Multiple solutions for a modified quasilinear Schrödinger elliptic equation with a nonsquare diffusion term

Let Ω be a nonempty bounded open set of the real Euclidean space RN (N ≥ 2) with C1boundary ∂Ω, consider the multiple solutions for the following quasilinear Schrödinger elliptic equation with the p-Laplacian and nonsquare diffusion term:

where ∆pu = div(∇u│p-2 ∇u), N < p ≤ 2α, λ ≥ 0 is a parameter, f : Ω x R → R is a continuous function.

Equation (1) involves a quasilinear and nonconvex diffusion term ∆p( │u│ 2α) u 2α−2u. In the literature, it is referred as so-called modified nonlinear Schrödinger equation. For the case p = 2, the solution of (1) is related to standing wave solutions of the following quasilinear Schrödinger equation:

where z : R x Rn → C, V : Rn → R is a given potential, h and g are real functions, κ is a real constant. Putting z(t, x) = e−iβtu(x) in (2), where β R and u(x) > 0 is a real function, the quasilinear equation (2) reduces to the following modified elliptic form:

If h(s) = s, then (3) turns into a superfluid film equation in plasma physics

Kurihara [8] used this equation to model the time evolution of the condensate wave function in superfluid film. Moreover, if h(s) = (1 + s)1/2, equation (3) is transformed to the following elliptic form:

which is a model of the self-channeling of a high-power ultrashort laser in matter [7, 16]. Many mathematical methods, such as dual approach [3, 22, 24,25,26], iterative techniques [19,27,28,29,30], fixed point theorem [5,14,21], variational methods [6,15,23] and normal boundary intersection method [9,10], have been employed to solve the properties and control problems for various differential equations. In particular, by using a constrained minimization argument Poppenberg et al. [15] established the existence of positive ground state solution for quasilinear Schrödinger equation (4). Colin and Jeanjean [3], João Marcos and Severo [6] studied the existence of positive solutions for (4) by the change of variables. The Nehari method and the symmetric mountain pass lemma were also used to establish the existence of solutions in [2, 4, 11]. In [13], Liu et al. studied the following quasilinear Schrödinger equation:

where λ ≥ 0, 4α < p + 1 < 4αN/(N − 2), α ≥ 1/2, V ∈ C(RN ) and (V) There exists V0 > 0 such that V (x) ≥ V0 in RN . Moreover, V (x) → as │x│ → ∞, or more generally, for every M > 0, meas {x RN : V (x) ≤ M} < , where “meas” denotes the Lebesgue measure in RN .

Condition (V) is an essential assumption, which guarantees that the embedding E ‹→ Ls(RN ) is compact for 2 ≤ s < 2N/(N − 2), where

is a subspace of W 1,2(RN ) with the norm

Clearly, assumption (V) fails to hold for a general continuous and bounded function. Thus, if the potential V (x) fails to satisfy (V), whether the multiple solutions of problem (5) still exist or not? In order to answer this question, in this paper, we investigate the more general modified nonlinear Schrödinger equation (1) and get a positive answer, i.e., if the potential V (x) is a general continuous and bounded function, then there exist the multiple solutions to the quasilinear Schrödinger elliptic equation with the p-Laplacian and nonsquare diffusion term (1) under suitable growth conditions.

The rest of this paper is organized as follows. In Section 2, with help of a change of variables, we set up the variational framework for problem (1) and give some lemmas of the functional associated with problem (1). In Sections 3 and 4, by using Riccer’s critical point theorem we give the proof of main results.

Let E = W 1,p(Ω) (p ≥ 1) be the Sobolev spaces with the norm

We focus on the existence of nontrivial weak solutions of problem (1). A function u is called a weak solution of problem (1) if u ∈ W 1,p(Ω) and for any ϕ ∈ C∞(Ω), one has

where F (x, u) = ∫0 u f (t, ξ) dξ. But we notice that the natural functional of problem (1)

may not be well defined and not Gâteaux-differentiable in the corresponding Sobolev space E.

Thus, inspired by [12], we define a function h by

Let u = h(v), then

Moreover, the corresponding energy functional J(v) is well defined on W 1,p(Ω). Since C0∞(Ω) is dense in W 1,p(Ω), if v ∈ W 1,p(Ω) is a critical point of the functional J, i.e, for any ϕ ∈ W 1,p(Ω),

then v is a weak solution of the equation

Thus, from (6) and (7) it is easy to know that u = h(v) is a weak solution of problem (1). As the result, it is sufficient to consider the existence of solutions of (7) in W 1,p(Ω).

The following lemma can be found in [2]:

Lemma 1. The function h(t) enjoys the following properties:

(h1) h ∈ C2 is uniquely defined, odd, increasing and invertible in R; (h2) 0 < h′(t) ≤ 1 for all t ∈ R;

(h3) |h(t)| ≤ |t| for all t ∈ R;

(h4) limt→0 h(t)/t = 1;

(h5) |h(t)| ≤ (2α)1/(2pα)|t|1/(2α) for all t ∈ R;

(h6) h(t)/2 ≤ αth′(t) ≤ αh(t), t ≥ 0, αh(t) ≤ αth′(t)| ≤ h(t)/2, t ≤ 0;

(h7) There exists a ∈ (0, (2α)1/(2pα)] such that h(t)t−1/(2α) → a as t → +∞;

(h8) There exists b0 > 0 such that

(h9) For each τ > 0, there exists

m ∈ N, such that |h(τt)| ≤ χ(τ )|h(t)| for all t ∈ R; (h10) h2(t)/2 ≤ αth′(t)h(t) ≤ αh2(t) for all t ∈ R.

Notice that p > N , W 1,p(Ω) ‹ → C(Ω) is compact. Thus, there exists a constant c > 0 such that

where ǁuǁ ∞ = supx Ω u(x) .

Different from [4, 11, 13], the following assumption on potential is adopted in this paper:

(V) V ∈ C(Ω), and there exist two constants V0, V1 > 0 such that V0 ≤ V (x) ≤ V1, x ∈ Ω.

Now define two functionals Φ, Ψ : E → R as follows:

For any v, w ∈ E, we have Φ, Ψ ∈ C1(E, R) and

Lemma 2. For fixed r > 0 with Φ(v) ≤ r, v ϵ E, there exists a constant Q > 0 independent of r such that

Proof. Let v = 0, otherwise, the conclusion holds. In the following, we argue by contradiction to prove (9).

Suppose that there exists a sequence {vn} ⊂ E satisfying vn /= 0 for all n ∈ N such that

Set wn = vn/ǁvnǁ, then ǁwnǁ = 1. Noticing the compactness of embedding E ‹→ Ls for s ∈ [1, +∞) up to a subsequence, we have wn(x) w(x) in E, wn(x) → w(x) in Ls(Ω) for s ∈ [1, +∞) and wn(x) → w(x) a.e. on Ω. It follows from (10) and

that

Now according to the strategy in [26], we claim that for any ε > 0, there exists a constant τ > 0 independent of n such that meas(Bn := {x ∈ Ω: |vn| ≥ τ }) ≤ ε, where meas(·) denotes the standard Lebesgue measure.

In fact, if not, there exists ε0 > 0 such that meas(An) ≥ ε0, where An = {x ∈ Ω:

|vn| ≥ n}. By (h8) and the Fatou lemma we get

The above fact contradicts with the boundedness of {Φ(vn)} . Therefore, the above conclusion is valid.

Next, it follows from the Hölder inequality and the Sobolev embedding theorem that there exists ε small enough such that

where C1 is a constant, which is independent of ε.

On the other hand, noticing that if |vn(x)| ≤ τ , then |vn(x)|/τ ≤ 1, by (h8) we have

Thus, it follows from (h9) of Lemma 1, (12) and (10) that

Combining (11) and (13), one has

which implies that 1 ≤ 1/4, a contradiction. So the proof is completed.

Lemma 3. Assume that V (x) satisfies (V), then Φ′ is coercive, hemicontinuous and uniformly monotone.

Proof. Firstly, by (h4) and (h7) of Lemma 1 we have

which implies that for any sufficiently small ϵ > 0, there exists a constant Cє > 0 such that

On the other hand, for any v ∈ E with ǁvǁ > 1, (h10) of Lemma 1 and (14) yield

Notice that E ‹ →Ls for s ϵ [p, p∗) is continuous, then for any v ϵ E with v > 1, choose sufficiently small ε such that

It follows from N < p ≤ 2α, N ≥ 2, (15) and (16) that limǁvǁ→∞ Φ′(v), v / v = , which implies that Φ′ is coercive. The fact that Φ′ is hemicontinuous can be verified using standard arguments. In addition, with the help of Theorem 26(A) in [20], as well as J(v) = I(h(v)) and the inequality

we know that Φ′ exists and is continuous.

In this section, we show the existence of three solutions of (1). The main tool used for analysis is the Riccer’s critical point theorem [1, 18], which is given below for reader’s convenience.

Lemma 4. Let E be a separable and reflexive real Banach space, Φ : E → R be a continuously Gâteaux-differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on E∗, and Ψ : X → R be a continuously Gâteaux-differentiable functional whose Gâteaux derivative is compact.

Assume that

(i) lim ǁuǁ→+∞(Φ(z) + λΨ (z)) = +∞ for all λ ∈ (0, +∞);

(ii) There are r > 0, z0, z1 ∈ E such that Φ(z0) < r < Φ(z1);

Then there exist an open interval Λ ⊂ (0, ∞) and a positive real number ρ such that for each λ ∈ Λ, the equation

has at least three solutions in E whose norms are less than ρ.

Before stating our main results, we firstly denote two constants

where c, V0, V1 and Q are defined by (8), (V) and Lemma 1, Ω is the Lebesgue measure of Ω. Then some assumptions on F (x, s) to be used are also listed below:

(F1) There exist a function a(x) ∈ L1(Ω) and 0 < σ < p such that for all (x, s) ∈ Ω × R, F (x, s) ≤ a(x)(1 + |s|σ );

(F2) F (x, 0) = 0 for any x ∈ Ω;

(F3) There exists t0 ∈ R with |t0| > 1 such that

Now we state our main result here.

Theorem 1. Suppose (V) and (F1)–(F3) hold. Then there exist an open interval Λ (0, ∞) and a positive real number ρ > 0 such that for any λ ϵ Λ, the quasilinear elliptic equation (1) has at least three weak solutions whose norms are less than ρ.

Proof. By the definitions of Φ and Ψ we know that Ψ ′ is compact and Φ is weakly lower semicontinuous. Further, from Lemma 3 we know that (Φ′)−1 is well defined and continuous. Now we show that the hypotheses of Lemma 4 are fulfilled.

It follows from (F1), (7) and (15)–(16) that for any λ ≥ 0,

Since 0 < σ < p ≤ 2α, we have limǁvǁ→∞(Φ(z) + λΨ (z)) = ∞, and (i) is verified.

Now let z0 = 0, z1 = s0 = h−1(t0), |t0| > 1, then |t0| = |h(s0)|. We denote r = V0|Ω|/p, then

Thus, (ii) of Lemma 4 is satisfied.

On the other hand, from (F2) and (F3) we get Ω F (x, t0) dx ≥ 0 and

Next, we focus our attention on the case when v ∈ E with Φ(v) ≤ r. By (7) and (8) we have r ≥ Φ(v) ≥ Qǁvǁp ≥ Q(ǁvǁ∞/c)p, which implies that |v(x)| ≤ c(r/Q)1/p = c(V0|Ω|/(pQ))1/p = k for all x ∈ Ω. The above inequality and (h3) of Lemma 1 yield

From (17), (18) and (F3) it is easy to get that condition (iii) of Lemma 4 holds.

Thus, all the hypotheses of Lemma 4 are satisfied, and hence, according to Lemma 4, there exist an open interval Λ C (0, ∞ ) and a positive real number ρ > 0 such that for any λ ϵ Λ, the quasilinear elliptic equation (1) has at least three weak solutions whose norms are less than ρ.

Theorem 2. Assume (V), (F1)–(F2) and the following condition hold:

(F3∗) There exists a constant M > 0 such that F (x, z) ≤ 0, (x, |z|) ∈ Ω × [0, M ] and lim|z|→∞ F (x, z) > 0 for x ∈ Ω uniformly holds.

Then there exist an open interval Λ c (0, ∞ ) and a positive real number ρ > 0 such that for any λ ϵ Λ, the quasilinear elliptic equation (1) has at least three weak solutions whose norms are less than ρ.

Proof. By (F1), similar as the proof of Theorem 1, it is easy to know that hypothesis (i) of Lemma 4 holds. Thus, we only need to verify hypotheses (ii) and (iii). In fact, it follows from (F3∗) that for any x ∈ Ω, there exists a sufficiently large

such that F (x, t0) > 0. We take z0 = 0, z1 = s0 = h−1(t0), then 1 < t0 = h(s0) . Denote r = Q(M/c)p, we have

Thus, hypothesis (ii) of Lemma 4 is satisfied.

On the other hand, from (F2) and (F3∗) we have

Moreover, for Φ(v) ≤ r, v ∈ E, by (8) and Lemma 2 we have

The above inequality and (h3) of Lemma 1 show that

(18) and (19) show that condition (iii) of Lemma 4 holds. According to Lemma 4, the conclusion of Theorem 2 also holds.

In this section, we use an infinitely many critical points theorem to obtain the multiple solutions result of problem (1).

Let E be a reflexive real Banach space, Φ : E → R be a (strongly) continuous, coercive sequentially weakly lower semicontinuous and Gâteaux-differentiable functional, Ψ : E → R be a sequentially weakly upper semicontinuous and Gâteaux-differentiable functional.

For all r > infE Φ, let

and

Lemma 5. (See [17].) Suppose E, Φ, Ψ satisfy the above assumptions, then the following conclusions hold:

(i) If γ < +∞ then, for each λ ∈ (0, 1/γ), the following alternative holds: either the functional Φ − λΨ has a global minimum, or there exists a sequence {zn} of critical points (local minima) of Φ − λΨ such that limn→+∞ Φ(zn) = +∞.

(ii) If δ < +∞, then for each λ ∈ (0, 1/δ), the following alternative holds: either there exists a global minimum of Φ, which is a local minimum of Φ − λΨ, or there exists a sequence {zn} of pairwise distinct critical points (local minima) of Φ λΨ with limn→+∞ Φ(zn) = infE Φ, which weakly converges to a global minimum of Φ.

Suppose f : Ω × R → R+ is continuous and denote

We state the result of the multiple solutions as follows:

Theorem 3. Assume that l/L < pQ/(cpV1 Ω ) hold. Then for any λ (V1 Ω /(pL), Q/(cpl), the quasilinear elliptic equation (1) has an unbounded sequence of weak solutions in W 1,p(Ω).

Proof. Firstly, for any v ∈ E, define

Then Φ : E → R is a continuous, coercive sequentially weakly lower semicontinuous and Gâteaux-differentiable functional, Ψ : E → R is a sequentially weakly upper semicontinuous and Gâteaux-differentiable functional.

Take λ ∈ (V1|Ω|/(pL), 1/(pcpl)), and let {κn} be a real sequence satisfying limn→∞ κn = ∞, and so we have

Let rn = Q(κn/c)p, n ϵ N, and consider Φ(z) < rn. According to (8) and Lemma 2, we have

Consequently, from (20) and (h3) of Lemma 1 one has

which implies that

Now we show that the functional Φ − λΨ is unbounded from below. To do this, we take a real sequence {en} such that limn→∞ en = +∞. Noticing (h8) of Lemma 1, we have h(en) ≥ b0 en1/(2α) → ∞, n → ∞, and then

Let wn(x) = en, n ∈ N, x ∈ Ω, then we have

and

We divide into two cases for L to prove that Φ − λΨ is unbounded from below.

Case 1. If L < + ∞ , choose 0 < ϵ < L - V1 Ω /(λp), then by (21) there exists N0 > 0 such that for any n > N0, we have

Thus,

It follows from the choice of ϵ that V1|Ω|/p − λ(L − ϵ) < 0, and then one gets limn→∞(Φ(wn) − λΨ (wn)) = −∞.

Case 2. If L = + , we can choose sufficiently large M0 > V1 Ω /(λp), and from (21) there exists NM0 > 0 such that for any n > NM0 , we have

Consequently,

It follows from the choice of M0 that

The above facts show that the functional Φ−λΨ is unbounded from below. According to (i) of Lemma 5, the functional Φ − λΨ admits a sequence {vn} of critical points, that is, {h(vn)} are exactly the weak solutions of the quasilinear elliptic equation (1).

It follows from Theorem 3 that we have the following corollary:

Corollary 1. Assume (V) holds, and l < + ∞ , L = + ∞ . Then for any λ ϵ (0, Q/(cpl)), the quasilinear elliptic equation (1) has an unbounded sequence of weak solutions in W 1,p(Ω).

Denote

then, with help of (h3)–(h4) of Lemma 1 and arguing as in the proof of Theorem 3, we easily obtain the following results:

Theorem 4. Assume (V) holds, and cpV1|Ω|l < LpQ. Then for any λ ∈ (V1|Ω|/(pL), Q/(cpl)), the quasilinear elliptic equation (1) has an unbounded sequence of weak solutions in W 1,p(Ω).

Theorem 5. Assume (V) holds, and l < + ∞ , L = + ∞ . Then for any λ ϵ (0, Q/(cpl)), the quasilinear elliptic equation (1) has an unbounded sequence of weak solutions in W 1,p(Ω).