Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations

Solitary Waves for Nonlinear Klein-Gordon-Maxwell and Schr¨ odinger-Maxwell Equations Teresa D’Aprile∗

Dimitri Mugnai†

Abstract In this paper we study the existence of radially symmetric solitary waves for nonlinear Klein-Gordon equations and nonlinear Schr¨ odinger equations coupled with Maxwell equations. The method relies on a variational approach and the solutions are obtained as mountain-pass critical points for the associated energy functional.

1

Introduction

This paper has been motivated by the search of nontrivial solutions for the following nonlinear equations of the Klein-Gordon type: ∂2ψ − ∆ψ + m2 ψ − |ψ|p−2 ψ = 0, x ∈ R3 , ∂t2

(1.1)

or of the Schr¨ odinger type: i~

~2 ∂ψ =− ∆ψ − |ψ|p−2 ψ, x ∈ R3 , ∂t 2m

(1.2)

where ~ > 0, m > 0, p > 2, ψ : R3 × R → C. In recent years many papers have been devoted to find standing waves of (1.1) or (1.2), i.e. solutions of the form ψ(x, t) = eiωt u(x), ω ∈ R. With this Ansatz the nonlinear Klein-Gordon equation, as well as the nonlinear Schr¨odinger equation, is reduced to a semilinear elliptic equation and existence theorems have been established whether u is radially symmetric and real (see [8], [9]), or u is non-radially symmetric and complex (see [12], [14]). In this paper we want to investigate the existence of nonlinear Klein-Gordon or Schr¨ odinger fields interacting with an electromagnetic field E − H; such a problem has been extensively pursued in the case of assigned electromagnetic fields (see [3], [4], [11]). Following the ideas already introduced in [5], [6], [7], [10], [13], [15], we do not assume that the electromagnetic field is assigned. Then we have to study a system of equations whose unknowns are the wave function ψ = ψ(x, t) and the gauge potentials A, Φ,

∗

A : R3 × R → R3 , Φ : R3 × R → R

Dipartimento Interuniversitario di Matematica, via E. Orabona 4, 70125 BARI (Italy), e-mail: [email protected] † Dipartimento di Matematica e Informatica, via Vanvitelli 1, 06123 PERUGIA (Italy), e-mail: [email protected]

1

which are related to E − H by the Maxwell equations “ ’ ∂A E = − ∇Φ + ∂t H = ∇ × A. Let us first consider equation (1.1). The Lagrangian density related to (1.1) is given by LKG

”Œ Œ • 1 ŒŒ ∂ψ ŒŒ2 1 2 2 2 = − |∇ψ| − m |ψ| + |ψ|p . Œ Œ 2 ∂t p

The interaction of ψ with the electromagnetic field is described by the minimal coupling rule, that is the formal substitution ∂ ∂ 7−→ + ieΦ, ∇ 7−→ ∇ − ieA, ∂t ∂t where e is the electric charge. Then the Lagrangian density becomes: Œ2 ”Œ • Œ 1 ŒŒ ∂ψ 1 2 2 2 Œ + ieψ֌ − |∇ψ − ieAψ| − m |ψ| + |ψ|p . LKGM = Œ 2 ∂t p If we set

ψ(x, t) = u(x, t)eiS(x,t) ,

where u, S : R3 × R → R, the Lagrangian density takes the form ” • › š 1 1 2 ut − |∇u|2 − |∇S − eA|2 − (St + eΦ)2 + m2 u2 + |u|p . LKGM = 2 p Now we consider the Lagrangian density of the electromagnetic field E − H 1 1 1 L0 = (|E|2 − |H|2 ) = |At + ∇Φ|2 − |∇ × A|2 . 2 2 2 The total action is given by Z Z S=

(1.3)

LKGM + L0 .

Making the variation of S with respect to u, S, Φ and A respectively, we get utt − ∆u + [|∇S − eA|2 − (St + eΦ)2 + m2 ]u − |u|p−2 u = 0, i ∂h (St + eΦ)u2 − div[(∇S − eA)u2 ] = 0, ∂t div(At + ∇Φ) = e(St + eΦ)u2 ,

(1.4) (1.5) (1.6)

∂ (At + ∇Φ) = e(∇S − eA)u2 . (1.7) ∂t We are interested in finding standing (or solitary) waves of (1.4)-(1.7), that is solutions having the form u = u(x), S = ωt, A = 0, Φ = Φ(x), ω ∈ R. ∇ × (∇ × A) +

Then the equations (1.5) and (1.7) are identically satisfied, while (1.4) and (1.6) become −∆u + [m2 − (ω + eΦ)2 ]u − |u|p−2 u = 0, 2

(1.8)

−∆Φ + e2 u2 Φ = −eωu2 .

(1.9)

In [6] the authors proved the existence of infinitely many symmetric solutions (un , Φn ) of (1.8)-(1.9) under the assumption 4 < p < 6, by using an equivariant version of the mountain pass theorem (see [1], [2]). The object of the first part of this paper is to extend this result as follows. Theorem A. If m > ω > 0 and 2 < p < 6 the system (1.8) − (1.9) has infinitely many radially symmetric solutions (un , Φn ), un 6≡ 0 and Φn 6≡ 0, with un ∈ H 1 (R3 ), Φn ∈ L6 (R3 ) and |∇Φn | ∈ L2 (R3 ). In the second part of the paper we study the Schr¨odinger equation for a particle in a electromagnetic field. Consider the Lagrangian associated to (1.2): ” • 1 ~2 1 ∂ψ 2 LS = i~ ψ − |∇ψ| + |ψ|p . 2 ∂t 2m p By using the formal substitution ∂ e e ∂ 7−→ + i Φ, ∇ 7−→ ∇ − i A, ∂t ∂t ~ ~ we obtain LSM Now take

” Œ2 • 1 ~2 ŒŒ e 1 ∂ψ Œ 2 = i~ ψ − eΦ|ψ| − Œ∇ψ − i Aψ Œ + |ψ|p . 2 ∂t 2m ~ p ψ(x, t) = u(x, t)eiS(x,t)/~ .

With this Ansatz the Lagrangian LSM becomes ’ ” “ • 1 1 1 ~2 2 2 i~uut − LSM = |∇u| − St + eΦ + |∇S − eA| u2 + |ψ|p . 2 2m 2m p RR 1 Proceeding as in [5], we consider the total action S = [LSM + 8π (|E|2 − |H|2 )] of the system “particle-electromagnetic field”. Then the Euler-Lagrange equations associated to the functional S = S(u, S, Φ, A) give rise to the following system of equations:  ‘ ~2 1 ∆u + St + eΦ + |∇S − eA|2 u − |u|p−2 u = 0, 2m 2m 1 ∂ 2 u + div[(∇S − eA)u2 ] = 0, ∂t m ’ “ ∂A 1 2 + ∇Φ , eu = − div 4π ∂t “ • ” ’ e 1 ∂ ∂A 2 (∇S − eA)u = + ∇Φ + ∇ × (∇ × A) . 2m 4π ∂t ∂t −

(1.10) (1.11) (1.12) (1.13)

If we look for solitary wave solutions in the electrostatic case, i.e.

u = u(x), S = ωt, Φ = Φ(x), A = 0, ω ∈ R, then (1.11) and (1.13) are identically satisfied, while (1.10) and (1.12) become −

~2 ∆u + eΦu − |u|p−2 u + ωu = 0, 2m 3

(1.14)

−∆Φ = 4πeu2 .

(1.15)

The existence of solutions of (1.14)-(1.15) was already studied for 4 < p < 6: in [5] existence of infinitely many radial solutions was proved, while in [12] existence of a non radially symmetric solution was established. In the second part of the paper we prove the following result. Theorem B. Let ω > 0 and 3 < p < 6. Then the system (1.14) − (1.15) has at least a radially symmetric solution (u, Φ), u = 6 0 and Φ 6= 0, with u ∈ H 1 (R3 ), Φ ∈ L6 (R3 ) and |∇Φ| ∈ L2 (R3 ).

2

Nonlinear Klein-Gordon Equations coupled with Maxwell Equations

In this section we will prove Theorem A. For sake of simplicity, assume e = 1 so that (1.8)(1.9) give rise to the following system in R3 : −∆u + [m2 − (ω + Φ)2 ]u − |u|p−2 u = 0,

(2.16)

−∆Φ + u2 Φ = −ωu2 .

(2.17)

We will always make the assumptions: a) m > ω > 0, b) 2 < p < 6. We note that q = 6 is the critical exponent for the Sobolev embedding H 1 (R3 ) ⊂ Lq (R3 ). It is clear that (2.16)-(2.17) are the Euler-Lagrange equations of the functional F : H 1 × D1,2 → R defined as Z  Z ‘ 1 1 F (u, Φ) = |u|p dx. |∇u|2 − |∇Φ|2 + [m2 − (ω + Φ)2 ]u2 dx − 2 R3 p R3 Here H 1 ≡ H 1 (R3 ) denotes the usual Sobolev space endowed with the norm kukH 1 ≡

’Z

R3

 ‘ “1/2 2 2 |∇u| + |u| dx

(2.18)

“1/2 |∇u| dx .

(2.19)

and D1,2 ≡ D1,2 (R3 ) is the completion of C0∞ (R3 , R) with respect to the norm kukD1,2 ≡

’Z

2

R3

The following two propositions hold. Proposition 2.1. The functional F belongs to C 1 (H 1 × D1,2 , R) and its critical points are the solutions of (2.16) − (2.17). (For the proof we refer to [6]). Proposition 2.2. For every u ∈ H 1 , there exists a unique Φ = Φ[u] ∈ D1,2 which solves (2.17). Furthermore 4

(i) Φ[u] ≤ 0; (ii) Φ[u] ≥ −ω in the set {x | u(x) 6= 0}; (iii) if u is radially symmetric, then Φ[u] is radial too. Proof. Fixed u ∈ H 1 , consider the following bilinear form on D1,2 : Z  € ∇ψ∇ψ + u2 φψ dx. a(φ, ψ) = R3

Obviously a(φ, φ) ≥ kφk2D1,2 . Observe that, since H 1 (R3 ) ⊂ L3 (R3 ), then u2 ∈ L3/2 (R3 ). On the other hand D1,2 is continuously embedded in L6 (R3 ), hence, by H¨older’s inequality, a(φ, ψ) ≤ kφkD1,2 kψkD1,2 + ku2 kL3/2 kφkL6 kψkL6 ≤ (1 + Ckuk2L3 )kφkD1,2 kψkD1,2 for some positive constant C, given by Sobolev inequality (see [19]). Therefore a defines an inner product, equivalent to the standard inner product in D1,2 . Moreover H 1 (R3 ) ⊂ L12/5 (R3 ), and then ŒZ Œ Œ Œ 2 Œ Œ ≤ ku2 k 6/5 kψkL6 ≤ ckuk2 12/5 kψkD1,2 . u (2.20) ψ dx L Œ Œ L R3

R Therefore the linear map ψ ∈ D1,2 7→ R3 u2 ψ dx is continuous. By Lax-Milgram’s Lemma we get the existence of a unique Φ ∈ D1,2 such that Z Z €  2 ∇Φ∇ψ + u Φψ dx = −ω u2 ψdx ∀ ψ ∈ D1,2 , R3

R3

i.e. Φ is the unique solution of (2.17). Furthermore Φ achieves the minimum “ “ Z ’  Z ’  ‘ ‘ 1 1 2 2 2 2 2 2 2 2 inf |∇φ| + u |φ| + ωu φ dx = |∇Φ| + u |Φ| + ωu Φ dx. 2 φ∈D 1,2 R3 R3 2

Note that also −|Φ| achieves such a minimum; then, by uniqueness, Φ = −|Φ| ≤ 0. Now let O(3) denote the group of rotations in R3 . Then for every g ∈ O(3) and f : R3 → R, set Tg (f )(x) = f (gx). Note that Tg does not change the norms in H 1 , D1,2 and Lp . In Lemma 4.2 of [6] it was proved that Tg Φ[u] = Φ[Tg u]. In this way, if u is radial, we get Tg Φ[u] = Φ[u]. Finally, following the same idea of [16], fixed u ∈ H 1 , if we multiply (2.17) by (ω+Φ[u])− ≡ − min{ω + Φ[u], 0}, which is an admissible test function, since ω > 0, we get Z Z 2 (ω + Φ[u])2 u2 dx = 0, − |DΦ[u]| dx − Φ[u]<−ω

Φ[u]<−ω

so that Φ[u] ≥ −ω when u 6= 0. Remark 2.1. The result (ii) of Proposition 2.2 can be strengthened in some cases. Indeed, take u in H 1 (R3 ) ∩ C ∞ radially symmetric such that u > 0 in B(0, R), u ≡ 0 in R3 \ B(0, R) for some R > 0. Then there results −ω ≤ Φ[u](x) ≤ 0 ∀ x ∈ R3 . 5

In fact, since Φ[u] solves (2.17), by standard regularity results for elliptic equations, u ∈ C ∞ implies Φ[u] ∈ C ∞ . By Proposition 2.2, Φ[u] is radial; moreover Φ[u] is harmonic outside B(0, R). Since Φ[u] ∈ D1,2 , then Φ[u](x) = −

c , |x|

|x| ≥ R,

˜ ˜ 0 (R) > 0 and Φ(r) ˜ ˜ for some c > 0. Setting Φ(r) = Φ[u](x) for |x| = r, it results Φ for > Φ(R) every r > R. Therefore the minimum of Φ[u] is achieved in B(0, R). Let x be a minimum point for Φ[u]. Then (2.17) implies Φ[u](x) =

−ωu2 (x) + ∆Φ[u](x) ≥ −ω. u2 (x)

In view of proposition 2.2 we can define the map Φ : H 1 −→ D1,2 which maps each u ∈ H 1 in the unique solution of (2.17). From standard arguments it results Φ ∈ C 1 (H 1 , D1,2 ) and from the very definition of Φ we get Fφ0 (u, Φ[u]) = 0

∀ u ∈ H 1.

(2.21)

Now let us consider the functional J : H 1 −→ R,

J(u) := F (u, Φ[u]).

By proposition 2.1, J ∈ C 1 (H 1 , R) and, by (2.21), J 0 (u) = Fu0 (u, Φ[u]). By definition of F , we obtain Z Z Z €  1 1 2 2 2 2 2 2 2 2 J(u) = |∇u| − |∇Φ[u]| + [m − ω ]u − u Φ[u] dx − ω |u|p dx. u Φ[u] − 2 R3 p 3 3 R R Multiplying both members of (2.17) by Φ[u] and integrating by parts, we obtain Z Z Z 2 2 2 |∇Φ[u]| dx + |u| |Φ[u]| dx = −ω |u|2 Φ[u] dx. R3

R3

R3

Using (2.22) the functional J may be written as Z Z  € 1 1 J(u) = |∇u|2 + [m2 − ω 2 ]u2 − ωu2 Φ[u] dx − |u|p dx. 2 R3 p R3

(2.22)

(2.23)

The next lemma states a relationship between the critical points of the functionals F and J (the proof can be found in [6]). Lemma 2.1. The following statements are equivalent: i) (u, Φ) ∈ H 1 × D1,2 is a critical point of F , ii) u is a critical point of J and Φ = Φ[u]. Then, in order to get solutions of (2.16)-(2.17), we look for critical points of J. 6

Theorem 2.1. Assume hypotheses a) and b). Then the functional J has infinitely many critical points un ∈ H 1 having a radial symmetry. Proof. Our aim is to apply the equavariant version of the Mountain-Pass Theorem (see [1], Theorem 2.13, or [17], Theorem 9.12). Since J is invariant under the group of translations, there is clearly a lack of compactness. In order to overcome this difficulty, we consider radially symmetric functions. More precisely we introduce the subspace Œ ‰ ˆ Hr1 = u ∈ H 1 Œ u(x) = u(|x|) . We divide the remaining part of the proof in three steps.

Step 1. Any critical point u ∈ Hr1 of JŒŒ 1 is also a critical point of J. Hr The proof can be found in [6].

Step 2. The functional JŒŒ 1 satisfies the Palais-Smale condition, i.e. Hr

0 (u ) → 0 any sequence {un }n ⊂ Hr1 such that J(un ) is bounded and J|H 1 n r

contains a convergent subsequence.

We remark that in [6] the authors proved the Palais-Smale condition for the functional J for 4 < p < 6. For the sake of simplicity, from now on we set Ω = m2 − ω 2 > 0. Let {un }n ⊂ Hr1 be such that 0 |J(un )| ≤ M, J|H 1 (un ) → 0 r for some constant M > 0. Then, using the form of J given in (2.23), p p ‘Z € ‘Z  pJ(un ) − J 0 (un )un = −1 −1 |∇un |2 + Ω|un |2 dx − ω u2n Φ[un ] dx 2 2 3 3 R R ‘Z € p  |∇un |2 + Ω|un |2 dx (2.24) −1 ≥ 2 R3 by Proposition 2.2. Moreover p 2

−1

and by assumption

‘Z

R3

€

 |∇un |2 + Ω|un |2 dx ≥ c1 kun k2

(2.25)

pJ(un ) − J 0 (un )un ≤ pM + c2 kun k

(2.26)

for some positive constants c1 and c2 . Combining (2.24), (2.25), (2.26), we deduce that {un }n is bounded in Hr1 . On the other hand, using equation (2.17), and proceeding as in (2.20), we get Z Z Z Z 2 2 2 2 ∇Φ[un ]| dx ≤ |∇Φ[un ]| dx + u2n Φ[un ]dx |un | |Φ[un ]| dx = −ω R3

R3

R3

≤ cωkun k2L12/5 kΦ[un ]kD1,2 ,

which implies that {Φ[un ]}n is bounded in D1,2 . Then, up to a subsequence, un * u in Hr1 7

R3

in D1,2 .

Φ[un ] * φ If L : Hr1 → (Hr1 )0 is defined as then

L(u) = −∆u + Ωu,

L(un ) = ωun Φ[un ] + |un |p−2 un + εn ,

where εn → 0 in (Hr1 )0 , that is

un = L−1 (ωun Φ[un ]) + L−1 (|un |p−2 un ) + L−1 (εn ). 3/2

Now note that {un Φ[un ]} is bounded in Lr ; in fact, by H¨older’s inequality, kun Φ[un ]kL3/2 ≤ kun kL2r kΦ[un ]kL6r ≤ ckun kL2r kΦ[un ]kD1,2 . r

0

Moreover {|un |p−2 un } is bounded in Lpr (where

1 p

+

1 p0

= 1). The immersions Hr1 ,→ L3r and 3/2

0

Hr1 ,→ Lrp are compact (see [8] or [18]) and thus, by duality, Lr and Lrp are compactly embedded in (Hr1 )0 . Then by standard arguments L−1 (ωun Φ[un ]) and L−1 (|un |p−2 un ) strongly converge in Hr1 . Then we conclude in Hr1 .

un → u

Step 3. The functional JŒŒ 1 satisfies the geometrical hypothesis of the equivariant version of Hr the Mountain Pass Theorem. First of all we observe that J(0) = 0. Moreover by Proposition 2.2 and (2.23) Z Z Z 1 Ω 1 J(u) ≥ |∇u|2 dx + |u|2 dx − |u|p dx. 2 R3 2 R3 p R3

The hypothesis 2 < p < 6 and the continuous embedding H 1 ⊂ Lp imply that there exists ρ > 0 small enough such that inf J(u) > 0. kukH 1 =ρ

Since J is even, the thesis of step 3 will follow if we prove that for every finite dimensional subset V of Hr1 it results lim J(u) = −∞. (2.27) u∈V,

kukH 1 →+∞

Let V be an m-dimensional subspace of Hr1 . For every u ∈ V , since by Proposition 2.2 Φ[u] ≥ −ω where u 6= 0, we get Z Z  € 1 1 1 |∇u|2 + Ω|u|2 + ω 2 u2 dx − J(u) ≤ |u|p ≤ ckuk2H 1 − kukpLp 2 R3 p R3 p

and (2.27) follows, since all norms in V are equivalent. Proof of Theorem A. Lemma 2.1 + Theorem 2.1.

Remark 2.2. In view of Remark 2.1 the existence of one nontrivial critical point for the functional J follows from the classical mountain pass theorem: more precisely, taken u ∈ Hr1 ∩ C ∞ as in Remark 2.1, since kΦ[u]k∞ ≤ ω, there results Z Z €  t2 tp 2 2 2 2 J(tu) ≤ |∇u| + Ω|u| + ω u dx − |u|p → −∞ as t → +∞. 2 R3 p R3 8

3

Nonlinear Schr¨ odinger Equations coupled with Maxwell Equations

For sake of simplicity assume ~ = m = e = 1 in (1.14)-(1.15). Then we are reduced to study the following system in R3 : 1 − ∆u + Φu + ωu − |u|p−2 u = 0, 2

(3.28)

−∆Φ = 4πu2 .

(3.29)

We will assume a’) ω > 0 b’) 3 < p < 6. Of course, (3.28)-(3.29) are the Euler-Lagrange equations of the functional F : H 1 × D1,2 → R defined as Z Z Z Z € 2  1 1 1 1 2 2 2 |∇Φ| dx + F(u, Φ) = |∇u| dx − Φu + ωu dx − |u|p dx, 4 R3 16π R3 2 R3 p R3 where H 1 and D1,2 are defined as in the previous section. It is easy to prove the analogous of Proposition 2.1, i.e. that F ∈ C 1 (H 1 × D1,2 , R) and that its critical points are solutions of (3.28)-(3.29). Moreover we have the following proposition.

Proposition 3.1. For every u ∈ H 1 there exists a unique Φ = Φ[u] ∈ D1,2 solution of (3.29), such that • Φ[u] ≥ 0; • Φ[tu] = t2 Φ[u] for every u ∈ H 1 and t ∈ R.

R Proof. Let us consider the linear map φ ∈ D1,2 7→ R3 u2 φ dx, which is continuous by (2.20). By Lax-Milgram’s Lemma we get the existence of a unique Φ ∈ D1,2 such that Z Z u2 φ dx ∀φ ∈ D1,2 , ∇Φ∇φ dx = 4π R3

R3

i.e. Φ is the unique solution of (3.29). Furthermore Φ achieves the minimum š Z › Z Z Z 1 1 2 2 2 u φ dx = u2 Φ dx. inf |∇φ| − 4π |∇Φ| dx − 4π 2 R3 φ∈D1,2 2 R3 R3 R3 Note that also |Φ| achieves such minimum, then, by uniqueness, Φ = |Φ| ≥ 0. Finally, −∆Φ[tu] = 4πt2 u2 = −t2 ∆Φ[u] = −∆(t2 Φ[u]), thus, by uniqueness, Φ[tu] = t2 Φ[u].

9

Proceeding as in the previous section we can define the map Φ : H 1 → D1,2 , which maps each u ∈ H 1 in the unique solution of (3.29). As before, Φ ∈ C 1 (H 1 , D1,2 ) and FΦ0 (u, Φ[u]) = 0 ∀u ∈ H 1 . Now consider the functional J : H 1 → R defined by J (u) = F(u, Φ[u]). J belongs to C 1 (H 1 , R) and satisfies J 0 (u) = Fu (u, Φ[u]). Using the definition of F and equation (3.29), we obtain Z Z Z Z ω 1 1 1 2 2 2 |∇u| dx + |u| dx + |u| Φ[u] dx − |u|p dx. J (u) = 4 R3 2 R3 4 R3 p R3 As before one can prove the following lemma. Lemma 3.1. The following statements are equivalent: i) (u, Φ) ∈ H 1 × D1,2 is a critical point of F, ii) u is a critical point of J and Φ = Φ[u]. Now we are ready to prove the existence result for equations (3.28)-(3.29). Theorem 3.1. Assume hypotheses a’) and b’). Then the functional J has a nontrivial critical point u ∈ H 1 having a radial symmetry. Proof. Let Hr1 be defined as in theorem 2.1. Step 1. Any critical point u ∈ Hr1 of JŒŒ

Hr1

is also a critical point of J .

The proof is as in theorem 2.1.

Step 2. The functional JŒŒ 1 satisfies the Palais-Smale condition. Hr Let {un }n ⊂ Hr1 be such that

0 J|H 1 (un ) → 0 r

|J (un )| ≤ M, for some constant M > 0. Then 0

pJ (un ) − J (un )un = +

’

p 1 − 4 2

“Z

R3

2

’

p 1 − 4 2

“Z

|un | Φ[un ] dx ≥

2

R3

’

|∇un | +

p 1 − 4 2

“Z

R3

p 2

‘ Z −1 ω

|un |2 dx

€  |∇un |2 + ω|un |2 dx

by Proposition 3.1. Moreover ’ “Z €  p 1 − |∇u|2 + ω|u|2 dx ≥ c1 kun k2 4 2 R3 10

R3

and by assumption pJ (un ) − J 0 (un )un ≤ pM + c2 kun kH 1 for some positive constants c1 and c2 . in Hr1 . We have thus proved that {un }n is bounded R 2 2 On the other hand, kΦ[un ]kD1,2 = 4π R3 u Φ[un ] dx, and then, using inequality (2.20), we easily deduce that {Φ[un ]}n is bounded in D1,2 . The remaining part of the proof follows as in Step 2 of Theorem 2.1, after replacing L with L : Hr1 → (Hr1 )0 defined as L(u) = − 12 ∆u + ωu. Step 3. The functional JŒŒ 1 satisfies the three geometrical hypothesis of the mountain pass Hr theorem. By Proposition 3.1 it results Z Z Z ω 1 1 2 2 J (u) ≥ |∇u| dx + |u| dx − |u|p dx. 4 R3 2 R3 p R3

Then, using the continuous embedding H 1 ⊂ Lp , we deduce that J has a strict local minimum in 0. We introduce the following notation: if u : R3 → R, we set uλ,α,β (x) = λβ u(λα x), λ > 0, α, β ∈ R. Now fix u ∈ Hr1 . We want to show that Φ[uλ,α,β ] = (Φ[u])λ,α,2(β−α) .

(3.30)

In fact −∆Φ[uλ,α,β ](x) = 4πu2λ,α,β (x) = 4πλ2β u2 (λα x) = −λ2β (∆Φ[u])(λα x) = −∆((Φ[u])λ,α,2(β−α) )(x). By uniqueness (see Proposition 3.1), (3.30) follows. Now take u 6≡ 0 in Hr1 and evaluate Z Z λ2β−α ω 2β−3α 2 J (uλ,α,β ) = u2 dx |∇u| dx + λ 4 2 3 3 R R Z Z λ4β−5α λβp−3α u2 Φ[u] dx − |u|p dx. + 4 p R3 R3

We want to prove that J (uλ,α,β ) < J (0) for some suitable choice of λ, α and β. For example assume  βp − 3α < 0,    βp − 3α < 2β − α, βp − 3α < 2β − 3α,    βp − 3α < 4β − 5α,

(3.31)

then it is clear that J (uλ,α,β ) → −∞ as λ → 0. So we look for a couple (α, β) which satisfies (3.31). From the third inequality we get β < 0. Combining the second and the fourth ones, we derive 4−p<

2α < p − 2. β 11

(3.32)

Such an inequality is satisfied by taking β = 2α, which also satisfies the first inequality in (3.31). In a similar way one can prove that if  βp − 3α > 0,    βp − 3α > 2β − α, (3.33) βp − 3α > 2β − 3α,    βp − 3α > 4β − 5α, then J (uλ,α,β ) → −∞ as λ → +∞ with the same choice β = 2α.

Remark 3.1. The condition p > 3 appears exactly in solving systems (3.31) or (3.33). More precisely if p ∈ (2, 3] there is no couple (α, β) which satisfies the inequality (3.32); then, since the systems (3.31) or (3.33) have no solutions, we cannot conclude J (uλ,α,β ) → −∞. Proof of Theorem B Lemma 3.1 + Theorem 3.1.

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