Electronic Journal of Qualitative Theory of Differential Equations 2008, No. 5, 1-16; http://www.math.u-szeged.hu/ejqtde/

Multiplicity of solutions for a class of quasilinear elliptic equations with concave and convex terms in ℝ ^{∗}

Uberlandio B. Severo ^{†}

Departamento de Matemática

Universidade Federal da Paraíba

58051-900, João Pessoa - PB - Brazil

Abstract

In this paper the Fountain theorem is employed to establish infinitely many solutions for the class of quasilinear Schrödinger equations −L_{p}u + V (x)|u|^{p−2}u = λ|u|^{q−2}u + μ|u|^{r−2}u in ℝ, where L_{p}u = (|u′|^{p−2}u′)′ + (|(u^{2})′|^{p−2}(u^{2})′)′u, λ, μ are real parameters, 1 < p < ∞, 1 < q < p, r > 2p and the potential V (x) is nonnegative and satisfies a suitable integrability condition.

AMS Subject Classification: 35J20, 35J60, 35Q55. Key words and phrases: Quasilinear elliptic equation; Schrödinger equation; Variational method; Fountain theorem; p-Laplacian.

1 Introduction

In this paper we establish multiple solutions for quasilinear elliptic equations of the form

−L_{p}u + V (x)|u|^{p−2}u = λ|u|^{q−2}u + μ|u|^{r−2}u,  (1.1)

u ∈ W^{1,p}(ℝ), where

L_{p}u := (|u′|^{p−2}u′)′ + (|(u^{2})′|^{p−2}(u^{2})′)′u,

λ, μ ∈ ℝ, 1 < p < ∞, 1 < q < p and r > 2p. We assume that the potential V (x) is nonnegative, locally bounded and satisfies the conditions (V_{1}) For some R_{0} ∈ (0, \frac{1}{2p}), there holds V (x) ≥ α > 0 for all x ∈ ℝ such that

∗Research partially supported by Millennium Institute for the Global Advancement of Brazilian Mathematics-IM-AGIMB.

†Tel.:55 83 3216 7434; Fax: 55 83 3216 7277; E-mail Address: uberlandio@mat.ufpb.br

|x| > R_{0}.

(V_{2}) ∫_{|x|≥R_{0}} V (x)^{−1/(p−1)}dx < ∞.

We work with the space of functions

X = {u ∈ W^{1,p}(ℝ) : ∫_{ℝ} V (x)|u|^{p} dx < ∞}

which is a separable and reflexive Banach Space when endowed with the norm

∥u∥^{p} = ∫_{ℝ} |u′|^{p} dx + ∫_{ℝ} V (x)|u|^{p} dx.

Notice that (1.1) is the Euler-Lagrange equation of the functional

F_{λ,μ}(u) = \frac{1}{p}∥u∥^{p} + \frac{2^{p−1}}{p} ∫_{ℝ} |u′|^{p}|u|^{p} dx − \frac{λ}{q} ∫_{ℝ} |u|^{q} dx − \frac{μ}{r} ∫_{ℝ} |u|^{r} dx.

The functional F_{λ,μ} is well defined and of class C^{1} on the space X (see Lemma 2.2) and we study the existence of solutions of (1.1) understood as critical points of F_{λ,μ}. The next theorem contains our main result:

Theorem 1.1 Under the assumptions (V_{1}) − (V_{2}) and supposing 1 < p < ∞, 1 < q < p and r > 2p we have

(a) for every μ > 0, λ ∈ ℝ, equation (1.1) has a sequence of solutions (u_{k}) such that F_{λ,μ}(u_{k}) → ∞ as k → ∞.

(b) for every λ > 0, μ ∈ ℝ, equation (1.1) has a sequence of solutions (v_{k}) such that F_{λ,μ}(v_{k}) < 0 and F_{λ,μ}(v_{k}) → 0 as k → ∞.

After the well-known results of Ambrosetti-Brezis-Cerami [4], problems involving elliptic equations with concave and convex type nonlinearities have been studied by several authors, see for example [2], [6], [9], [11] for semilinear problems and [3], [8],

[18] for quasilinear problems.

The study of problem (1.1) was in part motivated by the works of Bartsch-Willem [6], Poppenberg-Schmitt-Wang [16] and Ambrosetti-Wang [5]. In [6], Bartsch and Willem proved similar results to the Theorem 1.1 for the semilinear problem −Δu = λ|u|^{q−2}u + μ|u|^{r−2}u in an open bounded domain Ω ⊂ ℝ^{N} with Dirichlet boundary conditions and 1 < q < 2 < r < 2^{∗}. Poppenberg, Schmitt and Wang [16] studied the existence of a positive ground state solution for the quasilinear elliptic equation −u′′ + V (x)u − (u^{2})′′u = θ|u|^{p−1}u in ℝ, in the superlinear case by using a constrained minimization argument. Ambrosetti-Wang [5] consider the quasilinear problem −u′′ + [1 + εa(x)]u − k[1 + εb(x)](u^{2})′′u = [1 + εc(x)]u^{q} where q > 1, k, ε are real numbers and a, b, c are real-valued functions belonging to a certain class S. The authors use a variational method, together with a perturbation technique to prove that there exists k_{0} > 0 such that for k > −k_{0} and a, b, c belonging to S, the equation has a positive solution u ∈ H^{1}(ℝ), provided that |ε| is sufficiently small. Equations of type (1.1) were also studied in [1].

The special features of this class of problems in the present paper is that it is defined in ℝ, involves the p-Laplacian operator, the nonlinear term (|(u^{2})′|^{p−2}(u^{2})′)′u and concave and convex type nonlinearities. Here, we adapt an argument developed by Poppenberg-Schmitt-Wang [16]. Our main results complement and improve some of their results, in sense that we are considering a more general class of operators and nonlinearities, and we have allowed that the potential V vanishes in a bounded part of the domain, so that our results are new even in the semilinear case because we deal with a more general class of potential. To obtain multiplicity of solutions for (1.1), we will apply the Bartsch-Willem Fountain theorem as well as the Dual Fountain theorem, see [6], [19].

Equations like (1.1) in the case p = 2 model several physical phenomena. More explicitly, solutions of the equation

−Δu + V (x)u − kΔ(u^{2})u = g(u) in ℝ^{N}  (1.2)

are related to the existence of standing wave solutions for quasilinear Schrödinger equations of the form i∂_{t}z = −Δz + V (x)z − h(|z|^{2})z − kΔf(|z|^{2})f′(|z|^{2})z where V = V (x), x ∈ ℝ^{N}, is a given potencial, k is a real constant and f, h are real functions. For more details on physical motivations and applications, we refer to [5],

[16] and references therein.

There are several recent works on elliptic problems involving versions of (1.2), for instance, see [5], [10], [12], [13], [14], [15]. In [13], by a change of variables the quasilinear problem was transformed to a semilinear one and an Orlicz space framework was used as the working space, and they were able to prove the existence of positive solutions of (1.1) by the mountain-pass theorem. The same method of change of variables was used recently also in [10], but the usual Sobolev space H^{1}(ℝ^{N}) framework was used as the working space and they studied a different class of nonlinearities. In [14], the existence of both one-sign and nodal ground states of soliton type solutions were established by the Nehari method. But in these papers, the authors do not deal with the concave and convex case.

The organization of this work is as follows: Section 2 contains some preliminary results and we state the abstract theorems which we apply in this work. In Section 3, we establish the Palais-Smale condition and Section 4 presents the proof of the main theorem.

Notation. In this paper we make use of the following notation:

• ∫_{ℝ} denotes the integral on the line real and C, C_{0}, C_{1}, C_{2}, ... denote positive (possibly different) constants.

• For R > 0, I_{R} denotes the open interval (−R, R).

• For 1 ≤ p ≤ ∞, L^{p}(ℝ) denotes the usual Lebesgue space with norms

∥u∥_{p} = (∫_{ℝ} |u|^{p}dx)^{1/p} , 1 ≤ p < ∞;
∥u∥_{∞} = inf {C > 0 : |u(x)| ≤ C almost everywhere in ℝ} .

• For 1 ≤ p < ∞, W^{1,p}(ℝ) denotes the Sobolev space modeled on L^{p}(ℝ) with

norm

∥u∥_{1,p} = [∫_{ℝ}(|u′|^{p} + |u|^{p})dx]^{1/p} .

• By ⟨·, ·⟩ we denote the duality pairing between X and its dual X^{∗};

• We denote weak convergence in X by “ ⇀ ” and strong convergence by “ → ”.

2 Abstract framework

In this section we establish some properties of the space X and functional F_{λ,μ}. By the condition (V_{1}), it follows that the embedding X ↪ W^{1,p}(ℝ) is continuous. Indeed,

\begin{aligned}∥u∥_{1,p}^{p} & ≤ ∫_{ℝ} |u′|^{p} dx + ∫_{−R_{0}}^{R_{0}} |u|^{p} dx + α^{−1} ∫_{|x|>R_{0}} V (x)|u|^{p} dx \\ & ≤ max\{1, α^{−1}\}∥u∥^{p} + 2R_{0}∥u∥_{∞}^{p}. \end{aligned}

Since ∥u∥_{∞}^{p} ≤ p∥u∥_{1,p}^{p} , see for instance Brezis [7], we conclude that

∥u∥_{1,p} ≤ (\frac{max\{1, α^{−1}\}}{1 − 2pR_{0}})^{1/p} ∥u∥.

Furthermore, we have the following proposition: Proposition 2.1 Under the conditions (V_{1})−(V_{2}), the embedding from X into L^{s}(ℝ) is continuous for 1 ≤ s ≤ ∞ and compact for 1 ≤ s < ∞.

EJQTDE, 2008 No. 5, p. 4 Proof. Using Hölder’s inequality, we get

\begin{matrix}∫_{|x|≥R_{0}} |u| ≤ & (∫_{|x|≥R_{0}} V (x)|u|^{p} dx)^{1/p} (∫_{|x|≥R_{0}} V (x)^{−1/(p−1)}dx)^{p/(p−1)} \\ ≤ & C∥u∥. \end{matrix}

Hence

∥u∥_{1} = ∫_{−R_{0}}^{R_{0}} |u|dx + ∫_{|x|>R_{0}} |u|dx ≤ 2R_{0}∥u∥_{∞} + C∥u∥ ≤ C_{1}∥u∥.

Since X is immersed continuously in L^{∞}(ℝ), we can conclude by interpolation that the embedding from X into L^{s}(ℝ) is continuous for 1 ≤ s ≤ ∞. Now, we prove the compactness. Let (u_{n}) be a sequence in X satisfying ∥u_{n}∥ ≤ C. Thus, up to a subsequence,

u_{n} ⇀ u_{0} in X.

Without lost of generality, we can assume that u_{0} = 0. Let us show that u_{n} → 0 in L^{1}(ℝ). Given ϵ > 0, for R > 0 large enough we obtain

∫_{|x|≥R} V (x)^{−1/(p−1)}dx < (\frac{ϵ}{2C} )^{(p−1)/p} .

Hence

\begin{matrix}∫_{|x|≥R} |u_{n}|dx ≤ & (∫_{|x|≥R} V (x)|u_{n}|^{p} dx)^{1/p} (∫_{|x|≥R} V (x)^{−1/(p−1)}dx)^{p/(p−1)} \\ < & \frac{ϵ}{2C} ∥u_{n}∥ ≤ \frac{ϵ}{2}. \end{matrix}

On other hand, as the embedding W^{1,p}(I_{R}) ↪ L^{1}(I_{R}) is compact there exists n_{0} such that for all n ≥ n_{0}

∫_{−R}^{R} |u_{n}|dx < \frac{ϵ}{2}.

Thus, for all n ≥ n_{0}

∥u_{n}∥_{1} = ∫_{−R}^{R} |u_{n}|dx + ∫_{|x|≥R} |u_{n}|dx < ϵ

which implies that u_{n} → 0 in L^{1}(ℝ). From this convergence, if 1 < s < ∞ we obtain

∥u_{n}∥_{s}^{s} = ∫_{ℝ} |u_{n}|^{s−1}|u_{n}|dx ≤ ∥u_{n}∥_{∞}^{s−1}∥u_{n}∥_{1} ≤ C∥u_{n}∥_{1} → 0 as n → ∞,

and consequently

u_{n} → 0 in L^{s}(ℝ) for 1 ≤ s < ∞

and the proof is complete.

Lemma 2.2 The functional F_{λ,μ} : X → ℝ is of class C^{1} in X and for u, v ∈ X

\begin{aligned}⟨F′_{λ,μ}(u), v⟩ = & ∫_{ℝ} |u′|^{p−2}u′v′dx + ∫_{ℝ} V (x)|u|^{p−2}uvdx + 2^{p−1} ∫_{ℝ} |u|^{p}|u′|^{p−2}u′v′dx \\ & + 2^{p−1} ∫_{ℝ} |u′|^{p}|u|^{p−2}uv dx − λ ∫_{ℝ} |u|^{q−2}uv dx − μ ∫_{ℝ} |u|^{r−2}uvdx. \end{aligned}

Proof. It will be sufficient to show that Φ(u) := ∫_{ℝ} |u′|^{p}|u|^{p} dx is of class C^{1} in X, because for the other summands the proof is standard. Existence of Gateaux derivative: Let u, v ∈ X and 0 ≠ t ∈ ℝ. We have

\begin{aligned}\frac{Φ(u + tv) − Φ(u)}{t} = \frac{1}{t} ∫_{ℝ} [|u′ + tv′|^{p} & (|u + tv|^{p} − |u|^{p}) \\ & + (|u′ + tv′|^{p} − |u′|^{p})|u|^{p}]dx. \end{aligned}  (2.1)

Now, given x ∈ ℝ and 0 < |t| < 1, by the mean value theorem, there exist λ, θ ∈ (0, 1) such that

\begin{matrix}\frac{||u(x) + tv(x)|^{p} − |u(x)|^{p}|}{|t|} = & p|u(x) + λtv(x)|^{p−1}|v(x)| \\ ≤ & p (|u(x)| + |v(x)|)^{p−1} |v(x)| \end{matrix}

and

\begin{matrix}\frac{||u′(x) + tv′(x)|^{p} − |u′(x)|^{p}|}{|t|} = & p|u′(x) + θtv′(x)|^{p−1}|v′(x)| \\ ≤ & p (|u′(x)| + |v′(x)|)^{p−1} |v′(x)|. \end{matrix}

As (|u| + |v|)^{p−1} |v|, |u|^{p} ∈ L^{∞}(ℝ) and (|u′| + |v′|)^{p}, (|u′| + |v′|)^{p−1} |v′| ∈ L^{1}(ℝ), it follows, from (2.1) and Lebesgue’s dominated convergence theorem, that

⟨Φ′(u), v⟩ = p ∫_{ℝ} |u|^{p}|u′|^{p−2}u′v′dx + p ∫_{ℝ} |u′|^{p}|u|^{p−2}uv dx.

Continuity of the Gateaux derivative: We choose a sequence u_{n} → u in X and v ∈ X with ∥v∥ ≤ 1. By the Hölder’s inequality we have

|⟨Φ′(u_{n}), v⟩ − ⟨Φ′(u), v⟩|
≤ p ∫_{ℝ} ||u_{n}|^{p}|u′_{n}|^{p−2}u′_{n} − |u|^{p}|u′|^{p−2}u′| |v′|dx
+ p ∫_{ℝ} ||u′_{n}|^{p}|u_{n}|^{p−2}u_{n} − |u′|^{p}|u|^{p−2}u| |v|dx
≤ p (∫_{ℝ} ||u_{n}|^{p}|u′_{n}|^{p−2}u′_{n} − |u|^{p}|u′|^{p−2}u′|^{p/(p−1)} dx)^{(p−1)/p}
+ pC (∫_{ℝ} ||u′_{n}|^{p}|u_{n}|^{p−2}u_{n} − |u′|^{p}|u|^{p−2}u|^{p/(p−1)} dx)^{(p−1)/p} .

According to the Lebesgue’s dominated convergence theorem, the right hand side tends to 0 uniformly for ∥v∥ ≤ 1. Thus Φ′ : X → X^{∗} is continuous.

Let us consider the condition below:

(A_{1}) the compact group G acts isometrically on the Banach space X = \overline{⨁_{j=0}^{∞} X_{j}}, the spaces X_{j} are invariant and there exists a finite dimensional space V such that, for every j ∈ ℕ, X_{j} ≃ V and the action of G on V is admissible, that is, every continuous equivariant map ∂U → V^{k−1}, where U is an open bounded invariant neighborhood of 0 in V^{k}, k ≥ 2, has a zero.

From now on, we follow the notations:

Y_{k} := ⨁_{j=0}^{k} X_{j}, Z_{k} := \overline{⨁_{j=k}^{∞} X_{j}},

To prove the item (a) of the Theorem 1.1, we shall use the Fountain theorem of T. Bartsch [6] as given in [19, Theorem 3.6]:

Lemma 2.3 (Fountain Theorem) Under the assumption (A_{1}), let I ∈ C^{1}(X, ℝ) be an invariant functional. If, for every k ∈ ℕ, there exist ρ_{k} > r_{k} > 0 such that

(A_{2}) a_{k} := max_{u∈Y_{k},∥u∥=ρ_{k}} I(u) ≤ 0;

(A_{3}) b_{k} := min_{u∈Z_{k},∥u∥=r_{k}} I(u) → ∞, k → ∞;

(A_{4}) I satisfies the (P S)_{c} condition for every c > 0,

then I has an unbounded sequence of critical values.

For the item (b), we shall apply a dual version of the Fountain Theorem, see [6, Theorem 2] or [19, theorem 3.18].

Lemma 2.4 (Dual Fountain Theorem) Under the assumption (A_{1}), let I ∈ C^{1}(X, ℝ) be an invariant functional. Moreover, suppose that I satisfies the following conditions:

(B_{1}) for every k ≥ k_{0}, there exists R_{k} > 0 such that I(u) ≥ 0 for every u ∈ Z_{k} with ∥u∥ = R_{k};

(B_{2}) b_{k} := inf_{u∈Z_{k},∥u∥≤R_{k}} I(u) → 0 as k → ∞;

(B_{3}) for every k ≥ 1 there exists r_{k} ∈ (0, R_{k}) and d_{k} < 0 such that I(u) ≤ d_{k} for every u ∈ Y_{k} with ∥u∥ = r_{k}; (B_{4}) every sequence (u_{n}) ⊂ Y_{n} with I(u_{n}) < 0 bounded and (I|_{Y_{n}})′(u_{n}) → 0 as n → ∞

has a subsequence which converges to a critical point of I.

Then for each k ≥ k_{0}, I has a critical value c_{k} ∈ [b_{k}, d_{k}] and c_{k} → 0 as k → ∞.

Observe that (B_{2}) and (B_{3}) imply b_{k} ≤ d_{k} < 0.

3 The Palais-Smale condition

We begin this section by proving that any Palais-Smale sequence of the functional F_{λ,μ} is bounded.

Lemma 3.1 Any (P S)_{c} sequence in X, that is, satisfying F_{λ,μ}(u_{n}) → c and F′_{λ,μ}(u_{n}) → 0 is bounded.

Proof. We have for n large

F_{λ,μ}(u_{n}) − \frac{1}{r}⟨F′_{λ,μ}(u_{n}), u_{n}⟩ ≤ ∥u_{n}∥ + c + 1.

On the other hand, by Lemma 2.2,

\begin{aligned}F_{λ,μ}(u_{n}) − \frac{1}{r}⟨F′_{λ,μ}(u_{n}), u_{n}⟩ = & (\frac{1}{p} − \frac{1}{r}) ∥u_{n}∥^{p} + 2^{p−1} (\frac{1}{p} − \frac{2}{r}) ∫_{ℝ} |u′_{n}|^{p}|u_{n}|^{p} dx \\ & + λ (\frac{1}{r} − \frac{1}{q}) ∫_{ℝ} |u_{n}|^{q} dx. \end{aligned}

Thus,

∥u_{n}∥ + c + 1 ≥ (\frac{1}{p} − \frac{1}{r}) ∥u_{n}∥^{p} − C∥u_{n}∥^{q},

from which it follows that (u_{n}) is bounded since r > 2p and p > q.

Lemma 3.2 (The Palais-Smale condition) The functional F_{λ,μ} satisfies the (P S)_{c} condition for all c ∈ ℝ.

Proof. Let (u_{n}) be in X satisfying F_{λ,μ}(u_{n}) → c and F′_{λ,μ}(u_{n}) → 0. We will show that (u_{n}) has a convergent subsequence. By Lemma 3.1, (u_{n}) is bounded. Thus, up to a subsequence, u_{n} ⇀ u in X and using Proposition 2.1 u_{n} → u in L^{q}(ℝ) and in L^{s}(ℝ). Therefore, the Hölder’s inequality implies that

\begin{matrix}∫_{ℝ} |u_{n}|^{q−2}u_{n}(u_{n} − u) dx → 0 and \\ ∫_{ℝ} |u_{n}|^{r−2}u_{n}(u_{n} − u) dx → 0 as n → ∞. \end{matrix}  (3.1)

We have

o_{n}(1) =⟨F′_{λ,μ}(u_{n}), u_{n} − u⟩ + λ ∫_{ℝ} |u_{n}|^{q−2}u_{n}(u_{n} − u)dx
+ μ ∫_{ℝ} |u_{n}|^{r−2}u_{n}(u_{n} − u)dx
= ∫_{ℝ} |u′_{n}|^{p−2}u′_{n}(u′_{n} − u′)dx + ∫_{ℝ} V (x)|u_{n}|^{p−2}u_{n}(u_{n} − u)dx
+ 2^{p−1}[ ∫_{ℝ} |u_{n}|^{p}|u′_{n}|^{p−2}u′_{n}(u′_{n} − u′)dx + ∫_{ℝ} |u′_{n}|^{p}|u_{n}|^{p−2}u_{n}(u_{n} − u)dx]

and since ∫_{ℝ} |u′|^{p−2}u′(u′_{n} − u′)dx = o_{n}(1) and ∫_{ℝ} V (x)|u|^{p−2}u(u_{n} − u)dx = o_{n}(1) as n → ∞, the equality above can be rewritten as

o_{n}(1) = ∫_{ℝ}(|u′_{n}|^{p−2}u′_{n} − |u′|^{p−2}u′)(u′_{n} − u′)dx

+ ∫_{ℝ} V (x)(|u_{n}|^{p−2}u_{n} − |u|^{p−2}u)(u_{n} − u)dx

+ 2^{p−1}[ \underbrace{∫_{−R}^{R} |u_{n}|^{p}|u′_{n}|^{p−2}u′_{n}(u′_{n} − u′)dx}_{I_{1}} + \underbrace{∫_{|x|>R} |u_{n}|^{p}|u′_{n}|^{p−2}u′_{n}(u′_{n} − u′)dx}_{I_{2}} ]

+ 2^{p−1}[ \underbrace{∫_{−R}^{R} |u′_{n}|^{p}|u_{n}|^{p−2}u_{n}(u_{n} − u)dx}_{I_{3}} + \underbrace{∫_{|x|>R} |u′_{n}|^{p}|u_{n}|^{p−2}u_{n}(u_{n} − u)dx}_{I_{4}} ].

As |u|^{p}, |u′|^{p} ∈ L^{1}(ℝ), given ϵ > 0 there exists R > 0 such that

∫_{|x|>R} |u|^{p} dx < ϵ^{p} and ∫_{|x|>R} |u′|^{p} dx < ϵ^{p}.  (3.2)

By the convergence ∫_{ℝ} |u|^{p}|u′|^{p−2}u′(u′_{n} − u′) → 0 as n → ∞ we obtain

I_{1} = ∫_{−R}^{R}(|u_{n}|^{p}|u′_{n}|^{p−2}u′_{n} − |u|^{p}|u′|^{p−2}u′)(u′_{n} − u′)dx
= ∫_{−R}^{R}(|u_{n}|^{p} − |u|^{p})|u′|^{p−2}u′(u′_{n} − u′)dx

+ ∫_{−R}^{R} |u_{n}|^{p}(|u′_{n}|^{p−2}u′_{n} − |u′|^{p−2}u′)(u′_{n} − u′)dx.

Using the inequality

⟨|x|^{p−2}x − |y|^{p−2}y, x − y⟩ ≥ \{ \begin{matrix}C_{p}|x − y|^{p}, & if & p ≥ 2 \\ C_{p} \frac{|x − y|^{2}}{(|x| + |y|)^{2−p}} , & if & 1 < p < 2 \end{matrix}  (3.3)

where x, y ∈ ℝ^{N}, C_{p} > 0 and ⟨·, ·⟩ is the standard scalar product in ℝ^{N} (see Simon [17]), the second summand in I_{1} is nonnegative and applying the mean value theorem to the integrand of the first integral, we get

I_{1} ≥ ∫_{−R}^{R} |ξ_{n}|^{p−2}ξ_{n}(u_{n} − u)|u′|^{p−2}u′(u′_{n} − u′)dx + o_{n}(1),

where

min{u_{n}, u} ≤ ξ_{n} ≤ max{u_{n}, u}.

Notice that for all n and for some C > 0, |ξ_{n}(x)| ≤ C almost everywhere in ℝ because (u_{n}) is bounded in L^{∞}(ℝ). Observe that

| ∫_{−R}^{R} |ξ_{n}|^{p−2}ξ_{n}(u_{n} − u)|u′|^{p−2}u′(u′_{n} − u′)dx|
≤ ∫_{−R}^{R} |ξ_{n}|^{p−1}|u_{n} − u||u′|^{p−1}|u′_{n} − u′|dx
≤ C∥u_{n} − u∥_{L^{∞}(I_{R})} ∫_{−R}^{R} |u′|^{p−1}|u′_{n} − u′|dx

≤ C∥u_{n} − u∥_{L^{∞}(I_{R})} (∫_{−R}^{R} |u′|^{p} dx)^{(p−1)/p} (∫_{−R}^{R} |u′_{n} − u′|^{p} dx)^{1/p}
≤ C∥u_{n} − u∥_{L^{∞}(I_{R})} → 0 as n → ∞,

since the embedding W^{1,p}(ℝ) ↪ L^{∞}(I_{R}) is compact. Thus I_{1} ≥ o_{n}(1). Next, our purpose is to estimate the integral I_{2}. We have that

\begin{matrix}I_{2} = & ∫_{|x|>R} |u_{n}|^{p}|u′_{n}|^{p} dx − ∫_{|x|>R} |u_{n}|^{p}|u′_{n}|^{p−2}u′_{n}u′dx \\ ≥ & − ∫_{|x|>R} |u_{n}|^{p}|u′_{n}|^{p−2}u′_{n}u′dx. \end{matrix}

By (3.2), we can conclude that

\begin{matrix}|∫_{|x|>R} |u′_{n}|^{p}|u_{n}|^{p−2}u′_{n}u′dx| ≤ & ∥u_{n}∥^{p} (∫_{|x|>R} |u′_{n}|^{p} dx)^{(p−1)/p} (∫_{|x|>R} |u′|^{p} dx)^{1/p} \\ ≤ & Cϵ. \end{matrix}

Therefore I_{2} ≥ −Cϵ. For the integral I_{3}, we have

\begin{matrix}|I_{3}| ≤ ∫_{−R}^{R} |u′_{n}|^{p}|u_{n}|^{p−1}|u_{n} − u|dx ≤ & ∥u_{n} − u∥_{L^{∞}(I_{R})}∥u_{n}∥_{∞}^{p−1}∥u_{n}∥ \\ ≤ & C∥u_{n} − u∥_{L^{∞}(I_{R})} → 0 as n → ∞. \end{matrix}

Writing the integral I_{4} in the following way

I_{4} = ∫_{|x|>R} |u′_{n}|^{p}|u_{n}|^{p} dx − ∫_{|x|>R} u′_{n}|^{p}|u_{n}|^{p−2}u_{n}u dx,

similarly to I_{2}, we conclude that I_{4} ≥ −Cϵ, where ϵ > 0 was fixed previously. Summarizing, we obtain

\begin{aligned}2^{p}Cϵ + o_{n}(1) ≥ & ∫_{ℝ}(|u′_{n}|^{p−2}u′_{n} − |u′|^{p−2}u′)(u′_{n} − u′)dx \\ & + ∫_{ℝ} V (x)(|u_{n}|^{p−2}u_{n} − |u|^{p−2}u)(u_{n} − u)dx \end{aligned}

and by the inequality (3.3) we get

2^{p}C_{p}ϵ + o_{n}(1) ≥ \{ \begin{matrix}∫_{ℝ} |u′_{n} − u′|^{p} dx + ∫_{ℝ} V (x)|u_{n} − u|^{p} dx, & if p ≥ 2 \\ ∫_{ℝ} \frac{|u′_{n} − u′|^{2}}{(|u′_{n}| + |u′|)^{2−p}} dx + ∫_{ℝ} \frac{V (x)|u_{n} − u|^{2}}{(|u_{n}| + |u|)^{2−p}} dx, & if 1 < p < 2. \end{matrix}

Thus, if p ≥ 2 we deduce that ∥u_{n} − u∥^{p} → 0 as n → ∞ since ϵ > 0 was arbitrary. Therefore u_{n} → u in X. If 1 < p < 2 we also have

lim_{n→∞} ∫_{ℝ} \frac{|u′_{n} − u′|^{2}}{(|u′_{n}| + |u′|)^{2−p}} dx = 0 and lim_{n→∞} ∫_{ℝ} V (x)\frac{|u_{n} − u|^{2}}{(|u_{n}| + |u|)^{2−p}} dx = 0. (3.4)

Using the Hölder’s inequality and the fact that (u′_{n}) is bounded in L^{p}(ℝ) we get

\begin{matrix}∫_{ℝ} |u′_{n} − u′|^{p} dx ≤ & (∫_{ℝ} \frac{|u′_{n} − u′|^{2}}{(|u′_{n}| + |u′|)^{2−p}} dx)^{p/2} (∫_{ℝ} (|u′_{n}| + |u′|)^{p}dx)^{(2−p)/2} \\ ≤ & C (∫_{ℝ} \frac{|u′_{n} − u′|^{2}}{(|u′_{n}| + |u′|)^{2−p}} dx)^{p/2} → 0 as n → ∞. \end{matrix}

By similar arguments, we also have that ∫_{ℝ} V (x)|u_{n} − u|^{p}dx → 0 which shows that ∥u_{n} − u∥^{p} → 0 and the proof is complete.

4 Proof of the main result

Now, we are ready to prove the main theorem: Proof of Theorem 1.1.

(a) Let us verify that in Lemma 2.3 the conditions (A_{1}) − (A_{4}) are satisfied. Since X is a reflexive, separable Banach space, let us fix (e_{j})_{j=0}^{∞} a Schauder basis for X.

Thus X = \overline{⨁_{j=0}^{∞} X_{j}} where X_{j} = ℝe_{j} and on X we consider the antipodal action of ℤ_{2} which verifies the condition (A_{1}). Let us define

β_{k} := sup_{u∈Z_{k}\\{0\}} \frac{∥u∥_{r}}{∥u∥} .

By similar arguments as in [19, Lemma 3.8], we have that β_{k} → 0 as k → ∞. On Z_{k}, we get

F_{λ,μ}(u) ≥ \frac{1}{p}∥u∥^{p} − \frac{|λ|}{q}∥u∥_{q}^{q} − \frac{μ}{r}∥u∥_{r}^{r} ≥ \frac{1}{p}∥u∥^{p} − \frac{|λ|}{q}C∥u∥^{q} − \frac{μ}{r}β_{k}^{r}∥u∥^{r}.

Since q < p we have

\frac{|λ|}{q}C∥u∥^{q} ≤ \frac{1}{2p}∥u∥^{p}

for ∥u∥ ≥ R, R > 0 large. Thus, for ∥u∥ ≥ R we obtain

F_{λ,μ}(u) ≥ \frac{1}{2p}∥u∥^{p} − \frac{μ}{r}β_{k}^{r}∥u∥^{r}.

Choosing r_{k} := (μβ_{k}^{r})^{1/(p−r)}, it follows that r_{k} → ∞. Hence, there exists k_{0} such that r_{k} ≥ R for all k ≥ k_{0}. Thus, if u ∈ Z_{k} and ∥u∥ = r_{k} with k ≥ k_{0}, we obtain

F_{λ,μ}(u) ≥ (\frac{1}{2p} − \frac{1}{r}) (μβ_{k}^{r})^{p/(p−r)}

and since β_{k} → 0 as k → ∞, relation (A_{3}) is proved. The functional F_{λ,μ} is even and

\begin{matrix}F_{λ,μ}(u) ≤ & \frac{1}{p}∥u∥^{p} + \frac{2^{p−1}}{p} C∥u∥_{∞}^{p}∥u∥^{p} − \frac{λ}{q}∥u∥_{q}^{q} − \frac{μ}{r}∥u∥_{r}^{r} \\ ≤ & \frac{1}{p}∥u∥^{p} + \frac{2^{p−1}}{p} C∥u∥^{2p} − \frac{λ}{q}∥u∥_{q}^{q} − \frac{μ}{r}∥u∥_{r}^{r} \\ ≤ & \frac{1}{p}∥u∥^{p} + \frac{2^{p−1}}{p} C∥u∥^{2p} − \frac{λ}{q}C∥u∥^{q} − \frac{μ}{r}C∥u∥^{r} \end{matrix}

because on the finite-dimensional space Y_{k} all norms are equivalent. Since r > 2p the relation (A_{2}) is satisfied for every ρ_{k} > r_{k} > 0 large enough.

The condition (A_{4}) holds by Lemma 3.2. It suffices then to use the Fountain theorem and the item (a) is proved.

(b) In order to see (B_{1}) of Lemma 2.4 we set

α_{k} := sup_{u∈Z_{k}\\{0\}} \frac{∥u∥_{q}}{∥u∥} .

It follows easily from the Proposition 2.1 that α_{k} → 0 as k → ∞. We obtain for u ∈ Z_{k}

F_{λ,μ}(u) ≥ \frac{1}{p}∥u∥^{p} − \frac{λ}{q}α_{k}^{q}∥u∥^{q} − \frac{|μ|}{r}C∥u∥^{r}.

Since r > p, we have

\frac{|μ|}{r}C∥u∥^{r} ≤ \frac{1}{2p}∥u∥^{p}

for ∥u∥ ≤ R, R > 0 small. Thus,

F_{λ,μ}(u) ≥ \frac{1}{2p}∥u∥^{p} − \frac{λ}{q}α_{k}^{q}∥u∥^{q}.

Setting R_{k} := (\frac{2pλα_{k}^{q}}{q} )^{1/(p−q)}, we get

\frac{1}{2p}R_{k}^{p} = \frac{λ}{q}α_{k}^{q}R_{k}^{q}.

Clearly R_{k} → 0, so there exists k_{0} with R_{k} ≤ R when k ≥ k_{0}. Thus, if u ∈ Z_{k}, k ≥ k_{0} and satisfies ∥u∥ = R_{k} we have

F_{λ,μ}(u) ≥ \frac{1}{2p}∥u∥^{p} − \frac{λ}{q}α_{k}^{q}∥u∥^{q} = 0.

This proves (B_{1}). Next, (B_{2}) follows immediately from R_{k} → 0. To check (B_{3}), we observe that on the finite dimensional space Y_{k} all norms are equivalent. Hence, there exist C_{0}, C_{1} > 0 such that

\begin{matrix}F_{λ,μ}(u) ≤ & \frac{1}{p}∥u∥^{p} + C∥u∥^{2p} − \frac{λ}{q}C_{0}∥u∥^{q} − \frac{μ}{r}C_{1}∥u∥^{r} \\ = & ∥u∥^{q} (\frac{1}{p}∥u∥^{p−q} + C∥u∥^{2p−q} − \frac{λ}{q}C_{0} − \frac{μ}{r}C_{1}∥u∥^{r−q}) \end{matrix}

Since q < p < r and R_{k} → 0, taking r_{k} ∈ (0, R_{k}) with R_{k} sufficiently small, (B_{3}) is satisfied. This is precisely the point where λ > 0 enters. Finally, the condition (B_{4}) is showed similarly to Lemma 3.2.

In conclusion, we add some remarks about the behavior of solutions with respect to parameters λ and μ.

Remark 1 (a) For λ ∈ ℝ and μ ≤ 0 there are no solutions with positive energy. Moreover

inf {∥u∥ : u solves (1.1), F_{λ,μ}(u) > 0} → ∞ as μ → 0^{+}.

(b) For μ ∈ ℝ and λ ≤ 0 there are no solutions with negative energy. Moreover

sup {∥u∥ : u solves (1.1), F_{λ,μ}(u) < 0} → 0 as λ → 0^{+}.

Proof of (a): Let λ, μ ∈ ℝ. From F′_{λ,μ}(u) = 0 we obtain

λ∥u∥_{q}^{q} = −μ∥u∥_{r}^{r} + ∥u∥^{p} + 2^{p} ∫_{ℝ} |u|^{p}|u′|^{p} dx.

If F_{λ,μ}(u) ≥ 0, we have

(\frac{1}{p} − \frac{1}{q}) ∥u∥^{p} + μ (\frac{1}{q} − \frac{1}{r}) ∥u∥_{r}^{r} + 2^{p} (\frac{1}{2p} − \frac{1}{q}) ∫_{ℝ} |u|^{p}|u′|^{p} dx ≥ 0.

Since 1 < q < p and r > 2p, we see immediately that for μ ≤ 0 the only solution with nonnegative energy is u = 0. Now, if μ > 0, then there are constants c_{1}, c_{2} > 0 with

−c_{1}∥u∥^{p} + μ∥u∥^{r} ≥ 0,

hence

∥u∥^{r−p} ≥ μ^{−1}c_{1}/c_{2} → +∞ as μ → 0^{+},

and the result follows.

Proof of (b): Fix λ, μ ∈ ℝ. Similarly, from F′_{λ,μ}(u) = 0 we obtain

μ∥u∥_{r}^{r} = −λ∥u∥_{q}^{q} + ∥u∥^{p} + 2^{p} ∫_{ℝ} |u|^{p}|u′|^{p} dx.

Thus, if F_{λ,μ}(u) ≤ 0 we have

(\frac{1}{p} − \frac{1}{r}) ∥u∥^{p} + λ (\frac{1}{r} − \frac{1}{q}) ∥u∥_{q}^{q} + 2^{p} (\frac{1}{2p} − \frac{1}{r}) ∫_{ℝ} |u|^{p}|u′|^{p} dx ≤ 0.

This implies that for λ ≤ 0 the only solution with non-positive energy is u = 0. For μ > 0, there are constants c_{3}, c_{4} > 0 with

c_{3}∥u∥^{p} − λc_{4}∥u∥^{q} ≤ 0,

hence

∥u∥^{p−q} ≤ λc_{4}/c_{3} → 0 as λ → 0^{+}.

Remark 2 For λ > 0 small, it is easy to see that the functional F_{λ,μ} possesses the mountain-pass geometry. Thus, using the mountain-pass theorem (see [19, Theorem 2.10]) equation (1.1) has a mountain-pass type solution. This solution is nonnegative. Indeed, it is enough to work with the functional

J_{λ,μ}(u) = \frac{1}{p}∥u∥^{p} + \frac{2^{p−1}}{p} ∫_{ℝ} |u′|^{p}|u|^{p} dx − \frac{λ}{q} ∫_{ℝ}(u^{+})^{q} dx − \frac{μ}{r} ∫_{ℝ}(u^{+})^{r} dx

and if ⟨J′_{λ,μ}(u), v⟩ = 0 for all v ∈ X, setting v = u^{−}, where u^{−} = max{−u, 0}, we obtain that u^{−} = 0. Thus u ≥ 0.

Acknowledgements. This paper was written when the author was a PhD candidate at UNICAMP-Universidade Estadual de Campinas-SP-Brazil. The author wishes to express his thanks to professors Djairo G. de Figueiredo and J. M. do Ó.

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(Received November 27, 2007)