On a Semilinear Schrödinger Equation with Periodic Potential

A. Pankov and K. Pflüger

Preprint Nr. 23 A 96

A. Pankov: Department of Mathematics Vinnitsa Pedagogical Institute, Ukraine

email: pankov@apmth.rts.vinnica.ua

K. Pflüger: Institut für Mathematik I, Freie Universität Berlin email: pflueger@math.fu-berlin.de

1 Introduction

We study the semilinear Schrödinger equation

(P)  −Δu + V (x)u = f(x, u) in IR^{N}

with a periodic, continuous potential V and superquadratic, subcritical nonlinearity f , f(x, 0) = 0, with the same period in x as V. We look for nontrivial solutions of (P) having finite energy. There is a number of papers dealing with this problem. At first, the so-called definite case (when V is strictly positive) was considered. Using the Nehari variational principle, the first author [3] has proved an existence theorem for ground states, i.e. solutions having smallest energy among all nontrivial solutions. (Even the more general asymptotically almost periodic case was treated there). Later on P. Rabinowitz [4] has obtained the existence of a nontrivial solution (not necessary a ground state) under less restrictive assumptions on f. Moreover, in [1] V. Coti Zelati and P. Rabinowitz have proved the existence of infinitely many solutions, imposing some additional (perhaps, technical) conditions.

For general (indefinite) V it is known that the operator −Δ + V on L^{2}(IR^{N}) has a purely continuous spectrum consisting of closed disjoint intervals. In recent papers, C. Troestler and M. Willem [8], and W. Kryszewski and A. Szulkin [7] have proved the existence of a nontrivial solution of (P) under the assumption that 0 is in a gap of the spectrum of −Δ + V. Their proofs are based on a decomposition of H^{1}(IR^{N}) into two infinite dimensional subspaces and a generalized linking theorem which is applied to the energy functional associated to (P). This linking theorem requires the construction of a new degree theory in order to handle the lack of compactness in this problem. Moreover, in [7] it is stated that there exists infinitely many solutions provided the nonlinearity f is odd.

In this paper we present another proof for the existence of nontrivial solutions which is based on an approximation technique with periodic functions. Such an approach was used by P. Rabinowitz [4] in the positive case. We shall show that this procedure leads to a nontrivial solution of (P) without constructing any new degree theory.

Denote by F (x, u) = ∫_{0}^{u} f(x, ξ)dξ the primitive function of f. We make the following assumptions:

1^{o} f : IR^{N} × IR → IR and V : IR^{N} → IR are continuous and 1-periodic functions in each x_{i}, i = 1, …, N.

2^{o} |f(x, u)| ≤ C(1 + |u|^{p−1}), where 2 < p < 2N(N − 2).

3^{o} f(x, u) = o(|u|), as u → 0, uniformly in x.

4^{o} There exists γ > 2 such that 0 < γF (x, u) ≤ uf(x, u) for every u ≠ 0.

5^{o} 0 is in a spectral gap of the operator −Δ + V.

One of the main results of this paper is

Theorem 1 Under assumptions 1^{o}–5^{o} there exists a nontrivial solution of (P).

To prove this result, we consider a sequence of approximating problems in the next section and prove a general theorem which ensures the existence of a nontrivial limit. In Section 3 we apply the Linking Theorem of Rabinowitz to our approximating problems. This gives us a sequence of periodic solutions which leads to a solution of (P). In Section 4 we show that these solutions are asymptotically equal to a finite sum of H^{1}(IR^{N})- solutions. In the last Section 5 we prove the existence of a ground state solution by means of Nehari’s variational principle and periodic approximations. Additionally we prove that, on the cube of periods, these solutions look like a ground state when the period is large enough (see (18)).

It should be pointed out that all the results presented here have natural counterparts in the case of periodic Hamiltonian systems.

2 Approximation by periodic functions

Associated with (P) we study the approximate problem in cubes Q_{k} of IR^{N} with edge length k ∈ IN:

(P)_{k}  −Δu + V (x)u = f(x, u) in Q_{k} , u ∈ E_{k} = H_{per}^{1}(Q_{k}) ,

where H_{per}^{1}(Q_{k}) denotes the space of H^{1}(Q_{k})-functions which are k-periodic in x_{i}, i = 1, …, N. The operator −Δ + V on L_{per}^{2}(Q_{k}) has discrete spectrum with eigenvalues λ_{k,1} ≤ λ_{k,2} ≤ … → ∞ and it exists a finite

j(k) = min{i | λ_{k,i} > 0} .

Moreover, every eigenvalue λ_{k,i} is contained in the spectrum of −Δ + V on the whole space. This follows from the spectral decomposition theorem XIII.97 in [6]. Therefore, if (−α, β), α, β > 0 denots the spectral gap around 0, assumed in 5^{o}, we claim that λ_{k,i} ∉ (−α, β) for every k, i ∈ IN. We denote by ϕ_{k,i} the corresponding eigenfunctions.

Since every function u ∈ E_{k} is, by periodicity, also in E_{mk} for every m ∈ IN, we claim that every eigenvalue of −Δ + V on L_{per}^{2}(Q_{k}) is also an eigenvalue of this operator in L_{per}^{2}(Q_{mk}), m ∈ IN.

Now we define an orthogonal decomposition of E_{k} by

E_{k} = Y_{k} ⊕ Z_{k} , where Y_{k} = {ϕ_{k,1}, …, ϕ_{k,j(k)−1}} , Z_{k} = Y_{k}^{⊥} .

The associated energy functional to (P)_{k} is

J_{k}(u) = \frac{1}{2} ∫_{Q_{k}} |∇u|^{2} + V (x)|u|^{2} dx − ∫_{Q_{k}} F (x, u) dx

and the quadratic part of J_{k},

ℓ_{k}(u) = ∫_{Q_{k}} |∇u|^{2} + V u^{2}dx ,

is positive on Z_{k} and negative on Y_{k}. Moreover, we may define a new scalar product (·, ·)_{k} on E_{k} with corresponding norm ∥ · ∥_{k}, such that

∫_{Q_{k}} |∇u|^{2} + V (x)|u|^{2} dx = −∥u∥_{k}^{2} for u ∈ Y_{k} ,
∫_{Q_{k}} |∇u|^{2} + V (x)|u|^{2} dx = ∥u∥_{k}^{2} for u ∈ Z_{k} .

If we denote by P_{k} : E_{k} → Y_{k} and T_{k} : E_{k} → Z_{k} the orthogonal projections, our functional becomes

J_{k}(u) = \frac{1}{2}(∥T_{k}u∥_{k}^{2} − ∥P_{k}u∥_{k}^{2}) − ∫_{Q_{k}} F (x, u) dx.

From our assumptions it follows that J_{k} is C^{1} on E_{k} and its derivative is given by the formula

⟨J′_{k}(u), v⟩ = (T_{k}u, v)_{k} − (P_{k}u, v)_{k} − ∫_{Q_{k}} f(x, u)v dx .

Lemma 1 The norm ∥ · ∥_{k} is equivalent to the standard norm ∥ · ∥_{H^{1}} in H^{1}(Q_{k}), with constants not depending on k.

Proof. Let C_{0} = sup |V (x)| and (−α, β), α, β > 0 be the spectral gap of −Δ+V around 0. For u ∈ E_{k} we have

∥u∥_{k}^{2} ≤ ∫_{Q_{k}} |∇u|^{2} + C_{0}|u|^{2} dx ≤ C_{1}∥u∥_{H^{1}}^{2} .

On the other hand, for z ∈ Z_{k} and ε > 0

∥z∥_{k}^{2} = ε (∫_{Q_{k}} |∇z|^{2} + V |z|^{2}dx) + (1 − ε) (∫_{Q_{k}} |∇z|^{2} + V |z|^{2}dx)
≥ ε ∫_{Q_{k}} |∇z|^{2}dx − εC_{0} ∫_{Q_{k}} |z|^{2}dx + (1 − ε)β ∫_{Q_{k}} |z|^{2}dx .

For ε small enough this implies

∥z∥_{k}^{2} ≥ C_{2}∥z∥_{H^{1}}^{2} ,

where C_{2} only depends on C_{0}, β and ε. A similar argument for y ∈ Y_{k} shows that

∥y∥_{k}^{2} ≥ ε ∫_{Q_{k}} |∇y|^{2}dx − εC_{0} ∫_{Q_{k}} |y|^{2}dx + α ∫_{Q_{k}} |y|^{2}dx
≥ C_{3}∥y∥_{H^{1}}^{2}

with C_{3} depending on C_{0}, α and ε. □

Now we want to show that any sequence u_{k} of critical points of J_{k} has a nontrivial limit in H_{loc}^{1}(IR^{N}) which is a solution of (P), provided the sequence J_{k}(u_{k}) is bounded from above. To this end we need the following results concerning the behaviour of such sequences.

Lemma 2 Any critical point u_{k} of J_{k} satisfies the estimate ∥u_{k}∥_{k} ≤ C_{k} with a constant C_{k} > 0 only depending on the critical value.

Proof. From c_{k} = J_{k}(u_{k}) and J′_{k}(u_{k}) = 0 we obtain

c_{k} = J_{k}(u_{k}) − \frac{1}{2}⟨J′_{k}(u_{k}), u_{k}⟩
≥ (\frac{1}{2} − \frac{1}{γ}) ∫_{Q_{k}} f(x, u_{k})u_{k}dx .  (1)

From Assumptions 2^{o} − 4^{o} we have the inequalities

|f(x, u)|^{2} ≤ Cuf(x, u), if |u| ≤ 1,  (2)
|f(x, u)|^{p′} ≤ C|u|^{(p−1)(p′−1)}|f(x, u)| = Cuf(x, u), if |u| > 1,  (3)

where p′ = p/(p − 1) is the conjugate exponent. Denote B_{k} = {x ∈ Q_{k} | |u_{k}(x)| ≤ 1}. Then from (1)–(3) we get

c_{k} ≥ C_{1} (∫_{B_{k}} |f(x, u_{k})|^{2}dx + ∫_{Q_{k}\B_{k}} |f(x, u_{k})|^{p′}dx)  (4)

with C_{1} = (γ − 2)/(2γC). Hence

I_{1} := ∫_{B_{k}} |f(x, u_{k})|^{2}dx ≤ c_{k}/C_{1}, and I_{2} := ∫_{Q_{k}\B_{k}} |f(x, u_{k})|^{p′}dx ≤ c_{k}/C_{1} .

Now set y_{k} = P_{k}u_{k} and z_{k} = T_{k}u_{k}. From ⟨J′_{k}(u_{k}), y_{k}⟩ = 0 and Hölder’s inequality, we see that

∥y_{k}∥_{k}^{2} = − ∫_{Q_{k}} f(x, u_{k})y_{k}dx
≤ I_{1}^{1/2}∥y_{k}∥_{L^{2}} + I_{2}^{1/p′}∥y_{k}∥_{L^{p′}} ,

where | · |_{q} denotes the usual L^{q}-norm. Then the Sobolev embedding theorem implies

∥y_{k}∥_{k}^{2} ≤ C_{2} (I_{1}^{1/2} + I_{2}^{1/p′}) ∥y_{k}∥_{k} .

Clearly, the same argument works for z_{k} and we obtain

∥u∥_{k} ≤ 2C_{2} ((\frac{c_{k}}{C_{1}} )^{1/2} + (\frac{c_{k}}{C_{1}} )^{1/p′}) = C_{k}

with C_{1} and C_{2} independent of k. □

Lemma 3 There exists ε_{1} > 0 not depending on k such that ∥u_{k}∥_{k} ≥ ε_{1}, ∥u∥_{H^{1}} ≥ ε_{1} holds for any nontrivial critical point u_{k} of J_{k} and u of J. In addition, there exists a constant ε_{2} > 0 not depending on k such that J_{k}(u_{k}) ≥ ε_{2}, J(u) ≥ ε_{2} for any nontrivial critical point u_{k} of J_{k} and u of J.

Proof. Let ℓ_{k}(u) be the quadratic part of J_{k}. Since (−α, β) is in the spectral gap, there is a constant C_{0} = C_{0}(α, β) > 0, such that

|ℓ_{k}(u)| ≥ C_{0} ∥u∥_{L^{2}(Q_{k})}^{2} .

Let C_{1} = | min{V (x)}|. Then we obtain

|ℓ_{k}(u)| = ε | ∫_{Q_{k}} |∇u|^{2} + V u^{2}dx| + (1 − ε)| ∫_{Q_{k}} |∇u|^{2} + V u^{2}dx|
≥ ε ∫_{Q_{k}} |∇u|^{2}dx − εC_{1} ∥u∥_{L^{2}}^{2} + (1 − ε)C_{0} ∥u∥_{L^{2}}^{2}
≥ C_{2}∥u∥_{k}^{2}

for ε small enough, and C_{2} > 0 does not depend on k. Now let u_{k} be a nontrivial critical point of J_{k}. Then we claim from ⟨J′_{k}(u_{k}), u_{k}⟩ = 0 and Assumption 4^{o}, that ℓ_{k}(u_{k}) ≥ 0. We obtain

∥u∥_{k}^{2} ≤ \frac{1}{C_{2}} ℓ_{k}(u)
= \frac{1}{C_{2}} ∫_{Q_{k}} f(x, u_{k})u_{k} dx
≤ \frac{1}{C_{2}} (ε∥u∥_{k}^{2} + C_{ε}∥u∥_{k}^{p}) .

Consequently,

∥u∥_{k} ≥ (\frac{1 − ε/C_{2}}{C_{ε}} )^{\frac{1}{p−2}}  (5)

and ε can be choosen such that 1 − ε/C_{2} > 0. Again this estimate is independent of k and also holds for critical points of J. For the corresponding critical value we obtain (with s > 0 satisfying 1/s + 1/γ = 1/2, γ from Assumption 4^{o})

J_{k}(u_{k}) ≥ \frac{1}{2}ℓ_{k}(u_{k}) − \frac{1}{γ} ∫_{Q_{k}} f(x, u_{k})u_{k} dx
= \frac{1}{s}ℓ_{k}(u_{k}) + \frac{1}{γ} (\underbrace{ℓ_{k}(u_{k}) − ∫_{Q_{k}} f(x, u_{k})u_{k} dx}_{=0})

≥ \frac{C_{2}}{s} ∥u∥_{k}^{2} .

This shows, together with (5), that

J_{k}(u_{k}) ≥ \frac{C_{2}}{s} (\frac{1 − ε/C_{2}}{C_{ε}} )^{\frac{2}{p−2}}

with constants not depending on k. Clearly the same arguments work for J. □

Lemma 4 Let Q_{n} be the cube of edge length l_{n} → ∞ centered at the origin, and K_{r}(ξ) be the closed r-cube centered at the point ξ. Consider a sequence u_{n} ∈ H_{loc}^{1}(IR^{N}) of l_{n}-periodic functions such that ∥u_{n}∥_{H^{1}(Q_{n})} ≤ C. Assume that there is r > 0 such that

lim inf_{n→∞} (sup_{ξ} ∫_{K_{r}(ξ)} |u_{n}|^{2}dx) = 0.

Then ∥u_{n}∥_{L^{q}(Q_{n})} → 0 for q ∈ (2, 2N/(N − 2)).

Proof. Let us denote by Q′_{n} (resp. Q′′_{n}) the cube of edge length l_{n} + r (resp. l_{n} + 2r) centered at the origin. Choose a cut-of function χ_{n} ∈ C_{0}^{∞}(IR^{N}) such that 0 ≤ χ_{n}(x) ≤ 1, χ_{n} = 1 on Q′_{n}, supp χ_{n} ⊂ Q′′_{n} and |∇χ_{n}(x)| ≤ C , x ∈ IR^{N}, where C > 0 does not depend on n. Evidently, such a function exists.

Now we set v_{n} = χ_{n}u_{n}. For n being large enough it is easy to verify (using the periodicity of u_{n}) that

sup_{ξ} ∫_{K_{r}(ξ)} |v_{n}|^{2}dx = sup_{ξ} ∫_{K_{r}(ξ)} |u_{n}|^{2}dx

and v_{n} is bounded in H^{1}(IR^{N}). By the well-known concentration lemma of P.-L. Lions (see [2], Lemma I.1) v_{n} → 0 in L^{q}(IR^{N}), q ∈ (2, 2N/(N − 2)). Since u_{n} = v_{n} on Q_{n}, we get the assertion. □

Lemma 5 If a sequence u_{k} ∈ E_{k} is uniformly bounded in the H^{1}(Q_{k})-norm and satisfies J′_{k}(u_{k}) → 0, then the following alternative holds: Either

(i) ∥u_{k}∥_{k} → 0 as k → ∞; or

(ii) there exist numbers r, η > 0 and a sequence of points ξ_{k} ∈ IR^{N}, such that

lim inf_{k→∞} (∫_{K_{r}(ξ_{k})} |u_{k}|^{2}dx) ≥ η .

Proof. Assume that (ii) does not hold. Then we know from Lemma 4 that ∥u_{k}∥_{L^{q}(Q_{k})} → 0 for 2 < q < 2N/(N − 2). Set y_{k} = P_{k}u_{k}, z_{k} = T_{k}u_{k}. From ⟨J′_{k}(u_{k}), z_{k}⟩ ≤ ε_{k}∥z_{k}∥_{k} with ε_{k} → 0, Assumptions 2^{o}, 3^{o} and Hölder’s inequality, we get

∥z_{k}∥_{k}^{2} ≤ ∫_{Q_{k}} f(x, u_{k})z_{k} dx + ε_{k}∥z_{k}∥_{k}
≤ ε∥u_{k}∥_{L^{2}}∥z_{k}∥_{L^{2}} + C_{ε}∥u_{k}∥_{L^{p}}^{p−1}∥z_{k}∥_{L^{p}} + ε_{k}∥z_{k}∥_{k} .

This implies that ∥z_{k}∥_{k} → 0. In the same way it follows that ∥y_{k}∥_{k} → 0. □

Theorem 2 Let Assumptions 1^{o}–5^{o} be satisfied and let u_{k} ∈ E_{k} be a sequence of nontrivial critical points of J_{k} such that c_{k} = J_{k}(u_{k}) is bounded from above with a constant not depending on k. Then there exists a nontrivial solution u ∈ H^{1}(IR^{N}) of (P).

Proof. Since ∥u_{k}∥_{k} is uniformly bounded (by Lemma 2) and J_{k}(u_{k}) ≥ ε_{2} > 0 with ε_{2} not depending on k (Lemma 3), case (i) of Lemma 5 is not possible. From (ii) we obtain (at least for a subsequence)

∥u_{k}∥_{L^{2}(K_{r}(ξ_{k}))}^{2} ≥ η/2 .

Thus, we find a sequence of integer vectors b_{k} ∈ ZZ^{N} and a number s > 0, such that the sequence ũ_{k}, defined by ũ_{k}(x) = u_{k}(x + b_{k}), satisfies

∥ũ_{k}∥_{L^{2}(K_{s}(0))}^{2} ≥ η/2 .  (6)

Since V and f are invariant under translations by b_{k}, we have J_{k}(ũ_{k}) = J_{k}(u_{k}) and ∥J′_{k}(ũ_{k})∥ = ∥J′_{k}(u_{k})∥. Lemma 2 implies that the sequence ũ_{k} is uniformly bounded in ∥ · ∥_{k} and consequently converges weakly in H_{loc}^{1}(IR^{N}) to a function u ∈ H^{1}(IR^{N}). Then, for a subsequence, we have f(x, ũ_{k}) → f(x, u) in L_{loc}^{1}(IR^{N}) and we obtain for any test function φ ∈ C_{0}^{∞}(IR^{N})

⟨J′(u), φ⟩ = ∫_{IR^{N}} ∇u∇φ + V uφ − f(x, u)φ dx
= lim_{k→∞} ∫_{IR^{N}} ∇ũ_{k}∇φ + V ũ_{k}φ − f(x, ũ_{k})φ dx
= 0 .

Therefore, u is a weak solution of (P) and is not trivial in view of (6). □

3 Existence of periodic critical points

In this section we construct a sequence of critical points which satisfies the assumptions of Theorem 2.

Lemma 6 There exist real numbers δ, r > 0 not depending on k, such that

inf_{u∈N_{k}} J_{k}(u) ≥ δ , where N_{k} = {z ∈ Z_{k} | ∥z∥_{k} = r} .

Proof . For z ∈ Z_{k} we have

J_{k}(z) = \frac{1}{2}∥z∥_{k}^{2} − ∫_{Q_{k}} F (x, z) dx

and from our growth conditions 2^{o}, 3^{o} we claim that for every ε > 0 there is a constant C_{ε}, such that |F (x, u)| ≤ ε|u|^{2}+C_{ε}|u|^{p}. From the Sobolev embedding theorem we obtain a constant C, not depending on k, such that

∫_{Q_{k}} F (x, z) dx ≤ C (ε∥z∥_{k}^{2} + C_{ε}∥z∥_{k}^{p})

holds. Consequently, for ε and ∥z∥_{k} small enough, we obtain the assertion. □

Now for each k we fix a function z_{k} ∈ Z_{k} with ∥z_{k}∥_{k} = 1. For ρ > 0 we define the sets

M_{k} = {y + tz_{k} | ∥y + tz_{k}∥_{k} < ρ, t ≥ 0, y ∈ Y_{k}}.

Lemma 7 There exists a ρ > r not depending on k, such that

sup_{u∈∂M_{k}} J_{k}(u) = 0 .

Proof . Remark that

∂M_{k} = {y + tz_{k} | (∥y + tz_{k}∥_{k} = ρ, t ≥ 0) or (∥y + tz_{k}∥_{k} ≤ ρ, t = 0)} .

Since J_{k}(0) = 0, we have to show sup_{u∈∂M_{k}} J_{k}(u) ≤ 0. For y + tz_{k} ∈ M_{k} we have

J_{k}(y + tz_{k}) = \frac{1}{2}t^{2} − \frac{1}{2}∥y∥_{k}^{2} − ∫_{Q_{k}} F (x, y + tz_{k}) dx .

From our Assumptions 2^{o} − 4^{o} we claim that for every ε > 0 there is a constant C_{ε}, such that F (x, u) ≥ −ε|u|^{2} + C_{ε}|u|^{γ} where 2 < γ < 2N/(N − 2). It follows that

∫_{Q_{k}} F (x, y + tz_{k}) dx ≥ −ε∥y∥_{L^{2}}^{2} − εt^{2}∥z_{k}∥_{L^{2}}^{2} + C_{ε}∥y + tz_{k}∥_{L^{γ}}^{γ} .

From the Sobolev embedding theorem we obtain with a constant C not depending on k

J_{k}(y + tz_{k}) ≤ −\frac{1}{2}∥y∥_{k}^{2} + εC ∥y∥_{k}^{2} + \frac{1}{2}t^{2} + εCt^{2} − C_{ε}∥y + tz_{k}∥_{L^{γ}}^{γ} .  (7)

Now X_{k} = Y_{k} ⊕ IRz_{k} is continuously embedded in L^{γ}(Q_{k}) and there exists a continuous projection π_{k} : X_{k} → IRz_{k} with norm ∥π_{k}∥ = C_{k}, i. e. ∥tz_{k}∥_{L^{γ}} ≤ C_{k}∥y + tz_{k}∥_{L^{γ}}. Since z_{k} is not trivial, we clearly have C_{k} ≥ 1 for every k. Choosing ε in (7) so small that εC = 1/4, we get

J_{k}(y + tz_{k}) ≤ −\frac{1}{4}∥y∥_{k}^{2} + \frac{3}{4}t^{2} − C_{ε}Ct^{γ} ,

where we have used again the Sobolev embedding. Note that the right hand side of this estimate does not depend on k. Consequently, if t = 0, we have J_{k}(y) ≤ 0 and for ∥y + tz_{k}∥_{k} → ∞ we have J_{k}(y + tz_{k}) → −∞. This shows that we can choose ρ independent of k such that Lemma 7 holds. □

Lemma 8 For the set of deformations Γ_{k} = {h ∈ C^{0}(E_{k}, E_{k}) | h_{|∂M_{k}} = id} the numbers

c_{k} = inf_{h∈Γ_{k}} sup_{u∈M_{k}} J(h(u))

are critical values of J_{k} and satisfy 0 < δ ≤ c_{k} ≤ M < ∞.

Proof. Since Y_{k} is finite-dimensional and N_{k} and ∂M_{k} link, the first assertion follows from the Linking Theorem 4.6 of P. Rabinowitz [5]. This theorem also shows that c_{k} ≥ δ, where δ > 0 is independent of k by Lemma 2.

Since the constant ρ in the definition of M_{k} is also independent of k, and J_{k} is bounded on bounded sets, we obtain (taking h = id)

c_{k} ≤ sup_{u∈M_{k}} J_{k}(u) ≤ M ,

where M only depends on ρ. □

The proof of Theorem 1 now follows immediately from Lemma 8 and Theorem 2.

4 Further properties of periodic solutions

In this section we show that periodic solutions of (P) with large period look like a finite sum of finite energy solutions in the whole space, i. e. they converge to a sum of H^{1}(IR^{N})- solutions as the period tends to ∞. We state even a little more general assertion dealing with periodic “almost solutions”, namely, Theorem 3 below. It will be used, in full generality, to prove Theorem 4 of Section 5. This statement is the periodic analog of Proposition 4.2 of [7] and Proposition 2.13 of [1] (in the positive definite case).

We need the following additional assumption on f:

6^{o} There exist positive constants C and ε such that for all x ∈ IR^{N}

|f(x, u + v) − f(x, u)| ≤ C|v|(1 + |u|^{p−1}) if |v| ≤ ε .

Under this additional assumption we have

Theorem 3 Assume 1^{o}– 6^{o}. Let u_{k} ∈ E_{k} be a uniformly bounded sequence which satisfies J′_{k}(u_{k}) → 0 and c_{k} = J_{k}(u_{k}) → c > 0. Then there exist critical points u^{i}, i = 1, …, n of J and sequences b_{k}^{i} ∈ ZZ^{N} such that

∥u_{k} − ∑_{i=1}^{n} u^{i}(· + b_{k}^{i})∥_{k} → 0 and ∑_{i=1}^{n} J(u^{i}) = c .

Proof. We fix the following notations. Lemma 5 shows that there exist b_{k} ∈ ZZ^{N} such that ũ_{k} = u_{k}(· + b_{k}) → u weakly in H_{loc}^{1} and strongly in L_{loc}^{p} and u ∈ H^{1}(IR^{N}) is a nontrivial solution of (P). Let χ_{k} ∈ C_{0}^{∞} be a function such that 0 ≤ χ_{k} ≤ 1, χ_{k} = 1 on Q_{k−1}, χ_{k} = 0 outside of Q_{k} and |∇χ_{k}| ≤ C. Set

w_{k} = χ_{k}u and v_{k} = ũ_{k} − w_{k}

(w_{k} may be either viewed as a function in E_{k} or in H^{1}(IR^{N})). Since w_{k} → u in H^{1}(IR^{N}), we claim that

∥ũ_{k} − u∥_{k} → 0 iff ∥v_{k}∥_{k} → 0.  (8)

First we state the following assertions:

(i) For any ε > 0 there exists Ω ⊂ IR^{n} and k_{ε} ∈ IN such that for k ≥ k_{ε}

∥w_{k}∥_{H^{1}(IR^{N} \Ω)} ≤ ε, ∥w_{k}∥_{L^{∞}(IR^{N} \Ω)} ≤ ε and ∫_{IR^{N} \Ω} F (x, w_{k})dx ≤ ε .

(ii) J′_{k}(v_{k}) → 0 as k → ∞.

(iii) J_{k}(v_{k}) → c − J(u) as k → ∞.

(i) The L^{∞}- estimate follows from [4], the others are clear. (ii) Since J ∈ C^{1}, we have J′(w_{k}) → 0 and J_{k}(w_{k}) = J(w_{k}) → J(u). Since every function φ_{k} ∈ E_{k} can be extended (by periodicity and cut off) to a function φ̃_{k} ∈ H^{1}(IR^{N}) such that φ̃_{k} = φ_{k} on Q_{k} and ∥φ̃_{k}∥_{H^{1}(IR^{N})} ≤ C∥φ_{k}∥_{k} with C not depending on φ and k, we obtain

⟨J′_{k}(w_{k}), φ_{k}⟩ = ⟨J′(w_{k}), φ̃_{k}⟩ ≤ ε_{k}∥φ̃_{k}∥_{H^{1}(IR^{N})} ≤ ε_{k}C∥φ_{k}∥_{k}

for all φ_{k} ∈ E_{k}, where ε_{k} → 0. This shows J′_{k}(w_{k}) → 0. For v_{k} we have

⟨J′_{k}(v_{k}), φ_{k}⟩ = ⟨J′_{k}(ũ_{k}), φ_{k}⟩ − ⟨J′_{k}(w_{k}), φ_{k}⟩
+ ∫_{Q_{k}} [f(x, ũ_{k}) − f(x, w_{k}) − f(x, ũ_{k} − w_{k})] φ_{k} dx .  (9)

Since ũ_{k} → u locally in L^{p}, we get for any bounded domain Ω ⊂ IR^{N} and k large enough

|∫_{Ω} [f(x, ũ_{k}) − f(x, w_{k}) − f(x, ũ_{k} − w_{k})] φ_{k} dx| ≤ ε∥φ_{k}∥_{k} .

Since w_{k} ∈ H^{1}(IR^{N}), we also have

|∫_{Q_{k}\Ω} f(x, w_{k})φ_{k} dx| ≤ ε∥φ_{k}∥_{k}

for large k and appropriate Ω. Finally, from Assumption 6^{o} and (i) we get

∫_{Q_{k}\Ω} |f(x, v_{k} − w_{k}) − f(x, v_{k})| |φ_{k}| dx ≤ ∫_{Q_{k}\Ω} C(1 + |v_{k}|^{p−1})|w_{k}| |φ_{k}|dx

≤ C∥φ_{k}∥_{k} [∥w_{k}∥_{H^{1}(IR^{N} \Ω)} + ∥w_{k}∥_{L^{∞}(IR^{N} \Ω)}∥v_{k}∥_{H^{1}(IR^{N} \Ω)}^{p−1}]
≤ ε∥φ_{k}∥_{k} .

Thus (9) implies (ii). (iii) We have

J_{k}(ũ_{k}) = J_{k}(v_{k}) + J_{k}(w_{k}) + ∫_{Q_{k}} ∇v_{k}∇w_{k} + V v_{k}w_{k} dx
− ∫_{Q_{k}} F (x, v_{k} + w_{k}) − F (x, v_{k}) − F (x, w_{k}) dx .  (10)

For Ω ⊂ IR^{N} and k large enough, the first integral can be estimated by

∫_{Ω} ∇v_{k}∇u + V v_{k}u dx + C∥v_{k}∥_{k} ∥w_{k}∥_{H^{1}(IR^{N} \Ω)} .  (11)

Since v_{k} tends weakly to 0 in H_{loc}^{1} and ∥v_{k}∥_{k} is uniformly bounded, (i) implies that (11) tends to 0 as k → ∞. For the second integral we get

|∫_{Ω} F (x, v_{k} + w_{k}) − F (x, v_{k}) − F (x, w_{k}) dx| ≤ ε

if k is sufficiently large. By Hölder’s inequality we have

|∫_{IR^{N} \Ω} F (x, v_{k} + w_{k}) − F (x, v_{k}) dx|

≤ ∫_{Q_{k}\Ω} [|v_{k}| + |w_{k}| + C(|v_{k}| + |w_{k}|)^{p−1}] |w_{k}| dx
≤ C [∥v_{k}∥_{k} + ∥w_{k}∥_{k} + ∥v_{k}∥_{k}^{p−1} + ∥w_{k}∥_{k}^{p−1}] ∥w_{k}∥_{H^{1}(IR^{N} \Ω)}
≤ ε .

Together with the third estimate in (i) we obtain from (10)

c = lim_{k→∞} J_{k}(ũ_{k}) = lim_{k→∞} J_{k}(v_{k}) + J(u)

To complete the proof of the theorem we observe that any function φ ∈ E_{k}, φ ≠ 0 satisfies

J_{k}(φ) − \frac{1}{2}⟨J′_{k}(φ), φ⟩ ≥ (\frac{1}{2} − \frac{1}{γ} ) ∫_{Q_{k}} f(x, φ)φ dx > 0 ,  (12)

which implies

J_{k}(v_{k}) → c − J(u) ≥ 0, i. e. J(u) ≤ c .

If J(u) = c, we claim ∥v_{k}∥_{k} → 0, since otherwise we can use the same arguments as above for v_{k} instead of u_{k} and obtain a nontrivial solution v of (P) with J(v) ≤ 0, which is not possible in view of Lemma 3. If J(u) < c, then Lemma 3 shows J(u) ≥ ε_{2} > 0 and we can repeat the arguments above with u_{k} replaced by v_{k} and c replaced by c − J(u) ≤ c − ε_{2}. After a finite number of steps we are done. □

5 Ground states

In this section we use a different variational method to construct a nontrivial solution of (P), which in addition has minimal energy (ground state). To do this we need a more restrictive assumption on f, namely we assume that f is C^{1} in the second variable and instead of 4^{o} we assume

4^{′o} There exists 0 < θ < 1 such that 0 < y^{−1}f(x, y) ≤ θ\frac{∂f}{∂y} (x, y) for every y ≠ 0.

This implies in particular

0 < F (x, y) ≤ \frac{θ}{1 + θ}f(x, y)y  (13)

for y ≠ 0, and θ/(1 + θ) < 1/2. Moreover, setting σ = 1/θ, we obtain from 4^{′o}

|y|^{σ}y \frac{d}{dy} (|y|^{−σ}f(x, y)) = −σf(x, y) + y \frac{d}{dy} f(x, y) ≥ 0 for every y ≠ 0 .

Now if t > 0 is a real number, integration of \frac{d}{dy}(|y|^{−σ}f(x, y)) over the interval [u, tu] (resp. [tu, u]) leads to the inequalities

f(x, tu)u \{ \begin{matrix}≥ & t^{σ}f(x, u)u & if t ≥ 1 \\ ≤ & t^{σ}f(x, u)u & if t ≤ 1 \end{matrix} .  (14)

For k = 1, 2, … we define

I_{k}(u) = ⟨J′_{k}(u), u⟩ = ∫_{Q_{k}} |∇u|^{2} + V u^{2} dx − ∫_{Q_{k}} f(x, u)u dx

In a similar manner we define the functional I on H^{1}(IR^{N}). We consider the minimization problems

m_{k} = inf{J_{k}(u) | u ∈ S_{k}} , where S_{k} := {u ∈ E_{k} | u ≠ 0, I_{k}(u) = 0},  (15)
m = inf{J(u) | u ∈ S} , where S := {u ∈ H^{1}(IR^{N}) | u ≠ 0, I(u) = 0}.  (16)

Using the same arguments as in Lemma 3, we can show the following (note that we have only used the fact that ⟨J′_{k}u_{k}, u_{k}⟩ = 0 in the proof of Lemma 3):

Lemma 9 There exist numbers ε_{1}, ε_{2} > 0, such that ∥u_{k}∥_{k} ≥ ε_{1}, ∥u∥ ≥ ε_{1}, J_{k}(u_{k}) ≥ ε_{2} and J(u) ≥ ε_{2} for all u_{k} ∈ S_{k}, u ∈ S. □

Lemma 10 For each k, there exists a minimizer for (15) which is a nontrivial ground state solution of (P)_{k}.

Proof . From compactness of embeddings it follws that a minimizing sequence for (15) has a convergent subsequence with nontrivial limit (in view of Lemma 9).

Since u_{k} is a minimizer, there exists a Lagrange multiplier λ ∈ IR, such that J′_{k}(u_{k}) = λI′(u_{k}). Since ⟨J′_{k}(u_{k}), u_{k}⟩ = 0, we claim from 4^{′o}

⟨I′_{k}(u_{k}), u_{k}⟩ = ℓ_{k}(u_{k}) − ∫_{Q_{k}} f′_{y}(x, u_{k})u_{k}^{2}dx
≤ ℓ_{k}(u_{k}) − \frac{1}{θ} ∫_{Q_{k}} f(x, u_{k})u_{k}dx
= −\frac{1}{s} ∫_{Q_{k}} f(x, u_{k})u_{k}dx ,

where s > 0 is such that 1 + 1/s = 1/θ. Since u_{k} is not trivial, the right hand side is strictly less than 0 and it follows that λ must be zero. Hence, u_{k} is a solution of (P)_{k}. □

Theorem 4 Assume 1^{o}– 3^{o}, 4^{′o} and 5^{o} and let u_{k} ∈ S_{k} be the periodic solutions of (15). Then there is a nontrivial solution u of (P), which is a minimizer for (16). Moreover, if f satisfies

|f_{y}(x, y)| ≤ C(1 + |y|^{p−1}) ,  (17)

then there is a sequence b_{k} ∈ ZZ^{N} such that, along a subsequence,

∥u_{k} − u(· + b_{k})∥_{H^{1}(Q_{k})} → 0 .  (18)

Proof. First of all we derive a uniform bound for u_{k}. To do this we fix a positive function η ∈ C_{0}^{∞}(Q_{1}) which satisfies ℓ_{1}(η) > 0. Then inequality (14) shows that

I(tη) \{ \begin{matrix}≤ & t^{2}ℓ(η) − t^{σ+1} ∫_{IR^{N}} f(x, η)η dx & if t ≥ 1 \\ ≥ & t^{2}ℓ(η) − t^{σ+1} ∫_{IR^{N}} f(x, η)η dx & if t ≤ 1 , \end{matrix}  (19)

where σ = 1/θ > 1. Hence, there exists a number t > 0, such that

ℓ_{1}(tη) = ∫_{Q_{1}} f(x, tη)tη dx

and v_{1} := tη belongs to S_{1}. For k ≥ 2 we define

v_{k}(x) = \{ \begin{matrix}v_{1}(x) & , & x ∈ Q_{1} \\ 0 & , & x ∈ Q_{k} \ Q_{1} \end{matrix} .

Clearly,

v_{k} ∈ S_{k} and J_{k}(v_{k}) = J_{1}(v_{1}) = J(v_{1}) for all k = 1, 2, …

This shows J_{k}(u_{k}) ≤ J_{1}(v_{1}). Now Lemma 2 implies ∥u_{k}∥_{k} ≤ C with a constant not depending on k. As in the proof of Theorem 2 we find a sequence b_{k} ∈ ZZ^{N} such that a subsequence of ũ_{k} = u_{k}(· + b_{k}) converges locally to a function u ∈ H^{1}(IR^{N}), a nontrivial solution of (P).

To prove that u is a minimizer for (16) we first establish the following: For every v ∈ S and every ε > 0 there exists k_{ε} ∈ IN and v_{k} ∈ S_{k}, such that

J_{k}(v_{k}) ≤ J(v) + ε for all k ≥ k_{ε} .  (20)

Since C_{0}^{∞}(IR^{N}) is dense in H^{1}(IR^{N}) and J and I are continuous on H^{1}(IR^{N}), there exists a sequence η_{k} ∈ C_{0}^{∞}(IR^{N}), supp η_{k} ⊂ Q_{k} with the following properties:

η_{k} → v in the norm ∥ · ∥_{H^{1}} ,  (21)
J(η_{k}) → J(v),  I(η_{k}) → I(v) = 0 .  (22)

Now inequality (19) shows that if I(η_{k}) ≤ 0, then there exists 0 < t_{k} ≤ 1 such that I(t_{k}η_{k}) = 0, while if I(η_{k}) ≥ 0, then I(t_{k}η_{k}) = 0 for some t_{k} ≥ 1. Moreover we have the following estimates for t_{k}:

if I(η_{k}) = δ_{k} ≤ 0 : 1 ≥ t_{k} ≥ (\frac{ℓ(η_{k})}{∫_{IR^{N}} f(x, η_{k})η_{k} dx})^{\frac{1}{σ−1}} = (1 + \frac{δ_{k}}{∫_{IR^{N}} f(x, η_{k})η_{k} dx})^{\frac{1}{σ−1}}
if I(η_{k}) = δ_{k} ≥ 0 : 1 ≤ t_{k} ≤ (\frac{ℓ(η_{k})}{∫_{IR^{N}} f(x, η_{k})η_{k} dx})^{\frac{1}{σ−1}} = (1 + \frac{δ_{k}}{∫_{IR^{N}} f(x, η_{k})η_{k} dx})^{\frac{1}{σ−1}} .

Since ∫_{IR^{N}} f(x, v)v dx = ℓ(v) ≥ 2ε_{2} for v ∈ S (with ε_{2} > 0 from Lemma 3), we can choose k_{0} large enough such that η_{k} satisfies

∫_{IR^{N}} f(x, η_{k})η_{k} dx ≥ ε_{2} for all k ≥ k_{0} .

Since δ_{k} → 0, we obtain from the above inequalities

t_{k} → 1 and
J(t_{k}η_{k}) → J(v) as k → ∞ .

Now given ε > 0 we can choose k_{ε} large enough such that v_{k} := t_{k}η_{k} ∈ S_{k} and J_{k}(v_{k}) = J(v_{k}) ≤ J(u) + ε for all k ≥ k_{ε}, which proves (20).

In particular, (20) implies that

lim sup m_{k} ≤ J(v) + ε

for any v ∈ S and for any ε > 0; hence,

lim sup m_{k} ≤ m .

Since θ/(1 + θ) < 1/2, (13) implies that the function g(x, v) := [\frac{1}{2}f(x, v)v − F (x, v)] is nonnegative for every v. Moreover we have

J_{k}(v_{k}) = ∫_{Q_{k}} g(x, v_{k}) dx for every v_{k} ∈ S_{k}

and consequently

m_{k} = J_{k}(ũ_{k}) = ∫_{Q_{k}} g(x, ũ_{k}) dx

for any k ∈ IN. Since the integrand is nonnegative, we claim that for any bounded Ω ⊂ IR^{N} and k large enough

m_{k} ≥ ∫_{Ω} g(x, ũ_{k}) dx .

Passing to the limit, we obtain

lim inf m_{k} ≥ ∫_{Ω} g(x, u) dx .

Since Ω is an arbitrary domain and u is a nontrivial solution, this gives rise to the inequality

lim inf m_{k} ≥ J(u) ≥ m .

Therefore,

lim m_{k} = m = J(u)  (23)

and u is a ground state.

Thus, it remains to prove (18). Note that (17) implies Assumption 6^{o} from Section 4. Since J_{k}(u_{k}) → J(u) = m, the assertion of Theorem 3 holds with c = m, n = 1 and u^{1} = u. □

Acknowledgement. The present work was started when the authors participated in the Banach Center Workshop “Topological and Variational Methods in Nonlinear Analysis” (Warsaw, June, 1996), and continued during the visit of the first author to Humboldt University, Berlin (July, 1996). The first author was supported by “International Soros Science Education Program”, grant SPU 041048. The authors are grateful to all that organizations, which have made their joint work possible. We thank A. Szulkin and M. Willem for helpful discussions.

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