USA-Chile Workshop on Nonlinear Analysis, Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 215–223. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)

Gluing approximate solutions of minimum type on the Nehari manifold ^{∗}

Yanyan Li & Zhi-Qiang Wang

Abstract In the last decade or so, variational gluing methods have been widely used to construct homoclinic and heteroclinic type solutions of nonlinear elliptic equations and Hamiltonian systems. This note is concerned with the procedure of gluing mountain-pass type solutions. The first procedure to glue mountain-pass type solutions was developed through the work of Séré, and Coti Zelati - Rabinowitz. This procedure and its variants have been extensively used in many problems by now for nonlinear equations with superlinear nonlinearities. In this note we provide an alternative device to the by now standard procedure which allows us to glue minimizers on the Nehari manifold together as genuine, multi-bump type, solutions.

1 Introduction

In the last decade or so, variational gluing methods have been widely used to construct homoclinic and heteroclinic type solutions of nonlinear elliptic equations and Hamiltonian systems (see, e.g. Rabinowitz [7] and references therein). The idea is to first construct some basic solutions (or approximate solutions) which are characterized by minimax method and which are used as building blocks for construction of multi-bump type solutions. These multi-bump type solutions then are obtained by some gluing procedures and look roughly like sums of the basic solutions. The general idea is clear by now, though for different types of basic solutions one has to employ different procedures for the concrete problems. Different type of basic solutions have been glued together by various authors, which include minimizers and mountain-pass type solutions. In fact even cat > 1 solutions have been glued together, see for example Giannoni and Rabinowitz [4].

This note is concerned with the procedure of gluing mountain-pass type solutions. The first procedure to glue mountain-pass type solutions was developed through the work of Séré ([8] [9]) and Coti Zelati - Rabinowitz ([2] [3]), and this procedure and its variants have been extensively used in many problems by now for nonlinear equations with superlinear nonlinearities (see, e.g. Rabinowitz [7] and references therein). In these papers the basic solutions are mountainpass type solutions. On the other hand, under slightly stronger conditions these mountain-pass solutions can also be characterized as minimizers of a constrained problem, namely, minimizers on the Nehari manifold. In this paper we provide an alternative device to the by now standard procedure which allows us to glue minimizers on the Nehari manifold (or local minimizers, approximate local minimizers) together as genuine (multi-bump type) solutions. Though the new procedure is somewhat parallel to the original one for mountain-pass solutions there are still technical complications needed to be fixed. On the other hand, it seems the new device in gluing minimizers on Nehari manifold is simpler than those for gluing mountain-pass solutions in the full space. For instance, one step involved in [2] and [3] is to do a minimization problem on some annulus regions and to use elliptic estimates to achieve the smallness of certain map. This step has to be done on a case by case basis for ODEs, PDEs with subcritical exponents and PDEs with critical exponents and seems to be somewhat laborsome for PDE problems, especially for those involving critical exponents ([5] [6]). Our device given here will avoid this step and treat all problems uniformly.

∗Mathematics Subject Classifications: 35J20, 58E05.

Key words: luing, variational, minimax.

\begin{matrix}c \\ ◯ \end{matrix}2001 Southwest Texas State University.

Published January 8, 2001.

For simplicity we only present our device for an ODE problem to demonstrate the procedure. Although the results are not new, the procedure we use is different from the known one and may prove to be of advantage in dealing with some other problems with the presence of a Nehari manifold. The same device clearly works for analogous subcritical exponent periodic PDEs

−Δu + a(x)u = f(x, u), in ℝ^{N},

with suitable growth condition on f and periodic dependency in x; and presumably should also work for analogous critical exponent periodic PDEs.

2 An ODE problem

Consider

−u′′ + a(t)u = f(t, u), t ∈ ℝ  (1)

We look for homoclinic solutions of this equation, i.e.,solutions such that lim_{|t|→∞} u(t) = 0 and lim_{|t|→∞} u′(t) = 0. Assume

(f1) a(t) ∈ C(ℝ, ℝ) is T −periodic and min_{ℝ} a(t) > 0.

(f2) f(t, u) ∈ C(ℝ × ℝ, ℝ) is T −periodic in t.

(f3) f_{u}(t, 0) = 0 and |f_{u}(t, u)| ≤ C(1 + |u|^{p}) for some p > 1.

(f4) There is a θ > 1 such that f′(t, u)u^{2} ≥ θf(t, u)u for all t and u.

There is a variational formulation of the problem. Namely,

I(u) = \frac{1}{2} ∫_{ℝ}(|u̇|^{2} + au^{2})dt − ∫_{ℝ} F (t, u)dt

for u ∈ X := H^{1}(ℝ). Then critical points of I are solutions of (1). We use ∥ · ∥ to denote the norm in X.

There is alternative approach to the above, namely the Nehari manifold. Define

γ(u) := ∫_{ℝ} f(t, u)udt − ∫_{ℝ}(|u̇|^{2} + au^{2})dt,

and let

V = {u ∈ X \ {0} | γ(u) = 0}.

Then it is well known that under conditions (f_{1} - f_{4}), V is a C^{1} manifold and critical points of I on V are also critical points of I in X and therefore solutions of (1). We use the usual notations. I^{c} = {u ∈ V | I(u) ≤ c}, I_{c} = {u ∈ V | I(u) ≥ c}, I_{a}^{b} = I^{b} ∩ I_{a}, K = {u ∈ V | I′(u) = 0}, K^{c} = K ∩ I^{c}. For an integer j, τ_{j}u = u(t − j) the translation of u. Then for any j, τ_{j}w ∈ K^{c}.

Let

c := inf_{V} I(u),

the ground state energy of I. Using the following compactness results for (PS) sequences of I one easily gets that c is always achieved at some u which is a ground state solution of (1).

Proposition 2.1 Let (u_{n}) ⊂ V be such that I(u_{n}) → b and (I_{|V})′(u_{n}) → 0. Then there is an l ∈ N (depending on b), v_{1}, ..., v_{l} ∈ K \ {0}, a subsequence of u_{n} and corresponding (j_{i,n})_{i=1}^{l} ⊂ Z^{l} such that

∥u_{n} − ∑_{i=1}^{l} τ_{j_{i,n}}v_{i}∥ → 0, ∑_{i=1}^{l} I(v_{i}) = b,

and for i ≠ ℓ, |j_{i,n} − j_{ℓ,n}| → ∞.

This is just a reformulation of Prop. 2.31 in [3], since V is a natural constraint of I in the sense that (I_{|V})′(u) = 0 iff I′(u) = 0.

Due to the translation invariance of the problem, there may be many solutions on the energy level c. We shall assume

K_{c}^{c} has an isolated point, say, w.  (∗)

For an integer k ≥ 2, let j⃗ = (j_{1}, ⋯, j_{k}), a k-tuples of integers. We shall show that there are real solutions of (1) which roughly look like ∑_{i=1}^{k} τ_{j_{i}}w. More precisely, let

2r_{0} = min{ν, μ} > 0,

where ν = inf{∥u∥ | u ∈ K \ {0}} and μ = inf{∥u − w∥ | u ∈ K}.

Theorem 2.2 Assume (f1 - f4) and K_{c}^{c} has an isolated point. For 0 < α < \frac{c}{2} and 0 < r < r_{0} there is j_{0} > 0 such that for all k-tuples of integers j⃗ satisfying min_{i≠ℓ} |j_{i} − j_{ℓ}| > j_{0}

K_{kc−α}^{kc+α} ∩ N_{r}( ∑_{i=1}^{k} τ_{j_{i}}w) ≠ ∅.

Here, N_{r}(·) denotes the r-neighborhood in X.

The proof of Theorem 2.2 is based on an indirect argument with the basic idea going back to [8] [2] [3]. Our procedure below is somewhat different from the one used in the original argument ([8] [2] [3]), and in a way simpler. Step 1. First, for R > 0 we define a cut-off operator

T_{R}(u) = ρ(2R^{−1}|x|)u(x)

where ρ(t) = 1 for 0 ≤ t ≤ 1 and ρ(t) = 0 for t ≥ 2. With j⃗ = (j_{1}, ⋯, j_{k}) satisfying inf_{i≠ℓ} |j_{i} − j_{ℓ}| > 2R, for y = (y_{1}, ..., y_{k}) with y_{i} ≥ 0, i = 1, ..., k and ∑_{i=1}^{k} y_{i} = 1, we define

G_{0}(y) = b(y) ∑_{i=1}^{k} y_{i}τ_{j_{i}}T_{R}(w)

where b(y) > 0 is such that G_{0}(y) ∈ V. We fix a δ_{0} ∈ (0, 1/k) so that max_{y} γ(δ_{0}b(y)w) < 0 (which can be done due to (f_{3})) and define

Δ_{k} = {y = (y_{1}, ..., y_{k}) ∈ ℝ^{k}| ∑_{i=1}^{k} y_{i} = 1, y_{i} ≥ δ_{0}} ,

a (k − 1)-dimensional simplex. Then G_{0} ∈ C(Δ_{k}, V). By the explicit form of G_{0} we have, as R → ∞,

I(G_{0}(y)) = ∑_{i=1}^{k} I(b(y)y_{i}τ_{j_{i}}T_{R}(w)) = ∑_{i=1}^{k} I(b(y)y_{i}w) + o(1) ≤ kc + o(1).

So we get

lim_{R→∞} max_{Δ_{k}} I(G_{0}(y)) ≤ kc.  (2)

Note that I(G_{0}(y_{c})) ≥ kc, where y_{c} = (\frac{1}{k} , ..., \frac{1}{k}) the center of Δ_{k}.

Define

Γ = {G ∈ C(Δ_{k}, V) | G_{|∂Δ_{k}} = G_{0}} .

For j⃗ = (j_{1}, ..., j_{k}) satisfying inf_{i≠ℓ} |j_{i} − j_{ℓ}| > 2R, used for G_{0}, we define for any u ∈ V

u^{(i)}(x) = ρ(R^{−1}|x − j_{i}|)u(x), i = 1, ..., k.

Lemma 2.3 Given G_{0} as above with j⃗ = (j_{1}, ..., j_{k}) and R fixed, for any G ∈ Γ there exists y_{0} ∈ Δ_{k} such that

γ(G(y_{0})^{(i)}) = 0, i = 1, ..., k.

Proof. Regarding Δ_{k} as a part of an affine (k − 1)-plane which we denote by A^{k−1}, we see A^{k−1} − (\frac{1}{k} , ..., \frac{1}{k}) is a (k − 1)−plane passing through the origin in ℝ^{k} which we denote by Ã^{k−1}. For any G ∈ Γ we introduce a map from Δ̃_{k} = Δ_{k} − (\frac{1}{k} , ..., \frac{1}{k}) into Ã^{k−1} (with ỹ = y − (\frac{1}{k} , ..., \frac{1}{k})), by

h(ỹ) = (h_{1}, ..., h_{k}) := (γ(G(y)^{(1)}), ..., γ(G(y)^{(k)})).

Then the claim is proved if we can show that

deg(h, Δ̃_{k}, 0) ≠ 0.

Note that 0 is the center of Δ̃_{k}. But this degree only depends on G_{0} because all G agree with G_{0} on the boundary of Δ̃_{k}. Finally, we compute the above degree for G_{0} and we claim it is 1. We prove this by an induction in k. For k = 2 it is easy to check it by hand: when y_{1} = δ_{0} and y_{2} = 1 − δ_{0} we have ỹ_{1} = δ_{0} − \frac{1}{2} < 0 and ỹ_{2} = 1 − δ_{0} − \frac{1}{2} > 0; and h_{1} = γ(bδ_{0}τ_{j_{1}}T_{R}(w)) < 0 and therefore h_{2} = γ(b(1 − δ_{0})τ_{j_{2}}T_{R}(w)) > 0 for h_{1} + h_{2} = 0. At the other end point of Δ̃_{2} we have similar computations, which together shows that h is homotopy to the identity map.

Now for k ≥ 3, Δ̃_{k} has k faces (opposite to each vertex and denoted by F_{i}). On the ith-face F_{i}, if y = (y_{1}, ..., y_{k}) ∈ F_{i} then y_{i} = δ_{0} − \frac{1}{k}. And we get h_{i} = γ(bδ_{0}τ_{j_{i}}T_{R}(w)) < 0. Using this fact, we may first project (by radial scalings on Ã^{k−1}) the image of h to ϵΔ̃_{k} for some ϵ > small. Then using an expansion scaling we may have the image of ∂Δ̃_{k} into itself. We denote this operation by P, i.e., P h is a map from Δ̃_{k} → Ã^{k−1} such that P h(∂Δ̃_{k}) ⊂ ∂Δ̃_{k}. By the homotopy property, deg(h, Δ̃_{k}, 0) = deg(P h, Δ̃_{k}, 0). Note that taking y = 0 we see γ(g_{1}(y)) = ⋯ = γ(g_{k}(y)) = 0. By some standard properties of the degree (see, e.g. [1]),

deg(P h, Δ̃_{k}, 0) = deg(P h, ∂Δ̃_{k}, ∂Δ̃_{k}).

Now on F_{1} the center c_{1} has coordinates y_{1} = δ_{0} − \frac{1}{k} and for i = 2, ..., k, y_{i} = \frac{1−δ_{0}}{k−1} − \frac{1}{k}. Using this it is easy to see that c_{1} is not covered by P h(∪_{i=2}^{k}F_{i}), for if not c_{1} = P h(y) for some y ∈ F_{i} with i ≥ 2, then we have h_{i}(y) < 0 and therefore (P h)_{i}(y) < 0, this is a contradiction with y_{i} > 0 for c_{1}. By the excision property

deg(P h, ∂Δ̃_{k}, ∂Δ̃_{k}) = deg(P h, F_{1}, c_{1}).

However, this is what we would get from the (k − 1)-map. The induction is complete. ♢

We need another technical result.

Lemma 2.4 Let u ∈ V be such that u^{(i)} ∈ V for all i = 1, ..., k (obtained by using j⃗ = (j_{1}, ..., j_{k}) satisfying inf_{i≠ℓ} |j_{i} − j_{ℓ}| > 2R). Then I(u) ≥ kc.

Proof. First, we write W_{R} = ⋃_{i=1}^{k} B_{R}(j_{i}). Then

I(u) = \frac{1}{2} ∫ |∇u|^{2} + a|u|^{2} − ∫ F (x, u)

= ∑_{i=1}^{k} (\frac{1}{2} ∫_{B_{2R}(j_{i})} |∇u^{(i)}|^{2} + a|u^{(i)}|^{2} − ∫_{B_{2R}(j_{i})} F (x, u^{(i)}))

+\frac{1}{2} ∫_{ℝ\W_{2R}} |∇u|^{2} + a|u|^{2} − ∫_{ℝ\W_{2R}} F (x, u)

+ ∑_{i=1}^{k} \frac{1}{2} ∫_{B_{2R}(j_{i})\B_{R}(j_{i})} (|∇((1 − ρ)u)|^{2} + (1 − ρ)^{2}au^{2}

+2∇(ρu)∇((1 − ρ)u) + 2ρ(1 − ρ)au^{2})

− ∑_{i=1}^{k} ∫_{B_{2R}(j_{i})\B_{R}(j_{i})}(F (x, u) − F (x, u^{(i)}))

Using u ∈ V and u^{(i)} ∈ V for all i = 1, ..., k, we get

∫_{ℝ\W_{2R}} |∇u|^{2} + a|u|^{2} − ∫_{ℝ\W_{2R}} f(x, u)u

+ ∑_{i=1}^{k} ∫_{B_{2R}(j_{i})\B_{R}(j_{i})} (|∇((1 − ρ)u)|^{2} + (1 − ρ)^{2}au^{2}

+2∇(ρu)∇((1 − ρ)u) + 2ρ(1 − ρ)au^{2})
− ∑_{i=1}^{k} ∫_{B_{2R}(j_{i})\B_{R}(j_{i})}(f(x, u)u − f(x, u^{(i)})u^{(i)}) = 0 .

Bringing this into the earlier formula we have

I(u) ≥ kc + ∫_{ℝ\W_{2R}}(\frac{1}{2}f(x, u)u − F (x, u))

+ ∑_{i=1}^{k} ∫_{B_{2R}(j_{i})\B_{R}(j_{i})} (\frac{1}{2}f(x, u)u − \frac{1}{2}f(x, u^{(i)})u^{(i)}

−F (x, u) + F (x, u^{(i)}))

which implies I(u) ≥ kc since the last two terms on the right hand side are both non-negative. Indeed, by (f_{4}) we have

\frac{1}{2}f(x, u)u − F (x, u) ≥ 0,

and, writing g(t) = \frac{1}{2}f(x, tu)tu − F (x, tu) by the mean value theorem, we have for some ξ ∈ (0, 1),

\frac{1}{2}f(x, u)u − \frac{1}{2}f(x, u^{(i)})u^{(i)} − F (x, u) + F (x, u^{(i)})

= \frac{1}{2}f(x, u)u − \frac{1}{2}f(x, ρ(R^{−1}|x − j_{i}|)u)ρ(R^{−1}|x − j_{i}|)u − F (x, u)
+F (x, ρ(R^{−1}|x − j_{i}|)u)
= g(1) − g(ρ)
= g′(ρ + ξ(1 − ρ))(1 − ρ)
= \frac{1}{2} \{f′(x, [ξ + ρ(1 − ξ)]u)[ξ + ρ(1 − ξ)]u^{2} − f(x, [ξ + ρ(1 − ξ)]u)u\} (1 − ρ)
≥ 0 .

which completes the present proof ♢

Let z_{R} = b_{R} ∑_{i=1}^{k} τ_{j_{i}}T_{R}(w) with inf_{i≠ℓ} |j_{ℓ} − j_{i}| > 2R, where b_{R} > 0 is such that z_{R} ∈ V. Note b_{R} → 1 as R → ∞.

For any ϵ > 0, by choosing R > 0 large we may get, by (2), max_{Δ_{k}} I(G_{0}(y)) < kc + ϵ. Also we remark that when r_{0} > r > 0 is fixed, for all small ϵ and large R it holds that I(G_{0}(y)) ≥ kc − ϵ implies G_{0}(y) ∈ N\frac{r}{8} (z_{R}). We fix r > 0 now such that for all R ≥ 1 if G ∈ Γ satisfying ∥G(y) − G_{0}(y)∥ ≤ r then G(y)^{(i)} ≠ 0 for all y and i = 1, ..., k. Step 2. If we assume the conclusion of Theorem 2.2 is not true, using a deformation argument from a pseudo-negative gradient flow we deform G_{0} to a map G_{1} : Δ_{k} → V such that max_{Δ_{k}} I(G_{1}(y)) ≤ kc − ϵ, ∥G_{1}(y) − G_{0}(y)∥ ≤ r and G_{1}|_{∂Δ_{k}} = G_{0}. Then using Lemmas 2.3 and 2.4 we will have a contradiction. We need the following lemma.

Lemma 2.5 There exist δ_{r} > 0 and R_{r} > 0 such that for all R ≥ R_{r} and for all u ∈ N_{r}(z_{R}) \ N\frac{r}{8} (z_{R})

∥I′(u)∥ ≥ δ_{r}.

Proof. If the conclusion is not true, we would have a sequence R_{n} → ∞ and u_{n} ∈ N_{r}(z_{R}) \ N\frac{r}{8} (z_{R}) such that I′(u_{n}) → 0. Then (u_{n}) is a (PS)_{b} sequence for I with some b. By Proposition 2.1

∥u_{n} − ∑_{i=1}^{l} τ_{j_{i,n}}v_{i}∥ → 0

for some integer l and v_{1}, .., v_{l} ∈ K and |j_{i,n} − j_{ℓ,n}| → ∞ for i ≠ ℓ. Since as R_{n} → ∞, ∥z_{R_{n}} − ∑_{i=1}^{k} τ_{j_{i,Rn}} w∥ → 0, we get

∥ ∑_{i=1}^{l} τ_{j_{i,n}}v_{i} − ∑_{i=1}^{k} τ_{j_{i,Rn}} w∥ → 0.

From this it is easy to argue by using (*) that l = k, v_{i} = w for all i and for n large j_{i,n} = j_{i,R_{n}}. This is a contradiction to ∥u_{n} − z_{R_{n}}∥ ≥ \frac{r}{8}. ♢

Now we can finish the proof of our main theorem.

We take 0 < ϵ < \frac{3rδ_{r}}{8} and R ≥ R_{r} so that max_{Δ_{k}} I(G_{0}(y)) < kc + ϵ and that I(G_{0}(y)) ≥ kc − ϵ implies G_{0}(y) ∈ N\frac{r}{8} (z_{R}).

Next, choose ϵ < ϵ_{1} < c. Let

ϕ(u) = \frac{∥u − X \ N_{r}(z_{R})∥}{∥u − N\frac{r}{2} (z_{R})∥ + ∥u − X \ N_{r}(z_{R})∥}.

and let U be a locally Lipschitz pseudo-gradient vector field for I on V \ K such that

\begin{matrix}(i) & ∥U(u)∥ ≤ \frac{4ϵ_{1}}{∥I′(u)∥}, \\ (ii) & I′(u)U(u) ≥ 2ϵ_{1}. \end{matrix}

Let η be the flow given by the solution of

\frac{dη}{dt} = −ϕ(η)U(η), η(0, u) = u.

Let u = G_{0}(y) be such that I(u) ≥ kc−ϵ so that u ∈ N\frac{r}{8} (z_{R}). Using Proposition 2.1 we can show either (i) η(t, u) reaches ∂B_{r}(z_{R}) for some t ≤ 1 or (ii) η(t, u) remains in B_{r}(z_{R}) for t ∈ [0, 1]. If (i) occurs, in some time interval [t_{1}, t_{2}], η(t, u) reaches from ∂B\frac{r}{8} (z_{R}) to ∂B_{r}(z_{R}). Then it must reach I^{kc−ϵ} already in the time interval. Otherwise,

\frac{7r}{8} = ∥η(t_{2}, u) − η(t_{1}, u)∥ ≤ ∫_{t_{1}}^{t_{2}} ϕ(η)∥U(η(t, u)∥dt ≤ \frac{4ϵ_{1}}{δ_{r}} ∫_{t_{1}}^{t_{2}} ϕ(η)dt,

and

2ϵ ≥ I(η(t_{2}, u)) − I(η(t_{1}, u)) = ∫_{t_{1}}^{t_{2}} \frac{dI}{dt} (η(t, u))dt ≥ 2ϵ_{1} ∫_{t_{1}}^{t_{2}} ϕ(η)dt.

This implies ϵ ≥ \frac{7rδ_{r}}{8} , a contradiction. Thus if (i) occurs there is a unique σ(u) ≤ 1 such that I(η(σ(u), u)) = kc − ϵ. If (ii) occurs we may have either η(t, u) has to go from B\frac{r}{8} (z_{R}) to the boundary of B\frac{r}{2} (z_{R}) and similar argument shows that there is a unique σ(u) ≤ 1 such that I(η(σ(u), u)) = kc−ϵ, or η(t, u) stays in B\frac{r}{2} (z_{R}) for t ∈ [0, 1]. In the latter case if η(t, u) does not reach I^{kc−ϵ} we would have ϕ equal to 1 along η(t, u) and we have 2ϵ ≥ I(η(0, u)) − I(η(1, u)) ≥ 2ϵ_{1}, a contradiction. In both cases, we have ∥η(σ(u), u) − u∥ ≤ r. We get G_{1}(y) = η(σ(G_{0}(y)), G_{0}(y)) which is a continuous map from Δ_{k} into V and agrees with G_{0}(y) on ∂Δ_{k}. Moreover,

∥G_{1}(y) − G_{0}(y)∥ ≤ r.  (3)

To finish the proof of Theorem 2.2, let us produce a contradiction as follows. Applying Lemma 2.3 to G_{1}(y) we conclude that there exists y ∈ Δ_{k} such that

γ(G_{1}(y)^{(i)}) = 0, i = 1, ..., k.

Due to (3), we obtain G_{1}(y)^{(i)} ≠ 0 for i = 1, ..., k, i.e., G_{1}(y)^{(i)} ∈ V for i = 1, ..., k. Applying Lemma 2.4, we get a contradiction with max I(G_{1}(y)) ≤ kc−ϵ.

Acknowledgement. The authors want to thank P. Rabinowitz for his helpful comments.

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Yanyan Li Department of Mathematics, Rutgers University New Brunswick, NJ 08903 USA e-mail: yyli@math.rutgers.edu Zhi-Qiang Wang Department of Mathematics, Utah State University Logan, UT 84322 USA e-mail: wang@sunfs.math.usu.edu