reduced to the following equation −∆u + V (x)u − κ
r+1 ∆(|u|r+1 )|u|r−1 u = f (x, u) + λ|u|p−2 u. 2
Especially, if we take r = 2γ − 1 with γ > 12 , i.e., g 2 (u) = 1 + 2κγ 2 |u|2(2γ−1) , then we get the quasilinear Schr¨odinger equation with a parameter −∆u + V (x)u − κγ∆(|u|2γ )|u|2γ−2 u = f (x, u) + λ|u|p−2 u.
(1.8)
In [17], by a constrained minimization argument, Liu and Wang obtained the existence of solution for (1.8) with f (x, u) ≡ 0 and p ∈ (4γ − 1, 2γ2∗ − 1). Moreover, under some suitable conditions on f , Wu [29] obtained the existence of positive solutions, negative solutions and sequence of high energy solutions for (1.8) with λ = 0. Clearly, our result is also different from the above results.
2
Proof of Theorem 1.1 To begin with, we give some lemmas. Lemma 2.1 For the functions g, G and G−1 , the following properties hold
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the functions G(·) and G−1 (·) are strictly increasing and odd; G(s) ≤ g(s)s ≤ αG(s) for all s ≥ 0; αG(s) ≤ g(s)s ≤ G(s) for all s ≤ 0; g(G−1 (s)) ≥ g(0) = 1 for all s ∈ R; G−1 (s) is nonincreasing on (0, +∞) and nondecreasing on (−∞, 0); s 1 −1 |G (s)| ≤ g(0) |s| = |s| for all s ∈ R; |G−1 (s)| 1 g(G−1 (s)) ≤ g2 (0) |s| = |s| for all s ∈ R; −1 G (s)s G−1 (s)s −1 (s)|2 ≤ α g(G −1 (s)) for g(G−1 (s)) ≤ |G G−1 (s) 1 lim = g(0) = 1 and s |s|→0
−1 lim G s (s) |s|→∞
=
all s ∈ R;
1 , if g is bounded, g(∞)
0,
if g is unbounded;
(9) |s|α ≤ α β |G(s)| for all s ∈ R. Proof Properties (1)–(3) are obvious. By (2), we have G−1 (s) ′ s − G−1 (s)g(G−1 (s)) = ≤0 s g(G−1 (s))s2
for all s > 0 and
G−1 (s) ′ s
=
s − G−1 (s)g(G−1 (s)) ≥0 g(G−1 (s))s2
for all s < 0. Consequently, we obtain (4). By mean value theorem and (3), one has |G−1 (s)| = |G−1 (s) − G−1 (0)| =
1 1 |s| ≤ |s| g(G−1 (θs)) g(0)
for all s ∈ R, where θ ∈ (0, 1), i.e., (5) is proved. Obviously, (6) is a consequence of (3) and (5). Moreover, (7) is a consequence of (2). Using L’ Hospital’ s rule, we know that (8) is satisfied. Finally, by (2) we derive that G(s) ′ g(s)s2 − αG(s)s = ≤0 |s|α |s|α+2 g(s) α−1 s→+∞ |s|
for s > 0. Combining lim know that
= β > 0 we obtain |s|α ≤
|s|α ≤
α β G(s),
∀s > 0. By (1) we easily
α α G(−s) = |G(s)|, ∀s < 0. β β
Hence we obtain (9). This completes the proof. In order to study nontrivial solutions of (1.1), we define a function |t|p−2 t, if |t| ≤ M, φ(t) = M p−q |t|q−2 t, if |t| > M,
where M > 0. Set hλ (x, t) = λφ(t) + f (x, t) for all (x, t) ∈ RN × R. Then φ ∈ C(R, R), Rt φ(t)t ≥ qΦ(t) := q 0 φ(s)ds ≥ 0 and |φ(t)| ≤ M p−q |t|q−1 for all t ∈ R. Moreover, hλ (x, t) satisfies the following conditions. (h1 ) hλ ∈ C(RN ×R, R) and |hλ (x, t)| ≤ λM p−q |t|q−1 +C(1+|t|q−1) for all (x, t) ∈ RN ×R. (h2 ) hλ (x, t) = o(|t|) uniformly in x ∈ RN as |t| → 0. Rt (h3 ) hλ (x, t)t ≥ qHλ (x, t) := q 0 hλ (x, s)ds ≥ 0 for all (x, t) ∈ RN × R.
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By (h1 )–(h3 ), (V) and mountain pass theorem, using a standard argument (see [5, 27]), we can prove that the equation −∆v + V (x)
hλ (x, G−1 (v)) G−1 (v) − =0 g(G−1 (v)) g(G−1 (v))
(2.1)
has a nontrivial solution vλ ∈ E. Let Z Z Z 1 1 2 −1 2 ξλ (v) = |∇v| dx + V (x)|G (v)| dx − Hλ (x, G−1 (v))dx 2 RN 2 RN N R Z Z Z 1 1 2 −1 2 = |∇v| dx + V (x)|G (v)| dx − λ Φ(G−1 (v))dx 2 RN 2 RN RN Z − F (x, G−1 (v))dx. RN
Then
ξλ′ (vλ )
= 0 and ξλ (vλ ) = cλ := inf
sup ξλ (γ(t)), where
γ∈Γλ t∈[0,1]
Γλ := {γ ∈ C([0, 1], E) : γ(0) = 0, ξλ (γ(1)) < 0}. In the following, set ξ(v) =
1 2
Z
RN
|∇v|2 dx +
1 2
Z
RN
V (x)|G−1 (v)|2 dx −
Z
F (x, G−1 (v))dx
RN
and Γ := {γ ∈ C([0, 1], E) : γ(0) = 0, ξ(γ(1)) < 0} and c := inf sup ξ(γ(t)). Then Γ ⊂ Γλ and cλ ≤ c. γ∈Γ t∈[0,1]
2q cλ and there exists a constant A > 0 Lemma 2.2 The solution vλ satisfies k∇vλ k22 ≤ q−2α 2 independent of λ such that k∇vλ k2 ≤ A. Proof By Lemma 2.1 (2) we have |g(G−1 (vλ ))G−1 (vλ )| ≤ α|vλ |. By the fact that g is a even function and g ′ (s)s < (α − 1)g(s) for all s > 0 we have g ′ (s)s ≤ (α − 1)g(s) for all s ∈ R. Thereby, g ′ (G−1 (vλ ))G−1 (vλ ) |∇[g(G−1 (vλ ))G−1 (vλ )]| = 1 + |∇vλ | ≤ α|∇vλ |, g(G−1 (vλ ))
which implies that g(G−1 (vλ ))G−1 (vλ ) ∈ E. Hence Z Z 1 1 cλ = ξλ (vλ ) = |∇vλ |2 dx + V (x)|G−1 (vλ )|2 dx 2 RN 2 RN Z Z −λ Φ(G−1 (vλ ))dx − F (x, G−1 (vλ ))dx RN
RN
and 0 = hξλ′ (vλ ), g(G−1 (vλ ))G−1 (vλ )i Z Z g ′ (G−1 (vλ ))G−1 (vλ ) 2 = 1+ |∇v | dx + V (x)|G−1 (vλ )|2 dx λ −1 (v )) g(G N N λ R R Z Z −1 −1 −1 −λ φ(G (vλ ))G (vλ )dx − f (x, G (vλ ))G−1 (vλ )dx. RN
RN
Consequently, by (f3 ) we know that qcλ = qξλ (vλ ) = qξλ (vλ ) − hξλ′ (vλ ), g(G−1 (vλ ))G−1 (vλ )i
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Z q Z −α |∇vλ |2 dx + −1 V (x)|G−1 (vλ )|2 dx 2 2 RN RN Z +λ [φ(G−1 (vλ ))G−1 (vλ ) − qΦ(G−1 (vλ ))]dx RN Z + [f (x, G−1 (vλ ))G−1 (vλ ) − qF (x, G−1 (vλ ))]dx RN q Z ≥ −α |∇vλ |2 dx, 2 RN q
≥
2q q−2α cλ
which implies that k∇vλ k22 ≤ Denote
≤
2q q−2α c
g ∗ (x, s) = V (x)s − V (x) and
:= A > 0. This completes the proof.
G−1 (s) f (x, G−1 (s)) + −1 g(G (s)) g(G−1 (s))
1 p−q α α αq −2 φ(G−1 (s)) − M |s| s. g(G−1 (s)) α β q
g ∗∗ (s) = Then G∗ (x, s) :=
Z
s
g ∗ (x, τ )dτ =
0
and
G∗∗ (s) :=
Z
0
Consequently, ξλ (v) =
s
1 V (x)[s2 − |G−1 (s)|2 ] + F (x, G−1 (s)) 2
αα q 1 g ∗∗ (τ )dτ = Φ(G−1 (s)) − M p−q |s| α . q β q
Z Z [|∇v|2 + V (x)v 2 ]dx − G∗ (x, v)dx − λ G∗∗ (v)dx N N N R R R q Z q λ p−q α α |v| α dx. + M q β RN 1 2
Z
Lemma 2.3 The functions g ∗ (x, s), g ∗∗ (s), G∗ (x, s) and G∗∗ (s) enjoy the following properties under (f1 )–(f2 ) ∗ ∗ ∗∗ (1) lim g (x,s) = 0 and lim G s(x,s) = 0 uniformly in x ∈ RN , lim g s(s) = 0 and 2 s |s|→0 G∗∗ (s) = 0; s2 |s|→0 ∗ (2) lim g|s|(x,s) 2∗ −1 |s|→∞
|s|→0
|s|→0
lim
∗∗
lim G 2(s) ∗ |s|→∞ |s| Proof
= 0 and lim
|s|→∞
G∗ (x,s) |s|2∗
g∗∗ (s) 2∗ −1 |s|→∞ |s|
= 0 uniformly in x ∈ RN , lim
= 0 and
= 0. By (f1 )–(f2 ) we know that for any ε > 0, there exists Cε > 0 such that f (x, G−1 (s)) ε|G−1 (s)| Cε |G−1 (s)|q−1 g(G−1 (s)) ≤ g(G−1 (s)) + g(G−1 (s))
(2.2)
for all (x, s) ∈ RN × R. Consequently, Lemma 2.1 (3), (5) and (8) imply g ∗ (x, s) 1 f (x, G−1 (s)) lim = V (x) 1 − 2 + lim =0 s g (0) |s|→0 g(G−1 (s))s |s|→0
uniformly in x ∈ RN . Clearly,
q |φ(G−1 (s))| 1 p−q α α q −1 + M |s| α g(G−1 (s)) α β q M p−q |G−1 (s)|q−1 1 p−q α α q −1 ≤ + M |s| α . g(G−1 (s)) α β
|g ∗∗ (s)| ≤
(2.3)
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Combining with Lemma 2.1 (3) and (5) we have lim
= 0. Moreover, by (2.2), Lemma
2.1 (2), (3), (5) and (9) one has g ∗ (x, s) G−1 (s) s f (x, G−1 (s)) lim =0 ∗ −1 = −V (x) lim ∗ −1 + ∗ 2 −1 2 |s|→∞ |s| |s|→∞ sg(G (s)) |s| |s|→∞ g(G−1 (s))|s|2 −1 lim
g∗∗ (s) 2∗ −1 |s| |s|→∞
uniformly in x ∈ RN . By (2.3), Lemma 2.1 (2) and (9) we obtain lim we have
uniformly in x ∈ RN and
lim
G∗ (x, s) = 0, s2 |s|→0
G∗ (x, s) =0 |s|→∞ |s|2∗
G∗∗ (s) = 0, s2 |s|→0
G∗∗ (s) = 0. ∗ |s|→∞ |s|2
lim
= 0. Similarly,
lim
lim
Hence, (1) and (2) hold. This completes the proof.
Lemma 2.4 There exist two constants B, D > 0 independent of λ such that kvλ kL∞ ≤ B(1 + λ)D . Proof Set T > 2, r > 0 and v˜λT := b(vλ ), where b : R → R is a smooth function satisfying b(s) = s for |s| ≤ T − 1, b(−s) = −b(s); b′ (s) = 0 for s ≥ T and b′ (s) is decreasing in [T − 1, T ]. This means that v˜λT = vλ for |vλ | ≤ T − 1; |˜ vλT | = |b(vλ )| ≤ |vλ | for T − 1 ≤ |vλ | ≤ T ; |˜ vλT | = CT > 0 for |vλ | ≥ T , where T − 1 ≤ CT ≤ T . Moreover, one has 0≤
sb′ (s) ≤ 1, ∀ s 6= 0. b(s)
Let ψ = vλ |˜ vλT |2r . Then ψ ∈ E. Hence, taking ψ as the test function, one has Z [g ∗ (x, vλ ) + λg ∗∗ (vλ )]ψdx N ZR Z α αq Z q λ |vλ | α −2 vλ ψdx = ∇vλ · ∇ψdx + V (x)vλ ψdx + M p−q α β N N N R R ZR Z Z T 2r 2 T 2r 2 ≥ (1 + r)|˜ vλ | |∇vλ | dx + |˜ vλ | |∇vλ | dx + V (x)|vλ |2 |˜ vλT |2r dx |vλ |≤T −1 |vλ |≥T RN Z + [|˜ vλT |2r + 2rvλ b(vλ )b′ (vλ )|˜ vλT |2r−2 ]|∇vλ |2 dx T −1
