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Signed and sign-changing solutions for a Kirchhoff-type equation in bounded domains
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Signed and sign-changing solutions for a Kirchhoff-type equation in bounded domains Sheng-Sen Lu Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, 300071, PR China
a r t i c l e
i n f o
Article history: Received 27 June 2015 Available online xxxx Submitted by J. Xiao Keywords: Kirchhoff-type equation Signed and sign-changing solutions Variational methods
a b s t r a c t The main concern of this article is a Kirchhoff-type equation of the form ⎛ −M ⎝
⎞
|∇u|
2⎠
Δu = λf (u),
Ω
where Ω is a bounded smooth domain in RN with N ≥ 3 and λ is a positive parameter. Under certain assumptions on M and f , the existence results of signed and sign-changing solutions are established for λ large, and when λ converges to infinity the asymptotic behavior of these solutions is also studied. The proofs are based on a careful study of the ground state and least energy nodal solutions of an auxiliary problem, which is constructed by making a refined truncation on M . Furthermore, we get the ground state and least energy nodal solutions, and prove the energy doubling property for all λ > 0 under more restricted assumptions on M and f . © 2015 Elsevier Inc. All rights reserved.
1. Introduction and main results In this paper, we consider the existence of signed and sign-changing solutions for the following problem ⎞ ⎛ ⎧ ⎪ ⎪ ⎨ −M ⎝ |∇u|2 ⎠ Δu = λf (u), in Ω, (Pλ ) ⎪ Ω ⎪ ⎩ u = 0, on ∂Ω, where Ω is a bounded smooth domain in RN with N ≥ 3, λ is a positive parameter, M : R+ → R+ and f : R → R are C 1 functions that satisfy some assumptions which will be stated later on. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jmaa.2015.07.033 0022-247X/© 2015 Elsevier Inc. All rights reserved.
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The class of problem (Pλ ) is called of Kirchhoff type because it comes from an important application in Physic and Engineering. Indeed, if we set M (t) = a + bt and λ = 1 in (Pλ ), then we get the following Kirchhoff problem ⎛ ⎞ − ⎝a + b |∇u|2 ⎠ Δu = f (u), (1.1) Ω
which is related to the stationary analogue of the equation ⎛ ⎞ L 2 2 2 E ∂ u P0 ∂u ⎠ ∂ u = 0, + ρ 2 −⎝ ∂t h 2L ∂x ∂x2 0
presented by G. Kirchhoff in [12]. For more mathematical and physical background, we refer readers to papers [2,3,5,15,18] and the references therein. In the last ten years, by classical variational method, there are many interesting results about the existence and nonexistence of solutions, sign-changing solutions, ground state solutions, the existence of positive solutions and positive ground states, least energy nodal solutions, multiplicity of solutions, semiclassical limit and concentrations of solution to Kirchhoff type problems, see e.g. [1,4,6–11,13,14,16,19–21] and the references therein. 2 One major difficulty that arises in the use of this technique is the growth of the operator M(u )=
t b 2 4 m0 u + 2 u , where M (t) := 0 M (s)ds and m0 , b > 0. This requires to impose 4-superlinear growth on the nonlinearity f , that is, f (t) = |t|p−2 t with p ∈ (4, 2∗ ). But 2∗ := N2N −2 ≤ 4 for N ≥ 4, so it is common to fix N = 3 to circumvent this difficulty in most cases. To the best of our knowledge, only article [21] considers the signed and sign-changing solutions for problem (1.1) with superlinear nonlinearity which just has 4-sublinear growth. Those results are available for bounded smooth domains of RN with N ≥ 3. In that paper, the authors adopt the variational method and invariant sets of descent flow. In [8], under certain assumptions on M and f , the authors show the existence of least energy nodal solution for (Pλ ) on bounded smooth domains Ω ⊂ R3 with λ = 1. This type problems include but are not restricted to the type (1.1), and much closer to problem (Pλ ) we consider here. However, they impose the 4-superlinear Ambrosetti–Rabinowitz type condition on f in that paper that we just try to drop in the present article. We would like to point out that it is the work [6] that gives us the primal idea for dealing with problem (Pλ ) with more general nonlinearity which may not have 4-superlinear growth. That paper concerns the following problem: ⎞ ⎛ ⎧ ⎪ ⎪ ⎨ −M ⎝ |∇u|2 ⎠ Δu = λf (x, u) + |u|2∗ −2 u, in Ω, ⎪ Ω ⎪ ⎩ u = 0, on ∂Ω, where Ω is a bounded smooth domain in RN with N ≥ 3, λ is a positive parameter, M : R+ → R+ is an increasing function with m0 := M (0) > 0, and f (x, t) satisfies the well-known Ambrosetti–Rabinowitz superlinear condition and other some fundamental assumptions. Based on the variational method, an appropriated truncation argument and a priori estimates, the author shows the existence result of positive solution for the above problem and study the asymptotic behavior of that solution when λ converges to infinity. Motivated by the articles [6,8,21], we shall investigate the existence of constant sign and sign-changing solutions for problem (Pλ ). The main differences are the following:
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(a) The truncation explored in [6] cannot be used here directly, because that truncated function is not regular enough even though M is a C 1 class function. Indeed, the truncation in the present paper is more refined and technical, see (2.2) and (2.3) in Section 2. In addition, we succeeded in proving that the nodal Nehari manifold Mλ,θ is nonempty by using an elementary method and more analysis instead of Miranda’s theorem which is adopted in [8], see Lemma 3.3 and its proof in Section 3. (b) Under certain assumptions on M and f , we get a constant sign solution uλ and a nodal solution wλ with the energy estimates Φλ (wλ ) > 2Φλ (uλ ) > 0 for λ large. However, the critical levels of the signed and sign-changing solutions given in [21] for the 4-sublinear case are all negative with no comparison between each other. Besides, the nodal solution wλ that we obtain in this paper has precisely two nodal domains, while [21] gives no information about the number of nodal domains of the sign-changing solution they obtain. (c) Here we also get a general existence result of least energy nodal solutions for problem (Pλ ) under more restricted assumptions on M and f , which covers and generalizes the result in [8], see Remark 5.2 in Section 5. Besides, the asymptotic behavior of the least energy nodal solutions of problem (Pλ ) when λ converges to infinity and the energy doubling property are studied, which are not observed in [8], see Theorem 1.2 below. Before stating our main results, we make the following hypotheses on the function M . The function M : R+ → R+ is of C 1 class and satisfies the following condition: (M1 ) The function M is increasing and m0 := M (0) > 0. A typical example of a function verifying the assumptions (M1 ) is given by M (t) = m0 + bt,
where m0 > 0 and b > 0.
In what follows, we also assume the nonlinearity f is of C 1 class and satisfies (f1 ) For all t = 0, f (t)t > 0, and there exists C > 0 such that ∗ |f (t)| ≤ C 1 + |t|2 −2 ,
∀t ∈ R,
where 2∗ := N2N −2 for N ≥ 3. ∗ (f2 ) f (t) = o(|t|), as t → 0, and f (t) = o(|t|2 −1 ), as t → ∞. f (t) (f3 ) There exists μ ∈ (2, 2∗ ) such that |t|μ−2 t is nondecreasing in |t| > 0. In view of (f1 ), we have that the functional Φλ : H01 (Ω) → R given by 1 (u2 ) − λ F (u) Φλ (u) := M 2 Ω
1
(t) := t M (s)ds, F (t) := t f (s)ds and u := is well defined, where M |∇u|2 2 is the standard norm 0 0 Ω on H01 (Ω). Moreover, Φλ ∈ C 1 (H01 (Ω), R) with the following derivative Φλ (u), ϕ = M u2 ∇u∇ϕ − λ f (u)ϕ. Ω
Ω
Thus, the weak solutions of problem (Pλ ) are precisely the critical points of Φλ . We say that u ∈ H01 (Ω) is a least energy nodal solution of problem (Pλ ) if u is a weak solution of problem (Pλ ) with u± = 0 and
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Φλ (u) = inf{Φλ (v) | v ± = 0, Φλ (v) = 0}, where u+ (x) = max{u(x), 0} and u− (x) = min{u(x), 0}. Our main results are as follows: Theorem 1.1. Assume that the function M satisfies (M1 ), and (f1 )–(f3 ) hold. Then there exists λ∗ > 0 such that problem (Pλ ) has a constant sign solution uλ and a nodal solution wλ for all λ ≥ λ∗ . Moreover, for all λ ≥ λ∗ , wλ has precisely two nodal domains, and we have Φλ (wλ ) > 2Φλ (uλ ) > 0, and uλ → 0,
wλ → 0,
as λ → +∞.
Theorem 1.2. Under the assumptions of Theorem 1.1, furthermore, the following conditions: (M2 ) The function t →
M (t) t
μ−2 2
is decreasing in t ∈ (0, +∞),
and (f4 ) lim
t→∞
F (t) = +∞, |t|μ
are satisfied for μ given in (f3 ). Then problem (Pλ ) has at least one ground state solution uλ and one least energy nodal solution wλ for all λ > 0. Moreover, uλ has constant sign and wλ has precisely two nodal domains with the energy estimates Φλ (wλ ) > 2Φλ (uλ ) > 0, and the asymptotic behaviors uλ → 0,
wλ → 0,
as λ → +∞.
Throughout this paper, we use standard notations. For simplicity, we write Ω h to mean the Lebesgue integral of h(x) over a domain Ω ⊂ RN . Lp (Ω) (1 ≤ p < +∞) is the usual Lebesgue space with the standard norm | · |p . We use “→” and “” to denote the strong and weak convergence in the related function space respectively. C, C1 , C2 , . . . denote various positive constants whose exact values are not important unless specified. We use “:=” to denote definitions and denote a subsequence of a sequence {un} as {un } to simplify the notation unless specified. The rest of this paper is organized as follows. In Section 2, an auxiliary problem is constructed by making a refined truncation on M and its result is given at the same time. In Section 3, some technical lemmas are shown and proved which shall be used in the proof of the auxiliary result in Section 4. In Sections 4 and 5, the proofs of Theorem 2.1 and the main results of this paper are completed respectively. 2. The auxiliary problem and its result In this section, we shall construct an auxiliary problem by making a truncation on M , and also intend to show the existence results of ground state and least energy nodal solutions for the auxiliary problem.
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Since we intend to work with N ≥ 3, without loss of generality, we shall make a truncation on M as follows. From (M1 ), there exists t1 > 0 such that m0 < M (t1 ) <
1 μm0 , 2
where μ is given in (f3 ). Moreover, there are t2 > 0 and K > 0 such that M (t) ≤ K,
∀t ∈ [0, t2 ].
1 Let δ := min{t1 , t2 , 2K (μ − 2)m0 } and θ := M (δ), we have
m0 ≤ M (t) ≤ θ <
1 μm0 , 2
∀t ∈ [0, δ],
and then, a straightforward computation shows that M (t) t
μ−2 2
is decreasing in t ∈ (0, δ].
(2.1)
Now, we set ⎧ ⎨ t, m(t) :=
2 ⎩ τ + (δ − τ ) arctan π
π t−τ · 2 δ−τ
t ∈ [0, τ ],
t ∈ [τ, +∞),
,
(2.2)
where τ ∈ (0, δ) is a fixed constant, and Mθ (t) := M (m(t)),
t ∈ [0, +∞).
(2.3)
It is easy to see that m(t) is a C 2 function with m (t) > 0 and m (t) ≤ 0,
∀t ≥ 0,
(2.4)
which imply m(t) ∈ [0, δ)
and m (t)t ≤ m(t),
∀t ≥ 0.
(2.5)
From (M1 ), (2.1), (2.4) and (2.5), a simple calculation shows that Mθ is a C 1 function bounded by m0 from below and θ from above, and satisfies (M1 ) and (M2 ). Now, we can construct the auxiliary problem as follows: ⎛ ⎞ ⎧ ⎪ ⎪ ⎨ −M ⎝ |∇u|2 ⎠ Δu = λf (u), in Ω, θ (Tλ,θ ) ⎪ Ω ⎪ ⎩ u = 0, on ∂Ω, where Ω, f and λ are as in introduction. The corresponding functional of problem (Tλ,θ ) is given as follows Φλ,θ (u) :=
1 Mθ (u2 ) − λ 2
F (u), Ω
θ (t) := where M
t 0
Mθ (s)ds.
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We shall establish the following auxiliary result: Theorem 2.1. Under the assumptions of Theorem 1.1, problem (Tλ,θ ) has at least one ground state solution uλ,θ and one least energy nodal solution wλ,θ for any given θ as above and for any given λ > 0. Moreover, uλ,θ has constant sign and wλ,θ has precisely two nodal domains with the energy estimates Φλ,θ (wλ,θ ) > 2Φλ,θ (uλ,θ ) > 0,
(2.6)
and the asymptotic behaviors uλ,θ → 0
and
wλ,θ → 0,
as λ → +∞.
(2.7)
3. Technical lemmas In this section, we shall establish some useful preliminary results for problem (Tλ,θ ) which will be exploited in the proof of Theorem 2.1. Given ε > 0 and given q ∈ (2, 2∗ ], by (f2 ), there exists Cε,q > 0 such that ∗ f (t)t ≤ ε t2 + |t|2 + Cε,q |t|q .
(3.1)
From f ∈ C 1 (R) satisfying (f3 ), we get f (t)t2 − (μ − 1)f (t)t ≥ 0,
for all t ∈ R,
(3.2)
which implies 1 f (t)t − F (t) is nondecreasing in |t| and nonnegative. μ
(3.3)
Moreover, taking advantage of (f1 ) and (3.3), a straightforward computation yields that F (t) ≥ C1 |t|μ − C2 ,
∀t ∈ R,
for some C1 , C2 > 0, which implies F (t) = +∞. t→∞ t2 lim
(3.4)
In addition, from (M2 ) we have 2Mθ (t)t < (μ − 2)Mθ (t),
∀t > 0,
(3.5)
which implies 1 1 Mθ (t) − Mθ (t)t is increasing and positive for t > 0. 2 μ For the proof of Theorem 2.1, we define the Nehari manifold of (Tλ,θ ) given as Nλ,θ := u ∈ H01 (Ω) \ {0} | Φλ,θ (u), u = 0 ,
(3.6)
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and the corresponding nodal Nehari manifold Mλ,θ := w ∈ H01 (Ω) | w± = 0, Φλ,θ (w), w± = 0 . Obviously, Mλ,θ ⊂ Nλ,θ contains all sign-changing solutions of (Tλ,θ ). Furthermore, two candidate critical levels can be defined as follows cλ,θ :=
inf Φλ,θ (u),
cλ,θ :=
u∈Nλ,θ
inf
w∈Mλ,θ
Φλ,θ (w).
Lemma 3.1. (i) There exists ρ := ρλ > 0 such that u ≥ ρ,
∀u ∈ Nλ,θ ,
and w± ≥ ρ,
∀w ∈ Mλ,θ .
(ii) For all u ∈ Nλ,θ we have Φλ,θ (u) ≥
θ m0 − 2 μ
u2 .
(iii) If {un } and {wn } are a bounded sequence in Nλ,θ and Mλ,θ respectively, then we have lim inf |un |q > 0 n→∞
and
lim inf |wn± |q > 0, n→∞
for any q ∈ (2, 2∗ ). Proof. The proof is similar to that in [8] by noting that m0 ≤ Mθ (t) ≤ θ < 12 μm0 . 2 The next lemma tries to infer geometrical information of Φλ,θ with respect to Nλ,θ , which can also be used to show that Nλ,θ = ∅. Lemma 3.2. For any u ∈ H01 (Ω) \ {0}, there is a unique tu > 0 such that tu u ∈ Nλ,θ . Moreover, Φλ,θ (tu u) > Φλ,θ (tu) for any t ≥ 0 and t = tu . Proof. Since Mθ (t) ≤ θ, one could obtain the result by a standard argument. 2 Corollary 3.1. Suppose that u ∈ H01 (Ω) \ {0} with Φλ,θ (u), u ≤ 0, then there is a unique tu ∈ (0, 1] such that tu u ∈ Nλ,θ . Proof. The existence and uniqueness of tu follow from Lemma 3.2 immediately, so we just need to show tu ∈ (0, 1]. Note that g (1) ≤ 0 where g(t) := Φλ,θ (tu). Due to the fact that
g (t) tμ−1
and g (tu ) = 0, is decreasing in t > 0, then we have tu ∈ (0, 1].
2
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The following lemma is a version of Lemma 3.2 for the case of Φλ,θ with respect to Mλ,θ . Before stating the lemma, we need define a suitable function and its gradient vector field which are related to functional Φλ,θ and will be involved in particular in the application of the deformation lemma. Indeed, for each v ∈ H01 (Ω) with v ± = 0 we consider hvλ : [0, +∞) × [0, +∞) → R given by hvλ (t, s) = Φλ,θ (tv + + sv − ) and its gradient Ψv : [0, +∞) × [0, +∞) → R2 is defined by v
Ψ (t, s) =
(Ψv1 (t, s), Ψv2 (t, s))
=
∂hv ∂hvλ (t, s), λ (t, s) ∂t ∂s
= Φλ,θ (tv + + sv − ), v + , Φλ,θ (tv + + sv − ), v − , for every (t, s) ∈ [0, +∞) × [0, +∞). Furthermore, we consider the Hessian matrix of hv , i.e. ∂Ψv v
∂t (t, s) ∂Ψv 2 ∂t (t, s) 1
(Ψ ) (t, s) =
∂Ψv 1 ∂s (t, s) , ∂Ψv 2 ∂s (t, s)
for every (t, s) ∈ [0, +∞) × [0, +∞). For the proof of the following lemma, we also need a useful observation. Let us observe that, from (M1 ), we obtain θ (t) + M θ (s), θ (t + s) ≥ M M
(3.7)
for all t, s ≥ 0. Then we have Φλ,θ (ϕ + ψ) ≥
1 Mθ (ϕ2 ) − λ 2
F (ϕ) Ω
1 2 + M θ (ψ ) − λ 2
F (ψ) = Φλ,θ (ϕ) + Φλ,θ (ψ),
(3.8)
Ω
for any ϕ, ψ ∈ H01 (Ω) with suppt(ϕ) ∩ suppt(ψ) = ∅. Lemma 3.3. Suppose that w ∈ H01 (Ω) with w± = 0, then there is a unique pair (t, s) ∈ (0, +∞) × (0, +∞) such that tw+ + sw− ∈ Mλ,θ . In particular, Mλ,θ = ∅. Moreover, for all t, s ≥ 0 such that (t, s) = (t, s), we have w + − Φλ,θ (tw+ + sw− ) = hw λ (t, s) > hλ (t, s) = Φλ,θ (tw + sw ).
Proof. From (M1 ), (3.1), (3.3) and Sobolev embedding we have ∗ ∗ ελC 2 + 2 λ m0 − t w − (ε + Cε,2∗ ) t2 |w+ |22∗ ≥ 2 μ μ ∗ ∗ ελC m0 λ − s2 w− 2 − (ε + Cε,2∗ ) s2 |w− |22∗ , + 2 μ μ
hw λ (t, s)
for some C > 0. Thus, hw λ (t, s) is positive for sufficiently small t, s ≥ 0.
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On the other hand, by Mθ (t) ≤ θ we obtain hw λ (t, s) ≤
θ 2 + 2 t w − t2 λ 2
Ω
θ + s2 w− 2 − s2 λ 2
F (tw+ ) 2 (tw+ )
Ω
w+
2
F (sw− ) 2 (sw− )
w−
2
,
which implies lim
|(t,s)|→+∞
hw λ (t, s) = −∞,
by using (3.4). Therefore, there exists a pair (t, s) such that hw (t, s) = max hw λ (t, s). t,s≥0
We claim that t, s > 0. Indeed, by (3.8) we have w w w w hw λ (t, s) ≥ hλ (t, s) ≥ hλ (t, 0) + h (0, s) > hλ (t, 0),
for sufficiently small s > 0, which implies s > 0. Similarly, we get t > 0. Now it is easy to see that tw+ + sw− ∈ Mλ,θ , due to the fact that tw+ + sw− ∈ Mλ,θ
⇔
Ψw (t, s) = 0,
t, s > 0.
Next we try to give the uniqueness of (t, s). Without loss of generality, we may assume that w ∈ Mλ,θ and (a, b) ∈ (0, +∞) × (0, +∞) such that w = aw+ + bw− ∈ Mλ,θ . The proof shall be completed by showing that (a, b) = (1, 1). Actually, without loss of generality, we suppose b ≤ a. Then, repeating some arguments explored in [8] for the proof of Lemma 2.4, item (a) in that paper, we obtain 1 ≤ b ≤ a ≤ 1, by using w ∈ Mλ,θ , w ∈ Mλ,θ , (M2 ) and (f3 ).
i.e.
a = b = 1,
2
Corollary 3.2. If w ∈ H01 (Ω) with w± = 0 and Φλ,θ (w), w± ≤ 0, then there is a unique pair (t, s) ∈ (0, 1] × (0, 1] such that tw+ + sw− ∈ Mλ,θ . Proof. Lemma 3.3 yields the existence and uniqueness of pair (t, s), and the claim (t, s) ∈ (0, 1] × (0, 1] holds by repeating some arguments explored in [8] for the proof of Lemma 2.4, item (a) in that paper. 2
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Corollary 3.3. If w ∈ Mλ,θ , then (i) For all t, s ≥ 0 such that (t, s) = (1, 1), we have w Φλ,θ (w) = hw λ (1, 1) > hλ (t, s);
(ii) det (Ψw ) (1, 1) > 0. Proof. Item (i) is an immediate consequence of Lemma 3.3. Let us prove item (ii). Consider the notations + − + Ψw 1 (t, s) = Φλ,θ (tw + sw ), w ,
+ − − Ψw 2 (t, s) = Φλ,θ (tw + sw ), w .
Thus Ψw 1 (t, s)
− 2
= tMθ (t w + s w )w − λ 2
+ 2
2
+ 2
f (tw+ )w+ , Ω
and 2 + 2 2 − 2 − 2 Ψw 2 (t, s) = sMθ (t w + s w )w − λ
f (sw− )w− ,
Ω
and note that ∂Ψw 1 (t, s) = Mθ (t2 w+ 2 + s2 w− 2 )w+ 2 ∂t + 2t2 Mθ (t2 w+ 2 + s2 w− 2 )w+ 4 − λ
f (tw+ )(w+ )2 ,
Ω
implies ∂Ψw 1 (1, 1) = Mθ (w2 )w+ 2 + 2Mθ (w2 )w+ 4 − λ ∂t
f (w+ )(w+ )2 .
Ω
Using (3.5) and Φ (w), w+ = 0 in the expression of ∂Ψw 1 (1, 1) < (μ − 1)Mθ (w2 )w+ 2 − λ ∂t =λ
∂Ψw 1 ∂t (1, 1)
we obtain
f (w+ )(w+ )2 − 2Mθ (w2 )w+ 2 w− 2
Ω
(μ − 1)f (w+ )w+ − f (w+ )(w+ )2 − 2Mθ (w2 )w+ 2 w− 2 ,
Ω
and then, from (3.2) we get ∂Ψw 1 (1, 1) < −2Mθ (w2 )w+ 2 w− 2 . ∂t
(3.9)
Arguing on the same way we conclude ∂Ψw 1 (1, 1) < −2Mθ (w2 )w+ 2 w− 2 . ∂s
(3.10)
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Since ∂Ψw ∂Ψw 1 2 (1, 1) = (1, 1) = 2Mθ (w2 )|w+ 2 w− 2 , ∂s ∂t and considering (3.9) and (3.10), we conclude that
det (Ψw ) (1, 1) > 0, and the item (ii) is proved. 2 Remark 3.1. For any w ∈ Mλ,θ , by item (i) of Corollary 3.3 and (3.8) we have Φλ,θ (w) ≥ Φλ,θ (tw+ w+ + tw− w− ) ≥ Φλ,θ (tw+ w+ ) + Φλ,θ (tw− w− ), where tw+ and tw− are given by applying Lemma 3.2 to w+ and w− respectively. Recall the definition of cλ,θ , it is easy to see that Φλ,θ (w) ≥ 2 cλ,θ . Then we conclude that cλ,θ :=
inf
w∈Mλ,θ
Φλ,θ (w) ≥ 2 cλ,θ .
Moreover, if cλ,θ is achieved, then cλ,θ > 2 cλ,θ . cλ,θ and the The next lemma tries to build a relationship between the minimizers (if there exists) of critical points of Φλ,θ , and then show more information about the minimizers. cλ,θ is attained by some u ∈ Nλ,θ , then u is a critical point of Φλ,θ . Moreover, the Lemma 3.4. Suppose that minimizer u has constant sign. Proof. Let G(v) := Mθ (v2 )v2 − λ
Ω
f (v)v, then for v ∈ Nλ,θ we have
G (v), v = 2Mθ (v2 )v2 + 2Mθ (v2 )v4 − λ < μMθ (v2 )v2 − λ = −λ
f (v)v 2 + f (v)v
Ω
f (v)v 2 + f (v)v
Ω
f (v)v 2 − (μ − 1)f (v)v ≤ 0,
Ω
by taking advantage of (3.5), G(v) = 0 and (3.2). Thus, Nλ,θ has a C 1 structure and is a manifold. cλ,θ is attained by some u ∈ Nλ,θ , and then u ∈ Nλ,θ is a critical point of Φλ,θ |Nλ,θ . Moreover, there If exists a Lagrange multiplier ν such that Φ (u), v = ν G (u), v ,
∀v ∈ H01 (Ω).
Let v = u, by G (u), u = 0, we have ν = 0, then u is a critical point of Φλ,θ . If u± = 0, then u ∈ Mλ,θ . By Remark 3.1 and Lemma 3.1 we get a contradiction cλ,θ = Φλ,θ (u) ≥ cλ,θ ≥ 2 cλ,θ > 0. Thus, the minimizer u has constant sign. 2
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The following lemma is a version of Lemma 3.4 for the case of cλ,θ with respect to Mλ,θ , the proof of which is more complicated. Lemma 3.5. Suppose that cλ,θ is attained by w ∈ Mλ,θ , then w is a critical point of Φλ,θ . Moreover, the minimizer w has precisely two nodal domains. Proof. The proof of the first part is similar to that in [8], so we omit it here. For the proof of the claim that w has exactly two nodal domains, we assume by contradiction that w = v1 + v2 + v3 with vi = 0, v1 ≥ 0, v2 ≤ 0,
suppt(vi ) ∩ suppt(vj ) = ∅,
i = j (i, j = 1, 2, 3).
Setting u = v1 + v2 , we see that u± = 0. Moreover, using the fact that Φλ,θ (w) = 0, by (M1 ) we get ≤ 0. Consequently, by Corollary 3.2, there exist t, s ∈ (0, 1] such that tu+ + su− ∈ Mλ,θ , i.e. tv1 + sv2 ∈ Mλ,θ , let v := tv1 + sv2 and so Φλ,θ (v) ≥ cλ,θ . From Φλ,θ (w), w = 0, (3.3), (3.6) and Φλ,θ (v), v = 0, we have a contradiction Φλ,θ (u), u±
cλ,θ = Φλ,θ (w) = Φλ,θ (w) −
1 Φ (w), w μ λ,θ
1 1 2 Mθ (w22 )w2 + λ = M θ (w ) − 2 μ +λ
Ω
Ω
+λ
Ω
1 f (tv1 )tv1 − F (tv1 ) μ
Ω
1 f (sv2 )sv2 − F (sv2 ) μ
Ω
=
1 f (v1 )v1 − F (v1 ) μ
1 1 f (v2 )v2 − F (v2 ) + λ f (v3 )v3 − F (v3 ) μ μ
1 1 2 Mθ (v22 )v2 + λ > M θ (v ) − 2 μ
1 1 Mθ (v2 ) − Mθ (v22 )v2 + λ 2 μ
1 f (v)v − F (v) μ
Ω
1 = Φλ,θ (v) − Φλ,θ (v), v = Φλ,θ (v) ≥ cλ,θ . μ Thus, the minimizer w has precisely two nodal domains. 2 At last, we intend to give an estimate for cλ,θ as λ → +∞, which shall play a vital role in the proof of Theorem 2.1 and Theorem 1.1. To be more precise, the estimate can give the asymptotic behavior of the ground state (least energy nodal solution) of problem (Tλ,θ ), and then show that the ground state (least energy nodal solution) of the auxiliary problem is a constant sign (sign-changing) solution of the original problem (Pλ ) for λ large. Lemma 3.6. cλ,θ is non-increasing in λ > 0 and
lim cλ,θ = 0.
λ→+∞
Proof. For any given λ2 > λ1 > 0, for any w ∈ Mλ1 ,θ , by Corollary 3.3 we have w Φλ1 ,θ (w) = hw λ1 (1, 1) ≥ hλ1 (t, s)
= hw λ2 (t, s) + (λ2 − λ1 ) Ω
F (tw+ + sw− ) > hw λ2 (t, s) ≥ cλ2 ,
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where t, s are given by applying Lemma 3.3 to w for the case Mλ2 ,θ . Thus we conclude that cλ1 ,θ ≥ cλ2 ,θ . Fix w ∈ M1,θ , then we obtain Φλ,θ (w), w±
=
Φ1,θ (w), w±
− (λ − 1)
f (w± )w± ≤ 0,
Ω
for any λ > 1. Thus, by Corollary 3.2, there exist tλ , sλ ∈ (0, 1] such that tλ w+ + sλ w− ∈ Mλ,θ . We claim that (tλ , sλ ) → (0, 0) as λ → +∞. Thus, cλ,θ ≤ Φλ,θ (tλ w+ + sλ w− ) ≤
1 2 + 2 Mθ (tλ w + s2λ w− 2 ) → 0, 2
as λ → +∞, and the proof is complete. Actually, we assume by contradiction that there exists λn → +∞ such that (tλn , sλn ) (0, 0). Due to the boundedness of {(tλn , sλn )}, up to a subsequence, there exists (α, β) ∈ [0, 1] × [0, 1] \ {(0, 0)} such that (tλn , sλn ) → (α, β),
as λn → +∞.
Let wλn := tλn w+ + sλn w− and w := αw+ + βw− , then wλn → w = 0
in H01 (Ω),
as λn → +∞,
which gives a contradiction Mθ (w2 )w2 =
lim
λn →+∞
Mθ (wλn 2 )wλn 2 =
lim
λn →+∞
λn
f (wλn )wλn = +∞. Ω
Thus, cλ,θ is non-increasing in λ > 0 and lim cλ,θ = 0. λ→+∞
2
4. Proof of Theorem 2.1 Recall the definition of cλ,θ in Section 3, it’s easy to see that cλ,θ is well defined from Lemma 3.3 and positive from Lemma 3.1. Thus, there exists a minimizing sequence {wn } in Mλ,θ which is bounded from Lemma 3.1 again. From the boundedness of {wn } and the compactness of the embedding H01 (Ω) → Lp (Ω) (2 < p < 2∗ ), without loss of generality, we can assume, up to a subsequence, that there exist w, w1 , w2 ∈ H01 (Ω) such that wn w,
wn+ w1 ,
wn− w2
in H01 (Ω),
wn → w,
wn+ → w1 ,
wn− → w2
in Lq (Ω), q ∈ (2, 2∗ ).
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Then, we have w+ = w1 ≥ and w− = w2 ≤ 0 by the continuity of the transformations w → w± from L (Ω) to Lq (Ω). Moreover, by the strong convergence of {wn± } and item (iii) of Lemma 3.1, we conclude that w± = 0 and consequently w = w+ + w− is sign-changing. At this point, we can prove that w ∈ Mλ,θ and Φλ,θ (w) = cλ,θ . Indeed, since f has a subcritical growth, we get q
f (wn± )wn± →
Ω
f (w± )w± ,
Ω
F (wn± ) →
Ω
F (w± ).
Ω
Thus, since Φλ,θ (wn ), wn± = 0, by (M1 ) we have Φλ,θ (w), w+ ≤ 0,
Φλ,θ (w), w− ≤ 0,
and then, by Corollary 3.2, there exists (t, s) ∈ (0, 1] × (0, 1] such that tw+ + sw− ∈ Mλ,θ . Noting that 1 1 Φλ,θ (w) − Φλ,θ (w), w ≤ lim inf Φλ,θ (wn ) − Φλ,θ (wn ), wn n→+∞ μ μ = lim inf Φλ,θ (wn ) = cλ,θ , n→+∞
we only need to show t = s = 1. Let w = tw+ + sw− , then Φλ,θ (w) = Φλ,θ (w) −
1 Φ (w), w μ λ,θ
1 1 2 Mθ w2 )w2 + λ = M θ (w ) − 2 μ +λ
1 f (tw+ )tw+ − F (tw+ ) μ
Ω
1 f (sw− )sw− − F (sw− ) . μ
Ω
If (t, s) = (1, 1), using w2 < w2 , (3.3) and (3.6), we have 1 1 2 Mθ w2 )w2 + λ Φλ,θ (w) < M θ (w ) − 2 μ +λ
1 f (w+ )w+ − F (w+ ) μ
Ω
1 f (w− )w− − F (w− ) μ
Ω
= Φλ,θ (w) −
1 Φ (w), w , μ λ,θ
which gives a contradiction that cλ,θ ≤ Φλ,θ (w) < cλ,θ . Thus, we conclude w ∈ Mλ,θ and Φλ,θ (w) = cλ,θ . And then, Lemma 3.5 yields that wλ,θ := w is a least energy nodal solution of problem (Tλ,θ ), which has precisely two nodal domains. In addition, the asymptotic behavior (2.7) of wλ,θ follows from Lemma 3.6 and Lemma 3.1.
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Similarly, we can prove that cλ,θ is positive and is attained by some uλ,θ ∈ Nλ,θ . And then, uλ,θ is a ground state solution of problem (Tλ,θ ) with definite sign by Lemma 3.4, and its asymptotic behavior (2.7) follows from Lemma 3.6, Remark 3.1 and Lemma 3.1. cλ,θ > 0 and Remark 3.1, Finally, we get the conclusion (2.6) by using Φλ,θ (wλ,θ ) = cλ,θ , Φλ,θ (uλ,θ ) = and the proof of Theorem 2.1 is complete. 2 5. Proof of the main results Proof of Theorem 1.1. Fixing θ, we have a ground state solution uλ,θ and a least energy nodal solution wλ,θ of problem (Tλ,θ ) given by Theorem 2.1. Moreover, by (2.7), there exists λ∗ > 0 such that uλ,θ 2 , wλ,θ 2 ≤ τ,
∀λ ≥ λ∗ ,
where τ ∈ (0, δ) is given in (2.2). Thus, for all λ ≥ λ∗ , we conclude that M (uλ,θ 2 ) = Mθ (uλ,θ 2 ) and M (wλ,θ 2 ) = Mθ (wλ,θ 2 ). Let uλ := uλ,θ and wλ := wλ,θ , then uλ and wλ are the desired solutions. 2 Proof of Theorem 1.2. The proof of Theorem 1.2 is similar to that of Theorem 2.1. As the proof of Theorem 2.1, we define the Nehari manifold of (Pλ ) given as Nλ := u ∈ H01 (Ω) \ {0} | Φλ (u), u = 0 , and the corresponding nodal Nehari manifold Mλ := w ∈ H01 (Ω) | w± = 0, Φλ (w), w± = 0 . Furthermore, two candidate critical levels also be defined as follows cλ := inf Φλ (u), u∈Nλ
cλ := inf Φλ (w). w∈Mλ
It’s not difficult to see that, all the preliminary results, corollaries and remark, and almost all the lemmas in Section 3, except item (ii) of Lemma 3.1, are valid here and only the proofs of Lemma 3.2 and Lemma 3.3 need some minor changes. To be more precise, what we only have to show in the proof of Lemma 3.2 for the case of Φλ is that lim g(t) = −∞,
(5.1)
t→+∞
where g(t) := Φλ (tu). Actually, from (M1 ) and (M2 ) we have M (t) ≤ M (1) + M (1)t
μ−2 2
,
which implies (t) ≤ M (1)t + 2 M (1)t μ2 . M μ Thus,
(5.2)
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⎡ g(t) ≤
M (1) u2 t2 − ⎣λ 2
Ω
⎤ F (tu) μ M (1) uμ ⎦ tμ |u| − |tu|μ μ
yields (5.1) by taking advantage of (f4 ). Similarly, by (5.2) and (f4 ) we can get lim
|(t,s)|→+∞
Φλ (tw+ + sw− ) = −∞,
which completes the proof of Lemma 3.3 for the case of Φλ . cλ is still positive. It is worth pointing out that although the item (ii) of Lemma 3.1 fails here in general, Indeed, from (3.3), (3.6) and item (i) of Lemma 3.1 we obtain 1 Φ (u), u μ λ 1 1 2 1 1 (u2 ) − M (u2 )u2 ≥ M (ρ ) − M (ρ2 )ρ2 ≥ M 2 μ 2 μ
Φλ (u) = Φλ (u) −
(5.3)
for all u ∈ Nλ , which yields cλ > 0. The major difficulty that we need to overcome in the proof of Theorem 1.2 is the boundedness of the minimizing sequence, which is caused by the fail of item (ii) in Lemma 3.1, more generally the coerciveness of Φλ on Nλ . Fortunately, we can get through this difficulty by a different way and more analysis, see e.g. [17]. Actually, we can prove a more general result about boundedness, that is Lemma 5.1. Under the assumption of Theorem 1.2, assume that {wn } ⊂ Nλ with then {wn } is bounded in
lim Φλ (wn ) = c > 0,
n→+∞
H01 (Ω).
For the proof of the boundedness, we assume by contradiction that wn → +∞, up to a subsequence, and consider vn :=
wn wn .
as n → +∞,
Passing to a subsequence, we may assume that
vn v
in H01 (Ω),
vn → v
in Lq (Ω), ∀q ∈ [2, 2∗ ),
vn → v
a.e. on Ω.
If v = 0, from (5.2) the equality (wn 2 ) 1M c + on (1) F (wn ) = − λ |vn |μ wn μ 2 wn μ |wn |μ Ω
implies the inequality M (1) 1 1 c + on (1) ≤ M (1) + −λ wn μ μ 2 wn μ−2 which gives a contradiction
Ω
F (wn ) |vn |μ , |wn |μ
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M (1) ≥λ μ
lim inf
n→+∞
Ω
17
F (wn vn ) μ μ |v| = +∞, |wn vn |
by taking advantage of (f4 ) and Fadou’s lemma. Thus we have v = 0 and vn → 0 in Lq (Ω) for q ∈ (2, 2∗ ). Fix q ∈ (2, 2∗ ) and an R > and (3.3), and using Lebesgue Dominated Convergence theorem, we have lim
2c m0
12
. From (3.1)
F (Rvn ) =
n→∞
Ω
lim F (Rvn ) = 0.
n→∞ Ω
By Lemma 3.2 we have Φλ (wn ) = Φλ (wn vn ) ≥ Φλ (Rvn ) ≥
m0 R 2 −λ 2
F (Rvn ). Ω 2
The left side of this inequality converges towards c and the right side tends to m02R > c, we also have a contradiction. Summing up the above, {wn } is bounded in H01 (Ω). At this point, we can complete the proof of Theorem 1.2 by repeating the argument in the proof of Theorem 2.1. 2 Remark 5.1. If the following property (M3 )
lim
t→+∞
1 1 M (t) − M (t)t 2 μ
= +∞
is satisfied, we can also get the coerciveness of Φλ on Nλ by using (M3 ) in (5.3), and then the boundedness of the minimizing sequence can be obtained easily. Unfortunately, (M3 ) is not always hold for functions which verify the hypotheses (M1 )–(M2 ). For example, let N = 3, μ = 4, M (t) = m0 + m0 arctan(t)t
and f (t) = 4t3 ln(1 + |t|) +
t4 sign(t), 1 + |t|
then a straightforward computation yields 1 m0 1 M (t) − M (t)t = arctan(t), 2 4 4 which is bounded. Moreover, assuming u ∈ Nλ , we have 1 1 Φλ (u) = m0 arctan(u2 ) + λ 4 4
Ω
|u|5 , 1 + |u|
which does not seem to give a positive answer to the coerciveness of Φλ on Nλ . Remark 5.2. It is not difficult to see that the main result of [8] is a special case of Theorem 1.2 in this paper by letting N = 3, μ = 4 and λ = 1. Acknowledgment The author would like to thank Professor Zhi-Qiang Wang for useful discussions and suggestions.
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References [1] C.O. Alves, G.M. Figueiredo, Nonlinear perturbations of a periodic Kirchhoff equation in RN , Nonlinear Anal. 75 (2012) 2750–2759. [2] P. D’Ancona, S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math. 108 (2) (1992) 247–262. [3] A. Arosio, S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc. 348 (1) (1996) 305–330. [4] A. Azzollini, The elliptic Kirchhoff equation in RN perturbed by a local nonlinearity, Differential Integral Equations 25 (5–6) (2012) 543–554. [5] S. Bernstein, Sur une class d’équations fonctionelles aux dérivées partielles, Bull. Acad. Sci. URRS Sér. 4 (1940) 1041–1053. [6] G.M. Figueiredo, Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument, J. Math. Anal. Appl. 401 (2013) 706–713. [7] G.M. Figueiredo, J.R.S. Junior, Multiplicity of solutions for a Kirchhoff equation with subcritical or critical growth, Differential Integral Equations 25 (9–10) (2012) 853–868. [8] G.M. Figueiredo, R.G. Nascimento, Existence of a nodal solution with minimal energy for a Kirchhoff equation, Math. Nachr. 288 (1) (2015) 48–60. [9] Y. He, et al., Concentrating bound states for Kirchhoff type problems in R3 involving critical Sobolev exponents, Adv. Nonlinear Stud. 14 (2014) 441–468. [10] X.M. He, W.M. Zhou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in R3 , J. Differential Equations 252 (2012) 1813–1834. [11] J.H. Jin, X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in RN , J. Math. Anal. Appl. 369 (2010) 564–574. [12] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. [13] G.B. Li, H.Y. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equation in R3 , J. Differential Equations 257 (2014) 566–600. [14] G.B. Li, H.Y. Ye, Existence of positive solutions for nonlinear Kirchhoff type problem in R3 with critical Sobolev exponent, Math. Methods Appl. Sci. 37 (2014) 2570–2584. [15] J.L. Lions, On some questions in boundary value problems of mathematical physics, in: Contemporary Development in Continuum Mechanics and Partial Differential Equations, in: North-Holland Math. Stud., vol. 30, North-Holland, Amsterdam, New York, 1978, pp. 284–346. [16] W. Liu, X.M. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations, J. Appl. Math. Comput. 39 (2012) 473–487. [17] Z.L. Liu, Z.-Q. Wang, On the Ambrosetti–Rabinowitz superlinear condition, Adv. Nonlinear Stud. 4 (2004) 563–574. zaev, A certain class of quasilinear hyperbolic equations, Mat. Sb. (N.S.) 96 (138) (1975) 152–166, 168 (in Russian). [18] S.I. Poho˘ [19] J. Wang, L.X. Tian, J.X. Xu, F.B. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations 253 (2012) 2314–2351. [20] X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger–Kirchhoff-type equation in RN , Nonlinear Anal. Real World Appl. 12 (2011) 1278–1287. [21] Z.T. Zhang, K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl. 317 (2006) 456–463.
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Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Signed and sign-changing solutions for a Kirchhoff-type equation in bounded domains Sheng-Sen Lu Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, 300071, PR China
a r t i c l e
i n f o
Article history: Received 27 June 2015 Available online xxxx Submitted by J. Xiao Keywords: Kirchhoff-type equation Signed and sign-changing solutions Variational methods
a b s t r a c t The main concern of this article is a Kirchhoff-type equation of the form ⎛ −M ⎝
⎞
|∇u|
2⎠
Δu = λf (u),
Ω
where Ω is a bounded smooth domain in RN with N ≥ 3 and λ is a positive parameter. Under certain assumptions on M and f , the existence results of signed and sign-changing solutions are established for λ large, and when λ converges to infinity the asymptotic behavior of these solutions is also studied. The proofs are based on a careful study of the ground state and least energy nodal solutions of an auxiliary problem, which is constructed by making a refined truncation on M . Furthermore, we get the ground state and least energy nodal solutions, and prove the energy doubling property for all λ > 0 under more restricted assumptions on M and f . © 2015 Elsevier Inc. All rights reserved.
1. Introduction and main results In this paper, we consider the existence of signed and sign-changing solutions for the following problem ⎞ ⎛ ⎧ ⎪ ⎪ ⎨ −M ⎝ |∇u|2 ⎠ Δu = λf (u), in Ω, (Pλ ) ⎪ Ω ⎪ ⎩ u = 0, on ∂Ω, where Ω is a bounded smooth domain in RN with N ≥ 3, λ is a positive parameter, M : R+ → R+ and f : R → R are C 1 functions that satisfy some assumptions which will be stated later on. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jmaa.2015.07.033 0022-247X/© 2015 Elsevier Inc. All rights reserved.
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The class of problem (Pλ ) is called of Kirchhoff type because it comes from an important application in Physic and Engineering. Indeed, if we set M (t) = a + bt and λ = 1 in (Pλ ), then we get the following Kirchhoff problem ⎛ ⎞ − ⎝a + b |∇u|2 ⎠ Δu = f (u), (1.1) Ω
which is related to the stationary analogue of the equation ⎛ ⎞ L 2 2 2 E ∂ u P0 ∂u ⎠ ∂ u = 0, + ρ 2 −⎝ ∂t h 2L ∂x ∂x2 0
presented by G. Kirchhoff in [12]. For more mathematical and physical background, we refer readers to papers [2,3,5,15,18] and the references therein. In the last ten years, by classical variational method, there are many interesting results about the existence and nonexistence of solutions, sign-changing solutions, ground state solutions, the existence of positive solutions and positive ground states, least energy nodal solutions, multiplicity of solutions, semiclassical limit and concentrations of solution to Kirchhoff type problems, see e.g. [1,4,6–11,13,14,16,19–21] and the references therein. 2 One major difficulty that arises in the use of this technique is the growth of the operator M(u )=
t b 2 4 m0 u + 2 u , where M (t) := 0 M (s)ds and m0 , b > 0. This requires to impose 4-superlinear growth on the nonlinearity f , that is, f (t) = |t|p−2 t with p ∈ (4, 2∗ ). But 2∗ := N2N −2 ≤ 4 for N ≥ 4, so it is common to fix N = 3 to circumvent this difficulty in most cases. To the best of our knowledge, only article [21] considers the signed and sign-changing solutions for problem (1.1) with superlinear nonlinearity which just has 4-sublinear growth. Those results are available for bounded smooth domains of RN with N ≥ 3. In that paper, the authors adopt the variational method and invariant sets of descent flow. In [8], under certain assumptions on M and f , the authors show the existence of least energy nodal solution for (Pλ ) on bounded smooth domains Ω ⊂ R3 with λ = 1. This type problems include but are not restricted to the type (1.1), and much closer to problem (Pλ ) we consider here. However, they impose the 4-superlinear Ambrosetti–Rabinowitz type condition on f in that paper that we just try to drop in the present article. We would like to point out that it is the work [6] that gives us the primal idea for dealing with problem (Pλ ) with more general nonlinearity which may not have 4-superlinear growth. That paper concerns the following problem: ⎞ ⎛ ⎧ ⎪ ⎪ ⎨ −M ⎝ |∇u|2 ⎠ Δu = λf (x, u) + |u|2∗ −2 u, in Ω, ⎪ Ω ⎪ ⎩ u = 0, on ∂Ω, where Ω is a bounded smooth domain in RN with N ≥ 3, λ is a positive parameter, M : R+ → R+ is an increasing function with m0 := M (0) > 0, and f (x, t) satisfies the well-known Ambrosetti–Rabinowitz superlinear condition and other some fundamental assumptions. Based on the variational method, an appropriated truncation argument and a priori estimates, the author shows the existence result of positive solution for the above problem and study the asymptotic behavior of that solution when λ converges to infinity. Motivated by the articles [6,8,21], we shall investigate the existence of constant sign and sign-changing solutions for problem (Pλ ). The main differences are the following:
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(a) The truncation explored in [6] cannot be used here directly, because that truncated function is not regular enough even though M is a C 1 class function. Indeed, the truncation in the present paper is more refined and technical, see (2.2) and (2.3) in Section 2. In addition, we succeeded in proving that the nodal Nehari manifold Mλ,θ is nonempty by using an elementary method and more analysis instead of Miranda’s theorem which is adopted in [8], see Lemma 3.3 and its proof in Section 3. (b) Under certain assumptions on M and f , we get a constant sign solution uλ and a nodal solution wλ with the energy estimates Φλ (wλ ) > 2Φλ (uλ ) > 0 for λ large. However, the critical levels of the signed and sign-changing solutions given in [21] for the 4-sublinear case are all negative with no comparison between each other. Besides, the nodal solution wλ that we obtain in this paper has precisely two nodal domains, while [21] gives no information about the number of nodal domains of the sign-changing solution they obtain. (c) Here we also get a general existence result of least energy nodal solutions for problem (Pλ ) under more restricted assumptions on M and f , which covers and generalizes the result in [8], see Remark 5.2 in Section 5. Besides, the asymptotic behavior of the least energy nodal solutions of problem (Pλ ) when λ converges to infinity and the energy doubling property are studied, which are not observed in [8], see Theorem 1.2 below. Before stating our main results, we make the following hypotheses on the function M . The function M : R+ → R+ is of C 1 class and satisfies the following condition: (M1 ) The function M is increasing and m0 := M (0) > 0. A typical example of a function verifying the assumptions (M1 ) is given by M (t) = m0 + bt,
where m0 > 0 and b > 0.
In what follows, we also assume the nonlinearity f is of C 1 class and satisfies (f1 ) For all t = 0, f (t)t > 0, and there exists C > 0 such that ∗ |f (t)| ≤ C 1 + |t|2 −2 ,
∀t ∈ R,
where 2∗ := N2N −2 for N ≥ 3. ∗ (f2 ) f (t) = o(|t|), as t → 0, and f (t) = o(|t|2 −1 ), as t → ∞. f (t) (f3 ) There exists μ ∈ (2, 2∗ ) such that |t|μ−2 t is nondecreasing in |t| > 0. In view of (f1 ), we have that the functional Φλ : H01 (Ω) → R given by 1 (u2 ) − λ F (u) Φλ (u) := M 2 Ω
1
(t) := t M (s)ds, F (t) := t f (s)ds and u := is well defined, where M |∇u|2 2 is the standard norm 0 0 Ω on H01 (Ω). Moreover, Φλ ∈ C 1 (H01 (Ω), R) with the following derivative Φλ (u), ϕ = M u2 ∇u∇ϕ − λ f (u)ϕ. Ω
Ω
Thus, the weak solutions of problem (Pλ ) are precisely the critical points of Φλ . We say that u ∈ H01 (Ω) is a least energy nodal solution of problem (Pλ ) if u is a weak solution of problem (Pλ ) with u± = 0 and
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Φλ (u) = inf{Φλ (v) | v ± = 0, Φλ (v) = 0}, where u+ (x) = max{u(x), 0} and u− (x) = min{u(x), 0}. Our main results are as follows: Theorem 1.1. Assume that the function M satisfies (M1 ), and (f1 )–(f3 ) hold. Then there exists λ∗ > 0 such that problem (Pλ ) has a constant sign solution uλ and a nodal solution wλ for all λ ≥ λ∗ . Moreover, for all λ ≥ λ∗ , wλ has precisely two nodal domains, and we have Φλ (wλ ) > 2Φλ (uλ ) > 0, and uλ → 0,
wλ → 0,
as λ → +∞.
Theorem 1.2. Under the assumptions of Theorem 1.1, furthermore, the following conditions: (M2 ) The function t →
M (t) t
μ−2 2
is decreasing in t ∈ (0, +∞),
and (f4 ) lim
t→∞
F (t) = +∞, |t|μ
are satisfied for μ given in (f3 ). Then problem (Pλ ) has at least one ground state solution uλ and one least energy nodal solution wλ for all λ > 0. Moreover, uλ has constant sign and wλ has precisely two nodal domains with the energy estimates Φλ (wλ ) > 2Φλ (uλ ) > 0, and the asymptotic behaviors uλ → 0,
wλ → 0,
as λ → +∞.
Throughout this paper, we use standard notations. For simplicity, we write Ω h to mean the Lebesgue integral of h(x) over a domain Ω ⊂ RN . Lp (Ω) (1 ≤ p < +∞) is the usual Lebesgue space with the standard norm | · |p . We use “→” and “” to denote the strong and weak convergence in the related function space respectively. C, C1 , C2 , . . . denote various positive constants whose exact values are not important unless specified. We use “:=” to denote definitions and denote a subsequence of a sequence {un} as {un } to simplify the notation unless specified. The rest of this paper is organized as follows. In Section 2, an auxiliary problem is constructed by making a refined truncation on M and its result is given at the same time. In Section 3, some technical lemmas are shown and proved which shall be used in the proof of the auxiliary result in Section 4. In Sections 4 and 5, the proofs of Theorem 2.1 and the main results of this paper are completed respectively. 2. The auxiliary problem and its result In this section, we shall construct an auxiliary problem by making a truncation on M , and also intend to show the existence results of ground state and least energy nodal solutions for the auxiliary problem.
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Since we intend to work with N ≥ 3, without loss of generality, we shall make a truncation on M as follows. From (M1 ), there exists t1 > 0 such that m0 < M (t1 ) <
1 μm0 , 2
where μ is given in (f3 ). Moreover, there are t2 > 0 and K > 0 such that M (t) ≤ K,
∀t ∈ [0, t2 ].
1 Let δ := min{t1 , t2 , 2K (μ − 2)m0 } and θ := M (δ), we have
m0 ≤ M (t) ≤ θ <
1 μm0 , 2
∀t ∈ [0, δ],
and then, a straightforward computation shows that M (t) t
μ−2 2
is decreasing in t ∈ (0, δ].
(2.1)
Now, we set ⎧ ⎨ t, m(t) :=
2 ⎩ τ + (δ − τ ) arctan π
π t−τ · 2 δ−τ
t ∈ [0, τ ],
t ∈ [τ, +∞),
,
(2.2)
where τ ∈ (0, δ) is a fixed constant, and Mθ (t) := M (m(t)),
t ∈ [0, +∞).
(2.3)
It is easy to see that m(t) is a C 2 function with m (t) > 0 and m (t) ≤ 0,
∀t ≥ 0,
(2.4)
which imply m(t) ∈ [0, δ)
and m (t)t ≤ m(t),
∀t ≥ 0.
(2.5)
From (M1 ), (2.1), (2.4) and (2.5), a simple calculation shows that Mθ is a C 1 function bounded by m0 from below and θ from above, and satisfies (M1 ) and (M2 ). Now, we can construct the auxiliary problem as follows: ⎛ ⎞ ⎧ ⎪ ⎪ ⎨ −M ⎝ |∇u|2 ⎠ Δu = λf (u), in Ω, θ (Tλ,θ ) ⎪ Ω ⎪ ⎩ u = 0, on ∂Ω, where Ω, f and λ are as in introduction. The corresponding functional of problem (Tλ,θ ) is given as follows Φλ,θ (u) :=
1 Mθ (u2 ) − λ 2
F (u), Ω
θ (t) := where M
t 0
Mθ (s)ds.
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We shall establish the following auxiliary result: Theorem 2.1. Under the assumptions of Theorem 1.1, problem (Tλ,θ ) has at least one ground state solution uλ,θ and one least energy nodal solution wλ,θ for any given θ as above and for any given λ > 0. Moreover, uλ,θ has constant sign and wλ,θ has precisely two nodal domains with the energy estimates Φλ,θ (wλ,θ ) > 2Φλ,θ (uλ,θ ) > 0,
(2.6)
and the asymptotic behaviors uλ,θ → 0
and
wλ,θ → 0,
as λ → +∞.
(2.7)
3. Technical lemmas In this section, we shall establish some useful preliminary results for problem (Tλ,θ ) which will be exploited in the proof of Theorem 2.1. Given ε > 0 and given q ∈ (2, 2∗ ], by (f2 ), there exists Cε,q > 0 such that ∗ f (t)t ≤ ε t2 + |t|2 + Cε,q |t|q .
(3.1)
From f ∈ C 1 (R) satisfying (f3 ), we get f (t)t2 − (μ − 1)f (t)t ≥ 0,
for all t ∈ R,
(3.2)
which implies 1 f (t)t − F (t) is nondecreasing in |t| and nonnegative. μ
(3.3)
Moreover, taking advantage of (f1 ) and (3.3), a straightforward computation yields that F (t) ≥ C1 |t|μ − C2 ,
∀t ∈ R,
for some C1 , C2 > 0, which implies F (t) = +∞. t→∞ t2 lim
(3.4)
In addition, from (M2 ) we have 2Mθ (t)t < (μ − 2)Mθ (t),
∀t > 0,
(3.5)
which implies 1 1 Mθ (t) − Mθ (t)t is increasing and positive for t > 0. 2 μ For the proof of Theorem 2.1, we define the Nehari manifold of (Tλ,θ ) given as Nλ,θ := u ∈ H01 (Ω) \ {0} | Φλ,θ (u), u = 0 ,
(3.6)
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and the corresponding nodal Nehari manifold Mλ,θ := w ∈ H01 (Ω) | w± = 0, Φλ,θ (w), w± = 0 . Obviously, Mλ,θ ⊂ Nλ,θ contains all sign-changing solutions of (Tλ,θ ). Furthermore, two candidate critical levels can be defined as follows cλ,θ :=
inf Φλ,θ (u),
cλ,θ :=
u∈Nλ,θ
inf
w∈Mλ,θ
Φλ,θ (w).
Lemma 3.1. (i) There exists ρ := ρλ > 0 such that u ≥ ρ,
∀u ∈ Nλ,θ ,
and w± ≥ ρ,
∀w ∈ Mλ,θ .
(ii) For all u ∈ Nλ,θ we have Φλ,θ (u) ≥
θ m0 − 2 μ
u2 .
(iii) If {un } and {wn } are a bounded sequence in Nλ,θ and Mλ,θ respectively, then we have lim inf |un |q > 0 n→∞
and
lim inf |wn± |q > 0, n→∞
for any q ∈ (2, 2∗ ). Proof. The proof is similar to that in [8] by noting that m0 ≤ Mθ (t) ≤ θ < 12 μm0 . 2 The next lemma tries to infer geometrical information of Φλ,θ with respect to Nλ,θ , which can also be used to show that Nλ,θ = ∅. Lemma 3.2. For any u ∈ H01 (Ω) \ {0}, there is a unique tu > 0 such that tu u ∈ Nλ,θ . Moreover, Φλ,θ (tu u) > Φλ,θ (tu) for any t ≥ 0 and t = tu . Proof. Since Mθ (t) ≤ θ, one could obtain the result by a standard argument. 2 Corollary 3.1. Suppose that u ∈ H01 (Ω) \ {0} with Φλ,θ (u), u ≤ 0, then there is a unique tu ∈ (0, 1] such that tu u ∈ Nλ,θ . Proof. The existence and uniqueness of tu follow from Lemma 3.2 immediately, so we just need to show tu ∈ (0, 1]. Note that g (1) ≤ 0 where g(t) := Φλ,θ (tu). Due to the fact that
g (t) tμ−1
and g (tu ) = 0, is decreasing in t > 0, then we have tu ∈ (0, 1].
2
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The following lemma is a version of Lemma 3.2 for the case of Φλ,θ with respect to Mλ,θ . Before stating the lemma, we need define a suitable function and its gradient vector field which are related to functional Φλ,θ and will be involved in particular in the application of the deformation lemma. Indeed, for each v ∈ H01 (Ω) with v ± = 0 we consider hvλ : [0, +∞) × [0, +∞) → R given by hvλ (t, s) = Φλ,θ (tv + + sv − ) and its gradient Ψv : [0, +∞) × [0, +∞) → R2 is defined by v
Ψ (t, s) =
(Ψv1 (t, s), Ψv2 (t, s))
=
∂hv ∂hvλ (t, s), λ (t, s) ∂t ∂s
= Φλ,θ (tv + + sv − ), v + , Φλ,θ (tv + + sv − ), v − , for every (t, s) ∈ [0, +∞) × [0, +∞). Furthermore, we consider the Hessian matrix of hv , i.e. ∂Ψv v
∂t (t, s) ∂Ψv 2 ∂t (t, s) 1
(Ψ ) (t, s) =
∂Ψv 1 ∂s (t, s) , ∂Ψv 2 ∂s (t, s)
for every (t, s) ∈ [0, +∞) × [0, +∞). For the proof of the following lemma, we also need a useful observation. Let us observe that, from (M1 ), we obtain θ (t) + M θ (s), θ (t + s) ≥ M M
(3.7)
for all t, s ≥ 0. Then we have Φλ,θ (ϕ + ψ) ≥
1 Mθ (ϕ2 ) − λ 2
F (ϕ) Ω
1 2 + M θ (ψ ) − λ 2
F (ψ) = Φλ,θ (ϕ) + Φλ,θ (ψ),
(3.8)
Ω
for any ϕ, ψ ∈ H01 (Ω) with suppt(ϕ) ∩ suppt(ψ) = ∅. Lemma 3.3. Suppose that w ∈ H01 (Ω) with w± = 0, then there is a unique pair (t, s) ∈ (0, +∞) × (0, +∞) such that tw+ + sw− ∈ Mλ,θ . In particular, Mλ,θ = ∅. Moreover, for all t, s ≥ 0 such that (t, s) = (t, s), we have w + − Φλ,θ (tw+ + sw− ) = hw λ (t, s) > hλ (t, s) = Φλ,θ (tw + sw ).
Proof. From (M1 ), (3.1), (3.3) and Sobolev embedding we have ∗ ∗ ελC 2 + 2 λ m0 − t w − (ε + Cε,2∗ ) t2 |w+ |22∗ ≥ 2 μ μ ∗ ∗ ελC m0 λ − s2 w− 2 − (ε + Cε,2∗ ) s2 |w− |22∗ , + 2 μ μ
hw λ (t, s)
for some C > 0. Thus, hw λ (t, s) is positive for sufficiently small t, s ≥ 0.
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On the other hand, by Mθ (t) ≤ θ we obtain hw λ (t, s) ≤
θ 2 + 2 t w − t2 λ 2
Ω
θ + s2 w− 2 − s2 λ 2
F (tw+ ) 2 (tw+ )
Ω
w+
2
F (sw− ) 2 (sw− )
w−
2
,
which implies lim
|(t,s)|→+∞
hw λ (t, s) = −∞,
by using (3.4). Therefore, there exists a pair (t, s) such that hw (t, s) = max hw λ (t, s). t,s≥0
We claim that t, s > 0. Indeed, by (3.8) we have w w w w hw λ (t, s) ≥ hλ (t, s) ≥ hλ (t, 0) + h (0, s) > hλ (t, 0),
for sufficiently small s > 0, which implies s > 0. Similarly, we get t > 0. Now it is easy to see that tw+ + sw− ∈ Mλ,θ , due to the fact that tw+ + sw− ∈ Mλ,θ
⇔
Ψw (t, s) = 0,
t, s > 0.
Next we try to give the uniqueness of (t, s). Without loss of generality, we may assume that w ∈ Mλ,θ and (a, b) ∈ (0, +∞) × (0, +∞) such that w = aw+ + bw− ∈ Mλ,θ . The proof shall be completed by showing that (a, b) = (1, 1). Actually, without loss of generality, we suppose b ≤ a. Then, repeating some arguments explored in [8] for the proof of Lemma 2.4, item (a) in that paper, we obtain 1 ≤ b ≤ a ≤ 1, by using w ∈ Mλ,θ , w ∈ Mλ,θ , (M2 ) and (f3 ).
i.e.
a = b = 1,
2
Corollary 3.2. If w ∈ H01 (Ω) with w± = 0 and Φλ,θ (w), w± ≤ 0, then there is a unique pair (t, s) ∈ (0, 1] × (0, 1] such that tw+ + sw− ∈ Mλ,θ . Proof. Lemma 3.3 yields the existence and uniqueness of pair (t, s), and the claim (t, s) ∈ (0, 1] × (0, 1] holds by repeating some arguments explored in [8] for the proof of Lemma 2.4, item (a) in that paper. 2
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Corollary 3.3. If w ∈ Mλ,θ , then (i) For all t, s ≥ 0 such that (t, s) = (1, 1), we have w Φλ,θ (w) = hw λ (1, 1) > hλ (t, s);
(ii) det (Ψw ) (1, 1) > 0. Proof. Item (i) is an immediate consequence of Lemma 3.3. Let us prove item (ii). Consider the notations + − + Ψw 1 (t, s) = Φλ,θ (tw + sw ), w ,
+ − − Ψw 2 (t, s) = Φλ,θ (tw + sw ), w .
Thus Ψw 1 (t, s)
− 2
= tMθ (t w + s w )w − λ 2
+ 2
2
+ 2
f (tw+ )w+ , Ω
and 2 + 2 2 − 2 − 2 Ψw 2 (t, s) = sMθ (t w + s w )w − λ
f (sw− )w− ,
Ω
and note that ∂Ψw 1 (t, s) = Mθ (t2 w+ 2 + s2 w− 2 )w+ 2 ∂t + 2t2 Mθ (t2 w+ 2 + s2 w− 2 )w+ 4 − λ
f (tw+ )(w+ )2 ,
Ω
implies ∂Ψw 1 (1, 1) = Mθ (w2 )w+ 2 + 2Mθ (w2 )w+ 4 − λ ∂t
f (w+ )(w+ )2 .
Ω
Using (3.5) and Φ (w), w+ = 0 in the expression of ∂Ψw 1 (1, 1) < (μ − 1)Mθ (w2 )w+ 2 − λ ∂t =λ
∂Ψw 1 ∂t (1, 1)
we obtain
f (w+ )(w+ )2 − 2Mθ (w2 )w+ 2 w− 2
Ω
(μ − 1)f (w+ )w+ − f (w+ )(w+ )2 − 2Mθ (w2 )w+ 2 w− 2 ,
Ω
and then, from (3.2) we get ∂Ψw 1 (1, 1) < −2Mθ (w2 )w+ 2 w− 2 . ∂t
(3.9)
Arguing on the same way we conclude ∂Ψw 1 (1, 1) < −2Mθ (w2 )w+ 2 w− 2 . ∂s
(3.10)
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Since ∂Ψw ∂Ψw 1 2 (1, 1) = (1, 1) = 2Mθ (w2 )|w+ 2 w− 2 , ∂s ∂t and considering (3.9) and (3.10), we conclude that
det (Ψw ) (1, 1) > 0, and the item (ii) is proved. 2 Remark 3.1. For any w ∈ Mλ,θ , by item (i) of Corollary 3.3 and (3.8) we have Φλ,θ (w) ≥ Φλ,θ (tw+ w+ + tw− w− ) ≥ Φλ,θ (tw+ w+ ) + Φλ,θ (tw− w− ), where tw+ and tw− are given by applying Lemma 3.2 to w+ and w− respectively. Recall the definition of cλ,θ , it is easy to see that Φλ,θ (w) ≥ 2 cλ,θ . Then we conclude that cλ,θ :=
inf
w∈Mλ,θ
Φλ,θ (w) ≥ 2 cλ,θ .
Moreover, if cλ,θ is achieved, then cλ,θ > 2 cλ,θ . cλ,θ and the The next lemma tries to build a relationship between the minimizers (if there exists) of critical points of Φλ,θ , and then show more information about the minimizers. cλ,θ is attained by some u ∈ Nλ,θ , then u is a critical point of Φλ,θ . Moreover, the Lemma 3.4. Suppose that minimizer u has constant sign. Proof. Let G(v) := Mθ (v2 )v2 − λ
Ω
f (v)v, then for v ∈ Nλ,θ we have
G (v), v = 2Mθ (v2 )v2 + 2Mθ (v2 )v4 − λ < μMθ (v2 )v2 − λ = −λ
f (v)v 2 + f (v)v
Ω
f (v)v 2 + f (v)v
Ω
f (v)v 2 − (μ − 1)f (v)v ≤ 0,
Ω
by taking advantage of (3.5), G(v) = 0 and (3.2). Thus, Nλ,θ has a C 1 structure and is a manifold. cλ,θ is attained by some u ∈ Nλ,θ , and then u ∈ Nλ,θ is a critical point of Φλ,θ |Nλ,θ . Moreover, there If exists a Lagrange multiplier ν such that Φ (u), v = ν G (u), v ,
∀v ∈ H01 (Ω).
Let v = u, by G (u), u = 0, we have ν = 0, then u is a critical point of Φλ,θ . If u± = 0, then u ∈ Mλ,θ . By Remark 3.1 and Lemma 3.1 we get a contradiction cλ,θ = Φλ,θ (u) ≥ cλ,θ ≥ 2 cλ,θ > 0. Thus, the minimizer u has constant sign. 2
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The following lemma is a version of Lemma 3.4 for the case of cλ,θ with respect to Mλ,θ , the proof of which is more complicated. Lemma 3.5. Suppose that cλ,θ is attained by w ∈ Mλ,θ , then w is a critical point of Φλ,θ . Moreover, the minimizer w has precisely two nodal domains. Proof. The proof of the first part is similar to that in [8], so we omit it here. For the proof of the claim that w has exactly two nodal domains, we assume by contradiction that w = v1 + v2 + v3 with vi = 0, v1 ≥ 0, v2 ≤ 0,
suppt(vi ) ∩ suppt(vj ) = ∅,
i = j (i, j = 1, 2, 3).
Setting u = v1 + v2 , we see that u± = 0. Moreover, using the fact that Φλ,θ (w) = 0, by (M1 ) we get ≤ 0. Consequently, by Corollary 3.2, there exist t, s ∈ (0, 1] such that tu+ + su− ∈ Mλ,θ , i.e. tv1 + sv2 ∈ Mλ,θ , let v := tv1 + sv2 and so Φλ,θ (v) ≥ cλ,θ . From Φλ,θ (w), w = 0, (3.3), (3.6) and Φλ,θ (v), v = 0, we have a contradiction Φλ,θ (u), u±
cλ,θ = Φλ,θ (w) = Φλ,θ (w) −
1 Φ (w), w μ λ,θ
1 1 2 Mθ (w22 )w2 + λ = M θ (w ) − 2 μ +λ
Ω
Ω
+λ
Ω
1 f (tv1 )tv1 − F (tv1 ) μ
Ω
1 f (sv2 )sv2 − F (sv2 ) μ
Ω
=
1 f (v1 )v1 − F (v1 ) μ
1 1 f (v2 )v2 − F (v2 ) + λ f (v3 )v3 − F (v3 ) μ μ
1 1 2 Mθ (v22 )v2 + λ > M θ (v ) − 2 μ
1 1 Mθ (v2 ) − Mθ (v22 )v2 + λ 2 μ
1 f (v)v − F (v) μ
Ω
1 = Φλ,θ (v) − Φλ,θ (v), v = Φλ,θ (v) ≥ cλ,θ . μ Thus, the minimizer w has precisely two nodal domains. 2 At last, we intend to give an estimate for cλ,θ as λ → +∞, which shall play a vital role in the proof of Theorem 2.1 and Theorem 1.1. To be more precise, the estimate can give the asymptotic behavior of the ground state (least energy nodal solution) of problem (Tλ,θ ), and then show that the ground state (least energy nodal solution) of the auxiliary problem is a constant sign (sign-changing) solution of the original problem (Pλ ) for λ large. Lemma 3.6. cλ,θ is non-increasing in λ > 0 and
lim cλ,θ = 0.
λ→+∞
Proof. For any given λ2 > λ1 > 0, for any w ∈ Mλ1 ,θ , by Corollary 3.3 we have w Φλ1 ,θ (w) = hw λ1 (1, 1) ≥ hλ1 (t, s)
= hw λ2 (t, s) + (λ2 − λ1 ) Ω
F (tw+ + sw− ) > hw λ2 (t, s) ≥ cλ2 ,
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where t, s are given by applying Lemma 3.3 to w for the case Mλ2 ,θ . Thus we conclude that cλ1 ,θ ≥ cλ2 ,θ . Fix w ∈ M1,θ , then we obtain Φλ,θ (w), w±
=
Φ1,θ (w), w±
− (λ − 1)
f (w± )w± ≤ 0,
Ω
for any λ > 1. Thus, by Corollary 3.2, there exist tλ , sλ ∈ (0, 1] such that tλ w+ + sλ w− ∈ Mλ,θ . We claim that (tλ , sλ ) → (0, 0) as λ → +∞. Thus, cλ,θ ≤ Φλ,θ (tλ w+ + sλ w− ) ≤
1 2 + 2 Mθ (tλ w + s2λ w− 2 ) → 0, 2
as λ → +∞, and the proof is complete. Actually, we assume by contradiction that there exists λn → +∞ such that (tλn , sλn ) (0, 0). Due to the boundedness of {(tλn , sλn )}, up to a subsequence, there exists (α, β) ∈ [0, 1] × [0, 1] \ {(0, 0)} such that (tλn , sλn ) → (α, β),
as λn → +∞.
Let wλn := tλn w+ + sλn w− and w := αw+ + βw− , then wλn → w = 0
in H01 (Ω),
as λn → +∞,
which gives a contradiction Mθ (w2 )w2 =
lim
λn →+∞
Mθ (wλn 2 )wλn 2 =
lim
λn →+∞
λn
f (wλn )wλn = +∞. Ω
Thus, cλ,θ is non-increasing in λ > 0 and lim cλ,θ = 0. λ→+∞
2
4. Proof of Theorem 2.1 Recall the definition of cλ,θ in Section 3, it’s easy to see that cλ,θ is well defined from Lemma 3.3 and positive from Lemma 3.1. Thus, there exists a minimizing sequence {wn } in Mλ,θ which is bounded from Lemma 3.1 again. From the boundedness of {wn } and the compactness of the embedding H01 (Ω) → Lp (Ω) (2 < p < 2∗ ), without loss of generality, we can assume, up to a subsequence, that there exist w, w1 , w2 ∈ H01 (Ω) such that wn w,
wn+ w1 ,
wn− w2
in H01 (Ω),
wn → w,
wn+ → w1 ,
wn− → w2
in Lq (Ω), q ∈ (2, 2∗ ).
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Then, we have w+ = w1 ≥ and w− = w2 ≤ 0 by the continuity of the transformations w → w± from L (Ω) to Lq (Ω). Moreover, by the strong convergence of {wn± } and item (iii) of Lemma 3.1, we conclude that w± = 0 and consequently w = w+ + w− is sign-changing. At this point, we can prove that w ∈ Mλ,θ and Φλ,θ (w) = cλ,θ . Indeed, since f has a subcritical growth, we get q
f (wn± )wn± →
Ω
f (w± )w± ,
Ω
F (wn± ) →
Ω
F (w± ).
Ω
Thus, since Φλ,θ (wn ), wn± = 0, by (M1 ) we have Φλ,θ (w), w+ ≤ 0,
Φλ,θ (w), w− ≤ 0,
and then, by Corollary 3.2, there exists (t, s) ∈ (0, 1] × (0, 1] such that tw+ + sw− ∈ Mλ,θ . Noting that 1 1 Φλ,θ (w) − Φλ,θ (w), w ≤ lim inf Φλ,θ (wn ) − Φλ,θ (wn ), wn n→+∞ μ μ = lim inf Φλ,θ (wn ) = cλ,θ , n→+∞
we only need to show t = s = 1. Let w = tw+ + sw− , then Φλ,θ (w) = Φλ,θ (w) −
1 Φ (w), w μ λ,θ
1 1 2 Mθ w2 )w2 + λ = M θ (w ) − 2 μ +λ
1 f (tw+ )tw+ − F (tw+ ) μ
Ω
1 f (sw− )sw− − F (sw− ) . μ
Ω
If (t, s) = (1, 1), using w2 < w2 , (3.3) and (3.6), we have 1 1 2 Mθ w2 )w2 + λ Φλ,θ (w) < M θ (w ) − 2 μ +λ
1 f (w+ )w+ − F (w+ ) μ
Ω
1 f (w− )w− − F (w− ) μ
Ω
= Φλ,θ (w) −
1 Φ (w), w , μ λ,θ
which gives a contradiction that cλ,θ ≤ Φλ,θ (w) < cλ,θ . Thus, we conclude w ∈ Mλ,θ and Φλ,θ (w) = cλ,θ . And then, Lemma 3.5 yields that wλ,θ := w is a least energy nodal solution of problem (Tλ,θ ), which has precisely two nodal domains. In addition, the asymptotic behavior (2.7) of wλ,θ follows from Lemma 3.6 and Lemma 3.1.
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Similarly, we can prove that cλ,θ is positive and is attained by some uλ,θ ∈ Nλ,θ . And then, uλ,θ is a ground state solution of problem (Tλ,θ ) with definite sign by Lemma 3.4, and its asymptotic behavior (2.7) follows from Lemma 3.6, Remark 3.1 and Lemma 3.1. cλ,θ > 0 and Remark 3.1, Finally, we get the conclusion (2.6) by using Φλ,θ (wλ,θ ) = cλ,θ , Φλ,θ (uλ,θ ) = and the proof of Theorem 2.1 is complete. 2 5. Proof of the main results Proof of Theorem 1.1. Fixing θ, we have a ground state solution uλ,θ and a least energy nodal solution wλ,θ of problem (Tλ,θ ) given by Theorem 2.1. Moreover, by (2.7), there exists λ∗ > 0 such that uλ,θ 2 , wλ,θ 2 ≤ τ,
∀λ ≥ λ∗ ,
where τ ∈ (0, δ) is given in (2.2). Thus, for all λ ≥ λ∗ , we conclude that M (uλ,θ 2 ) = Mθ (uλ,θ 2 ) and M (wλ,θ 2 ) = Mθ (wλ,θ 2 ). Let uλ := uλ,θ and wλ := wλ,θ , then uλ and wλ are the desired solutions. 2 Proof of Theorem 1.2. The proof of Theorem 1.2 is similar to that of Theorem 2.1. As the proof of Theorem 2.1, we define the Nehari manifold of (Pλ ) given as Nλ := u ∈ H01 (Ω) \ {0} | Φλ (u), u = 0 , and the corresponding nodal Nehari manifold Mλ := w ∈ H01 (Ω) | w± = 0, Φλ (w), w± = 0 . Furthermore, two candidate critical levels also be defined as follows cλ := inf Φλ (u), u∈Nλ
cλ := inf Φλ (w). w∈Mλ
It’s not difficult to see that, all the preliminary results, corollaries and remark, and almost all the lemmas in Section 3, except item (ii) of Lemma 3.1, are valid here and only the proofs of Lemma 3.2 and Lemma 3.3 need some minor changes. To be more precise, what we only have to show in the proof of Lemma 3.2 for the case of Φλ is that lim g(t) = −∞,
(5.1)
t→+∞
where g(t) := Φλ (tu). Actually, from (M1 ) and (M2 ) we have M (t) ≤ M (1) + M (1)t
μ−2 2
,
which implies (t) ≤ M (1)t + 2 M (1)t μ2 . M μ Thus,
(5.2)
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⎡ g(t) ≤
M (1) u2 t2 − ⎣λ 2
Ω
⎤ F (tu) μ M (1) uμ ⎦ tμ |u| − |tu|μ μ
yields (5.1) by taking advantage of (f4 ). Similarly, by (5.2) and (f4 ) we can get lim
|(t,s)|→+∞
Φλ (tw+ + sw− ) = −∞,
which completes the proof of Lemma 3.3 for the case of Φλ . cλ is still positive. It is worth pointing out that although the item (ii) of Lemma 3.1 fails here in general, Indeed, from (3.3), (3.6) and item (i) of Lemma 3.1 we obtain 1 Φ (u), u μ λ 1 1 2 1 1 (u2 ) − M (u2 )u2 ≥ M (ρ ) − M (ρ2 )ρ2 ≥ M 2 μ 2 μ
Φλ (u) = Φλ (u) −
(5.3)
for all u ∈ Nλ , which yields cλ > 0. The major difficulty that we need to overcome in the proof of Theorem 1.2 is the boundedness of the minimizing sequence, which is caused by the fail of item (ii) in Lemma 3.1, more generally the coerciveness of Φλ on Nλ . Fortunately, we can get through this difficulty by a different way and more analysis, see e.g. [17]. Actually, we can prove a more general result about boundedness, that is Lemma 5.1. Under the assumption of Theorem 1.2, assume that {wn } ⊂ Nλ with then {wn } is bounded in
lim Φλ (wn ) = c > 0,
n→+∞
H01 (Ω).
For the proof of the boundedness, we assume by contradiction that wn → +∞, up to a subsequence, and consider vn :=
wn wn .
as n → +∞,
Passing to a subsequence, we may assume that
vn v
in H01 (Ω),
vn → v
in Lq (Ω), ∀q ∈ [2, 2∗ ),
vn → v
a.e. on Ω.
If v = 0, from (5.2) the equality (wn 2 ) 1M c + on (1) F (wn ) = − λ |vn |μ wn μ 2 wn μ |wn |μ Ω
implies the inequality M (1) 1 1 c + on (1) ≤ M (1) + −λ wn μ μ 2 wn μ−2 which gives a contradiction
Ω
F (wn ) |vn |μ , |wn |μ
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M (1) ≥λ μ
lim inf
n→+∞
Ω
17
F (wn vn ) μ μ |v| = +∞, |wn vn |
by taking advantage of (f4 ) and Fadou’s lemma. Thus we have v = 0 and vn → 0 in Lq (Ω) for q ∈ (2, 2∗ ). Fix q ∈ (2, 2∗ ) and an R > and (3.3), and using Lebesgue Dominated Convergence theorem, we have lim
2c m0
12
. From (3.1)
F (Rvn ) =
n→∞
Ω
lim F (Rvn ) = 0.
n→∞ Ω
By Lemma 3.2 we have Φλ (wn ) = Φλ (wn vn ) ≥ Φλ (Rvn ) ≥
m0 R 2 −λ 2
F (Rvn ). Ω 2
The left side of this inequality converges towards c and the right side tends to m02R > c, we also have a contradiction. Summing up the above, {wn } is bounded in H01 (Ω). At this point, we can complete the proof of Theorem 1.2 by repeating the argument in the proof of Theorem 2.1. 2 Remark 5.1. If the following property (M3 )
lim
t→+∞
1 1 M (t) − M (t)t 2 μ
= +∞
is satisfied, we can also get the coerciveness of Φλ on Nλ by using (M3 ) in (5.3), and then the boundedness of the minimizing sequence can be obtained easily. Unfortunately, (M3 ) is not always hold for functions which verify the hypotheses (M1 )–(M2 ). For example, let N = 3, μ = 4, M (t) = m0 + m0 arctan(t)t
and f (t) = 4t3 ln(1 + |t|) +
t4 sign(t), 1 + |t|
then a straightforward computation yields 1 m0 1 M (t) − M (t)t = arctan(t), 2 4 4 which is bounded. Moreover, assuming u ∈ Nλ , we have 1 1 Φλ (u) = m0 arctan(u2 ) + λ 4 4
Ω
|u|5 , 1 + |u|
which does not seem to give a positive answer to the coerciveness of Φλ on Nλ . Remark 5.2. It is not difficult to see that the main result of [8] is a special case of Theorem 1.2 in this paper by letting N = 3, μ = 4 and λ = 1. Acknowledgment The author would like to thank Professor Zhi-Qiang Wang for useful discussions and suggestions.
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