On existence and multiplicity for Schrödinger–Poisson systems involving weighted sublinear nonlinearities

On existence and multiplicity for Schrödinger–Poisson systems involving weighted sublinear nonlinearities

Sara Barile

Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, 70125 Bari, Italy

Received 19 November 2016, appeared 3 April 2017 Communicated by Dimitri Mugnai

Abstract. We deal with existence and multiplicity for the following class of nonhomo- geneous Schrödinger–Poisson systems

(−_∆u+V(x)u+K(x)φ(x)u= f(x,u) +g(x) inR³,

−_∆φ=K(x)u² inR³,

where V,K: R³→_R⁺ are suitable potentials and f :R³×_R→_Rsatisfies sublinear growth assumptions involving a finite number of positive weightsW_i,i=1, . . . ,rwith r ≥ 1. By exploiting compact embeddings of the functional space on which we work in every weighted space L_W^wi

i(R³), wi ∈ (1, 2), we establish existence by means of a generalized Weierstrass theorem. Moreover, we prove multiplicity of solutions if f is odd in u and g(x) ≡ 0 thanks to a variant of the symmetric mountain pass theorem stated by R. Kajikiya for subquadratic functionals.

Keywords: Schrödinger–Poisson systems, sublinear nonlinearities, variational meth- ods, compact embeddings.

2010 Mathematics Subject Classification: 35J20, 35J47, 35J50, 35Q40, 35Q55.

1 Introduction

In this paper we consider the following class of Schrödinger–Poisson systems (also called Schrödinger–Maxwell systems) in both nonhomogeneous case g(x)6≡0, namely

(−_∆u+V(x)u+K(x)φ(x)u= f(x,u) +g(x) inR³,

−_∆φ=K(x)u² inR³, (P_g)

and in the homogeneous caseg(x)≡0, that is

(−_∆u+V(x)u+K(x)φ(x)u= f(x,u) inR³,

−_∆φ= K(x)u² inR³. (P₀)

BEmail: sara.barile@uniba.it

This class of systems has a strong physical meaning since it arises in several applications from mathematical physics, in particular in quantum mechanics models where it describes the mu- tual interactions of charged particles in the electrostatic case (see e.g. [5,6] and references therein for more detailed physical aspects). For this reason, many authors have devoted their attention to systems of this type and they have widely studied them by using variational meth- ods under various conditions on the potentials V(x) and K(x) and the nonlinearity f(x,u) especially when it is superlinear or asymptotically linear at infinity inu. On the contrary, up to now, there is no extensive literature dealing with the case of nonlinearities f(x,u)sublinear at infinity especially involving suitable weights and this motivates our work. Let us start with the homogeneous caseg(x)≡0.

In 2012, Sun [12] proved the existence of infinitely many small negative energy solutions to (P₀) in the caseK(x)≡1 by a variant fountain theorem established in [16] under the following assumptions

(V⁰) V∈C(_R³,R)satisfies inf_x∈R3V(x)≥a >0 withaa real constant;

(V⁰⁰) for any M > 0, meas{x ∈ _R³ : V(x) ≤ M}< +_∞ where meas denotes the Lebesgue measure onR³;

(F⁰) F(x,u) = W₁(x)|u|^w1 where F(x,u) = Ru

0 f(x,t)dt,W₁ :R³ →_Ris a positive continu- ous function such thatW₁∈ L^2−2w1(_R³)with w₁ ∈(1, 2).

In particular, conditions (V⁰)–(V⁰⁰) imply a coercive condition on V which was first intro- duced by Bartsch and Wang [4] in order to overcome the loss of compactness due to the unboundedness of the domainR³. Clearly, thanks to (F⁰) only the one-weight nonlinearity

f(x,u) =w₁W₁(x)|u|^w1−1 is allowed.

In 2013, Liu, Guo and Zhang [8] generalized the results in Sun [12] since they showed for (P₀) withK(x)≡ 1 the existence of a nontrivial solution by minimization arguments [10]

and the multiplicity of solutions with negative energy which goes to zero by a symmetric mountain pass lemma based on genus properties in critical point theory (see Salvatore [11]) by removing assumption(V⁰⁰)and relaxing assumption(F⁰)with the following

|f(x,u)| ≤w₁W₁(x)|u|^w1−1+w₂W₂(x)|u|^w2−1for a.e. x∈_R³ and for allu∈_R

with W₁ ∈ L^2−2w1(_R³) and W₁ > 0, W2 ∈ L³(_R³) and W2 ≥ 0 where w₁ ∈ (1, 2) and w2 ∈ [4/3, 2). This assumption makes indefinite nonlinearities f(x,u) can be also considered and the presence of the weights W₁ and W₂ ensures a good property of compactness for these

f(x,u).

In 2013, Lv [9] also generalized the result in Sun [12] by showing existence of a nontrivial solution by minimization arguments [10] and multiplicity of solutions with vanishing and negative energy levels by the dual fountain theorem [14] to (P₀) in the caseK(x)≡ 1 without the coercive assumption (V⁰⁰) and under only (V⁰) on V. This has been possible since the odd nonlinearity f(x,u) is supposed to satisfy suitable sublinear growth hypotheses which imply the existence of three weightsW_i ∈ L^2−2wi(_R³),W_i >0,i∈ {1, 2, 3}with w_i ∈(1, 2)and allow to recover compact embeddings of the functional space in the weighted space L^w_Wⁱ

i(_R³), i=1, 2, 3. Precisely,

|f(x,u)| ≤

∑

W_i(x)|u|^wi−1 _{for a.e.} x∈_R³ and for allu∈_R

so that the case of indefinite nonlinearities f(x,u) is also covered. This result improves and completes the paper by Liu, Guo and Zhang [8].

Few years later, in 2015 Ye and Tang [15] improved the results in Sun [12] since they showed the existence of infinitely many small solutions with small negative energy to (P₀) by means of a new version of symmetric mountain pass lemma developed by Kajikiya [7] with the non-negative potentialK ∈L²(_R³)∪L^∞(_R³)(see condition(K)below), under the weaker hypotheses

(V⁰) V ∈C(_R³,R)verifiesV(x)≥0 for everyx ∈_R³;

(V⁰⁰⁰) there exists M>0 such that meas{x∈_R³ :V(x)≤ M}<+_∞.

Besides a suitable local assumption, on the continuous and odd nonlinearity f(x,u) is as- sumed in particular the following sublinear growth condition

|f(x,u)| ≤W₁(x)|u|^w1−1+W₂(x)|u|^w2−1 for a.e. x∈_R³ and for allu ∈_R with weightsW₁ ∈ L^2−2w1(_R³),W₂∈ L^2−2w2(_R³),W₁,W₂ >0 andw₁,w₂∈ (1, 2).

Regarding the nonhomogeneous case g(x) 6≡ 0, Wang, Ma and Wang [13] in 2016 estab- lished only existence to (P_g) with non-negativeg ∈ L²(_R³), for a class of potentialsK(x)≥0 with K∈ L²(_R³)∪L^∞(_R³)(as in hypothesis(K)below), under conditions(V⁰)–(V⁰⁰)and the same sublinear growth condition assumed in [15] with two weightsW₁ andW₂. In this case the authors work without using for compactness this last condition.

The aim of this paper is to study (P_g) (resp. (P₀)) under more generic conditions in order to generalize or to give complementary results to the ones listed above. More precisely, we investigate existence (resp. multiplicity) of solutions to (P_g) (resp. (P₀)) under the following assumptions:

(V) V :R³→ _Ris a Lebesgue measurable function with ess inf_R3V(x)≥ a> 0 wherea is a real constant;

(K) K∈L²(_R³)∪ L^∞(_R³)andK(x)≥0 for a.e.x ∈_R³;

(f₁) f : R³×_R → _R is a Carathéodory function (i.e., f(·,s) is measurable on R³ for all s∈_Rand f(x,·)is continuous onRfor a.e.x∈_R³);

(f₂) there existsW_i ∈ L^2−2wi(_R³),W_i >0 (i∈ {1, . . . ,r}) with constant w_i ∈(1, 2)such that

|f(x,s)| ≤

∑

W_i(x)|s|^wi−1 _{for a.e.}x∈_R³and for alls∈_R;

(f₃) there existΩ⊂_R³ with meas(_Ω)>0,w_r+1∈ (1, 2),η>0 andδ >0 such that F(x,s)≥η|s|^wr+1 for a.e.x∈_Ωand for alls∈_R,|s| ≤δ where F(x,s) =Rs

0 f(x,t)dt;

(f₄) f(x,s) =−f(x,−s)_{for a.e.}x∈ _R³and for alls∈_R;

(G) g∈ L²(_R³)_.

Thus, we obtain the following results. For the definition of the functional spaces E_V and D^1,2(_R³)and of the energy functional J0which appear in next theorems, see Section2.

First, let us state the existence result for the nonhomogeneous case and for the homoge- neous case.

Theorem 1.1(Existence). Suppose that(V),(K),(f₁)and(f2)hold. Then, we get the following:

(i) (nonhomogeneous case g(x)6≡ 0) if in addition(G)holds, problem(P_g) admits at least a non- trivial weak solution(u,φ_u)∈E_V×D^1,2(_R³);

(ii) (homogeneous case g(x)≡0) if(f3)is also assumed, problem(P₀)possesses both a trivial weak solution and at least a non-trivial weak solution(u,φ_u)∈E_V×D^1,2(_R³).

Now, let us provide the multiplicity result obtained in the caseg(x)≡0.

Theorem 1.2(Multiplicity). Assume that(V),(K),(f₁),(f₂), (f₃),(f₄)hold. Then, problem(P₀) has a sequence{(u_k,φ_uk)} ⊂E_V×D^1,2(_R³)of non-trivial weak solutions such that

J₀(u_k,φ_uk) = ¹ 2

Z

R³ |∇u_k|²+V(x)|u_k|²dx−¹ 4

Z

R³|∇φ_uk|²dx + ¹

2 Z

R³K(x)φu_ku²_kdx−

Z

R³ F(x,u_k)dx→0 as k→+_∞.

Remark 1.3. Thanks to Remark 2.8 and the properties of J₀ and φ_u stated in Section 2, we remark that Theorem1.2 gives in particular the existence of a sequence{(u_k,φ_uk)}of critical points ofJ₀such that J₀(u_k,φ_uk)≤0, u_k 6=0 and thenφ_uk 6=0, lim_ku_k =0 from which we get lim_kφ_uk =0; consequently, lim_kJ₀(u_k,φ_uk) =0⁻.

Let us observe that, as concerns the existence result in the homogeneous caseg(x)≡0, we complete the papers by Sun [12] and by Ye and Tang [15] where no existence result has been stated. Moreover, we improve the existence of solutions to (P₀) for not necessarily constant potentialsK(x)by relaxing (V⁰)with (V)in Lv [9] and in Liu, Guo and Zhang [8].

Moreover, we generalize the existence of multiple solutions obtained in Sun [12], Liu, Guo and Zhang [8] and Lv [9] to (P₀) for K(x)≡1 to a more general class of potentials satisfying (K)thus providing the existence of infinitely many small solutions with small negative energy.

In the nonhomogeneous case g(x) 6≡ 0, we improve the existence result established by Wang, Ma and Wang [13] since we relax condition(V⁰)by(V), skip(V⁰⁰)and recover compact- ness by the different requirement(f₂) involving r weights. Furthermore, we do not impose any sign condition ong.

Remark 1.4. Let us observe that, from(f₂)by integration it follows that

|F(x,s)| ≤

∑

1

w_iW_i(x)|s|^wi for a.e.x ∈_R³and for alls∈ _R. (1.1) The paper is organized as follows: in Section2 we introduce the variational formulation of the problem and we recall a generalized version of Weierstrass theorem, Mazur theorem and a convexity criterion. Moreover, we recall a variant of the symmetric mountain pass theorem for “subquadratic” problems stated in [7]. In Section3we prove Theorem1.1and in Section4 we show Theorem1.2.

2 Variational tools

In order to introduce the variational structure of the problem, let E = H¹(_R³)be the usual Sobolev space endowed with the standard scalar product

(u,v)_E =

Z

R³(∇u· ∇v+uv)dx and the corresponding norm

kuk_E = (u,u)^1/2_E = Z

R³ |∇u|²+|u|²dx ¹₂

with dual space (E⁰,k · k_E0). Moreover, letD^1,2(_R³)be the completion ofC₀^∞(_R³)with respect to the norm

kuk_D = kuk_D1,2(_R³)= Z

R³|∇u|²dx ¹₂

.

We denote by L^s(_R³), 1<s< +_∞, the Lebesgue space endowed with the norm

|u|_s =|u|_Ls(_R³)= _Z

R³|u|^sdx ¹_s

. Moreover, let us introduce

E_V =

u∈E: Z

R³ |∇u|²+V(x)|u|²dx< _∞

. By assumption(V), EVis a Hilbert space endowed with the scalar product

(u,v)_V= (u,v)_EV =

Z

R³(∇u· ∇v+V(x)uv)dx and the related norm

kuk_V= (u,v)^1/2_V = Z

R³ |∇u|²+V(x)|u|²dx ¹₂

with dual space(E_V⁰ ,k · k_E⁰

V). From now on, let 1<s <_∞and L^s_V(_R³) =

u∈ L^s(_R³): Z

R³V(x)|u|^sdx<_∞

endowed with the norm

|u|_s,V = _Z

R³V(x)|u|^sdx ¹_s

.

Clearly, E_V = E∩L²_V(_R³)and by (V)we have that E_V ,→ E. Moreover, the following conti- nuous embeddings hold

E_V,→ L^s(_R³) for any s∈[2, 6] and D^1,2(_R³),→ L⁶(_R³) being 2^∗ =2N/(N−2) =6 for N=3.

From now on,candCwill denote real positive constants changing line from line.

At this point, we prove the following result which allows us to state the compact embed- ding of E_V in a weighted Lebesgue space with a specific weightW(x); the result will be ap- plied to the Lebesgue measurable weightW_i and to the constantw=w_i for any i∈ {1, . . . ,r} in assumption(f₂)_.

Proposition 2.1. Let1 < w < 2. Suppose that W is a positive function belonging to L^µ(_R³)with µ= _w²⁰ = ₂₋²_w. Under assumption(V), we get the following compact embedding

E_V,→,→L_W^w(_R³), where

L^w_W(_R³) =

u∈ L^w(_R³): Z

R³W(x)|u|^wdx <_∞

endowed with the norm

|u|_w,W = _Z

R³W(x)|u|^wdx _w¹

.

Proof. We adapt the arguments used in [9, Lemma 2.1] (see also [2, Remark 2.3] and [3, Propo- sition 2.2]). Let{un}be a sequence in EV such thatun *uin EV. Clearly,un−uis bounded inE_V, namely there exists a constant M>0 such thatku_n−uk_V ≤ M. SinceW ∈ L^µ(_R³)we have

for allε>0 there existsR_ε > 0 such that _Z

|x|≥Rε

|W(x)|^µdx ¹

µ

< ε.

Therefore, by Hölder’s inequality and Sobolev embeddings we get Z

|x|≥Rε

W(x)|u_n−u|^wdx≤ _Z

|x|≥Rε

|W(x)|^µdx ¹

µ _Z

|x|≥Rε

|u_n−u|²dx ^w₂

≤ε|u_n−u|^w₂ ≤ εc^wku_n−uk^w_V ≤εc^wM^w. (2.1) Now, settingE_V(B_Rε(0)) ={u|_BR

ε(0) :u∈ E_V}, since

E_V(B_Rε(0)),→ H¹(B_Rε(0)),→,→L_W^w(B_Rε(0)), fromu_n *uinE_Vwe deduceu_n|

BRε(0) *u_|

BRε(0) inE_V(B_Rε(0))and thenu_n→uinL^w_W(B_Rε(0)). Consequently,

for every ε>0 there existsnε ∈_Nsuch that for everyn>nε one has Z

|x|≤Rε

W(x)|u_n−u|^wdx <ε which, together with (2.1), implies

Z

R³W(x)|un−u|^wdx=

Z

|x|≤Rε

W(x)|un−u|^wdx+

Z

|x|≥Rε

W(x)|un−u|^wdx

< ε(1+M^wc^w) and thenun→uin L^w_W(_R³).

Let us point out that in Sun [12] and Wang, Ma and Wang [13] potential V satisfies stronger assumptions (V⁰)–(V⁰⁰). These conditions allow to prove that E_V ,→,→ L^s(_R³) for all 2≤s<6. Differently, here above in Proposition 2.1 we show that E_V ,→,→ L_W^w(_R³) _with w∈(1, 2). In the following (see Proposition2.3and Proposition4.1) we will exploit only this weaker result in order to overcome the lack of compactness due to the unboundedness of the domainR³.

Under our assumptions, it is not difficult to see that system (P_g) (resp. (P₀)) has a varia- tional structure, that is, it is possible to find its solutions by looking for critical points of the functional J_g∈C¹(E_V×D^1,2(_R³),R)(resp.J₀∈C¹(E_V×D^1,2(_R³),R)) defined by

Jg(u,φ) = ¹

2kuk²_V−¹ 4

Z

R³|∇φ|²dx+ ¹ 2

Z

R³K(x)φu²dx−

Z

R³F(x,u)dx−

Z

R³g(x)u dx (resp. J0(u,φ) = ¹

2kuk²_V−¹ 4

Z

R³|∇φ|²dx+ ¹ 2

Z

R³K(x)φu²dx−

Z

R³F(x,u)dx)

for every (u,φ)∈ E_V×D^1,2(_R³). But the functional J_g (resp.J₀) is strongly indefinite, namely it is unbounded from below and above on infinite dimensional subspaces. In order to remove its indefiniteness and to reduce to study a not strongly indefinite functional, we can use the following reduction method introduced in [5] (see also [6]). This method relies on the fact that, for every u ∈ E_V, the Lax–Milgram theorem implies the existence of a unique φ_u ∈ D^1,2(_R³) satisfying in the weak sense

−_∆φu=K(x)u² inR³.

It is well known thatφu can be written with the following integral formula φ_u(x) = ¹

4π Z

R³

K(y)u²(y)

|x−y| ^dy.

So, substituting φ = φ_u in Jg (resp. J0) it is possible to consider the functional Ig : EV → _R (resp. I₀ : E_V → R) defined by Ig(u) = Jg(u,φu)(resp. I₀(u) = J₀(u,φu)) for every u ∈ E_V. Now, by multiplying−_∆φu=K(x)u² byφ_uand integrating by parts we get

Z

R³K(x)φ_uu²dx=

Z

R³−(_∆φu)φ_udx=

Z

R³|∇φ_u|²dx, (2.2) then the reduced functional I_g(resp. I₀) takes the form for everyu∈ E_V

I_g(u) = ¹ 2

Z

R³ |∇u|²+V(x)|u|²dx+ ¹ 4

Z

R³K(x)φ_uu²dx−

Z

R³ F(x,u)dx−

Z

R³ g(x)u dx (resp.I₀(u) = ¹

2 Z

R³ |∇u|²+V(x)|u|²dx+¹ 4

Z

R³K(x)φ_uu²dx−

Z

R³F(x,u)dx). At the same time, problem (P_g) (resp. problem (P₀)) can be reduced to an equivalent single Schrödinger equation with a nonlocal term. Indeed, substituting φ = φ_u in (P_{g) (resp. (}P₀₎₎ we get the following equation

−_∆u+V(x)u+K(x)φ_u(x)u= f(x,u) +g(x) inR³ (S_g) (resp. −_∆u+V(x)u+K(x)φu(x)u= f(x,u) inR³.) (S₀) As we will prove in Proposition 2.3, Ig ∈ C¹(EV,R)(resp. I0 ∈ C¹(EV,R)) and every critical point of I_g (resp. I₀) corresponds to a solution u ∈ E_V to (S_g) (resp. (S₀)) and provides a solution(u,φ)∈ E_V×D^1,2(_R³)to (P_g) (resp. (P₀)).

Remark 2.2. Since by (K), it isK(x)≥0 for a.e.x∈_R³, we getφ_u≥0 for any u∈EV.

Now, as just noticed, by hypothesis(V)it isE_V ,→ H¹(_R³); this fact together with the well known continuity ofφ_u :H¹(_R³)→ D^1,2(_R³)implies alsoφ_u:E_V →D^1,2(_R³)is continuous.

Furthermore, let us observe that, ifK∈ L²(_R³)orK∈ L^∞(_R³), by (2.2), Hölder’s inequal- ity and Sobolev embeddings we obtain

kφ_uk²_D =

Z

R³K(x)φ_uu²dx

≤ _Z

R³(K(x))²dx

1/2 _Z

R³(φ_u)⁶dx

1/6 _Z

R³(u²)³dx 1/3

≤ |K|₂ckφ_uk_D|u|²₆ or

kφ_uk²_D =

Z

R³K(x)φ_uu²dx≤ |K|_{∞ Z}

R³(φ_u)⁶dx

1/6 _Z

R³(u²)^6/5dx 5/6

≤ |K|_∞ckφuk_D|u|²_12/5. Therefore, in the first case we get

kφ_uk_D ≤ |K|₂c|u|²₆ (2.3) while in the second

kφuk_D ≤ |K|_∞c|u|²_12/5. (2.4) At this point we can state the following variational principle and recover the compactness of the problem.

Proposition 2.3. Assume that (V), (K), (f₁), (f₂)and(G)hold. Then, the weak solutions of (P_g) (resp.(P₀)) are the critical points of the energy functional Ig :E_V →_R(resp. I₀: E_V→R) defined by

Ig(u) = ¹ 2

Z

R³ |∇u|²+V(x)|u|²dx+¹ 4

Z

R³K(x)φuu²dx−

Z

R³F(x,u)dx−

Z

R³g(x)u dx (resp. I₀(u) = ¹

2 Z

R³ |∇u|²+V(x)|u|²dx+¹ 4

Z

R³K(x)φ_uu²dx−

Z

R³F(x,u)dx)

for every u ∈ E_V. More precisely, Ig ∈ C¹(E_V,R)(resp. I₀ ∈ C¹(E_V,R)) and its derivative dIg : E_V →E⁰_V(resp. dI₀ :E_V →E⁰_V) is defined as

dI_g(u)[ζ] =

Z

R³[∇u· ∇ζ+V(x)uζ+K(x)φ_uuζ− f(x,u)ζ−g(x)ζ]dx (resp. dI₀(u)[ζ] =

Z

R³[∇u· ∇ζ+V(x)uζ+K(x)φuuζ− f(x,u)ζ]dx) (2.5) for all u,ζ ∈ E_V. Consequently, the pair(u,φ)∈ E_V×D^1,2(_R³)is a solution of problem(P_g)(resp.

(P₀)) if and only if u ∈E_V is a critical point of I_g(resp. I₀) andφ=φ_u. Moreover, the function u7−→ f(·_,u(·))is compact from E_Vto E⁰_V.

Proof. Let us start by showing that the functional Ig (resp. I0) is well defined and its Fréchet derivative given in (2.5) is a continuous operator fromE_V toE⁰_V. For the sake of completeness, we give here all the details of the proof. We define and study separately the following maps

ϕ_V(u) = ¹

2 kuk²_V, ϕ_K(u) =

Z

R³K(x)φ_uu²dx ϕ_F(u) =

Z

R³ F(x,u)dx and ϕ_g(u) =

Z

R³g(x)u dx.

Clearly, ϕ_V ∈C¹(E_V,R)since ϕ_V is continuous fromE_VtoRand its Gâteaux differential atu dϕ_V(u)[ζ] =

Z

R³∇u· ∇ζdx+

Z

R³V(x)uζdx is a linear continuous map on E_V.

Concerning the map ϕ_K, we need to show that ϕ_K ∈C¹(E_V,R)_with dϕ_K(u)[ζ] =

Z

R³K(x)φ_uuζdx for all u,ζ ∈E_V. (2.6) By Remark2.2, ifK ∈L²(_R³)it results

|ϕ_K(u)| ≤

Z

R³K(x)φ_uu²dx≤ |K|²₂c²|u|⁴₆ and, ifK∈ L^∞(_R³)one has

|ϕ_K(u)| ≤

Z

R³K(x)φ_uu²dx≤ |K|²_∞c²|u|⁴_12/5, then by Sobolev embeddings ϕ_K(u)∈_Rfor anyu∈ E_V.

Now we prove that Z

R³K(x)φuuζdx ∈_R for allu,ζ ∈E_V.

Indeed, ifK∈ L²(_R³), by Hölder’s inequality and (2.3) we get the following

Z

R³K(x)φ_uuζdx

≤

Z

R³K(x)φ_u|u| |ζ|dx ≤ |K|₂|φ_u|_{6 Z}

R³(|u| |ζ|)³dx _1/3

≤ |K|₂ckφuk_D Z

R³((|u|)³)²dx

1/6 Z

R³((|ζ|)³)²dx 1/6

≤ |K|²₂c²|u|₆|ζ|₆.

Similarly, ifK∈ L^∞(_R³)by Hölder’s inequality and (2.4) we obtain

Z

R³K(x)φuuζdx

≤

Z

R³K(x)φu|u| |ζ|dx≤ |K|_∞|φu|₆ Z

R³(|u| |ζ|)^6/5dx 5/6

≤ |K|_∞ckφ_uk_{D Z}

R³((|u|^6/5))²dx

5/12 _Z

R³((|ζ|^6/5))²dx 5/12

≤ |K|²_∞c²|u|_12/5|ζ|_12/5.

By Sobolev embeddings in both cases we have done. It is not difficult to find that the Gâteaux derivative of ϕ_K atu is as in (2.6) and it is linear and continuous fromE_V toR. It remains to prove thatdϕK is continuous fromEV toE_V⁰ , i.e.

kdϕ_K(u_n)−dϕ_K(u)k_E⁰

V →0 ifu_n →uinE_V. (2.7) First, observe that by adding and subtracting K(x)φ_unuζ in the integral we have

|(dϕ_K(u_n)−dϕ_K(u))[ζ]| ≤

Z

R³|K(x)φ_unu_nζ−K(x)φ_uuζ|dx

≤

Z

R³K(x)|un−u|φ_un|ζ|dx+

Z

R³K(x)|φ_un−φ_u||u| |ζ|dx. (2.8)

Now, ifK∈ L²(_R³), by Hölder’s inequality and Sobolev embeddings it follows Z

R³K(x)|u_n−u|φ_un|ζ|dx≤ |K|₂Ckφ_unk_Dku_n−uk_V|ζ|_V

Z

R³K(x)|φun−φu||u| |ζ|dx≤ |K|₂Ckφun−φuk_Dkuk_V|ζ|_V. Similarly, ifK∈ L^∞(_R³)we get

Z

R³K(x)|u_n−u|φ_un|ζ|dx≤ |K|_∞Ckφ_unk_Dku_n−uk_V|ζ|_V

Z

R³K(x)|φ_un−φ_u||u| |ζ|dx≤ |K|_∞Ckφ_un−φ_uk_Dkuk_V|ζ|_V.

Asun → u in EV, by the continuity of φu fromEV in D^1,2(_R³) ensured in Remark2.2 we get φ_un → φ_u as n → +_∞ and consequently the boundedness of φ_un in D^1,2(_R³); therefore, the right terms in these four inequalities above go to zero and by (2.8) the convergence in (2.7) follows.

Now, we have to prove that also ϕ_F∈C¹(E_V,R)with dϕ_F(u)[ζ] =

Z

R³ f(x,u)ζdx for allu,ζ ∈ E_V. (2.9) Let us point out that, by (1.1) in Remark1.4and Hölder’s inequality, we have

|ϕ_F(u)| ≤

Z

R³|F(x,u)|dx≤

∑

1 w_i

Z

R³W_i(x)|u|^widx≤

∑

1

w_i|W_i|_µi|u|^w₂ⁱ whereµ_i = _w²

i

0

= ₂₋²_w

i and similarly by(f₂)we obtain Z

R³|f(x,u)||ζ|dx≤

∑

Z

R³W_i(x)|u|^wi−1|ζ|dx≤

∑

|W_i|_µi|u|^w₂ⁱ⁻¹|ζ|₂.

Hence, by Sobolev embeddings it follows thatϕ_F(u)∈ _Randdϕ_F(u)[ζ]∈_Rfor allu,ζ ∈ E_V. Moreover, standard tools imply that the Gâteaux derivative of ϕ_F at u is as in (2.9) and it is linear and continuous fromE_V toR.

At this point, we have to prove thatdϕ_F is continuous fromE_V toE_V⁰ , i.e.

kdϕ_F(u_n)−dϕ_F(u)k_E⁰

V →0 ifu_n→uinE_V. (2.10) Indeed, by Hölder’s inequality and Sobolev embeddings,

|(dϕ_F(u_n)−dϕ_F(u))[ζ]| ≤

Z

R³|f(x,u_n)− f(x,u)||ζ|dx

≤ |f(·,u_n(·))− f(·,u(·))|₂|ζ|₂.

Now, by(f₂)we get for a.e.x ∈_R³

|f(x,u_n)− f(x,u)|² ≤₂ |f(x,u_n)|²+|f(x,u)|²

≤2





∑

W_i(x)|u_n|^(wi−1)

!₂ +

∑

W_i(x)|u|^(wi−1)

!₂



≤2 2

∑

(W_i(x))²|u_n|^2(wi−1)+2

∑

(W_i(x))²|u|^2(wi−1)

!

≤2²

∑

(W_i(x))²|u_n|^2(wi−1)+

∑

(W_i(x))²|u|^2(wi−1)

!

≤2²

∑

2^{2(wi−1)−1}(W_i(x))²|u_n−u|^2(wi−1) +

∑

(₂^{2(wi−1)−1}+₁)(W_i(x))²|u|^2(wi−1)

! . By Fatou’s lemma, it follows that

Z

R³lim inf

n→+_∞ c

∑

(W_i(x))²|un−u|^2(wi−1)+

∑

(W_i(x))²|u|^2(wi−1)

!

− |f(x,un)− f(x,u)|²

! dx

≤ lim inf

n→+_∞ Z

R³ c

∑

(W_i(x))²|u_n−u|^2(wi−1)+

∑

(W_i(x))²|u|^2(wi−1)

!

− |f(x,u_n)− f(x,u)|²

!

dx. (2.11)

Now, we observe that, sinceu_n→uinE_V it isu_n(x)→u(x)a.e.x∈_R³, therefore (W_i(x))²|u_n(x)−u(x)|^2(wi−1)→0 a.e. x∈_R³and for alli=1, . . . ,r and also by(f₁)

|f(x,un(x))− f(x,u(x))|² →0 a.e.x ∈_R³. On the other hand, by Hölder’s inequality and Sobolev embeddings we get

Z

R³(W_i(x))²|un−u|^2(wi−1)dx ≤ |W_i|²_µi|un−u|²₂^(wi−1) for alli=1, . . . ,r

and, since u_n → uin L²(_R³)by continuous embeddings, also the left-hand side term goes to zero asn→+_∞for everyi=1, . . . ,r. Consequently, (2.11) implies

c Z

R³

∑

(W_i(x))²|u|^2(wi−1)dx≤c Z

R³

∑

(W_i(x))²|u|^2(wi−1)dx +lim inf

n→+_∞

−

Z

R³|f(x,u_n)− f(x,u)|²dx

from which it follows that

0≤ −_{lim sup}

n→+_∞

_Z

R³|f(x,u_n)− f(x,u)|²dx

and therefore

0≤lim inf

n→+_∞

_Z

R³|f(x,u_n)− f(x,u)|²dx

≤lim sup

n→+_∞

_Z

R³|f(x,u_n)− f(x,u)|²dx

≤0.

Hence,

|f(·,u_n(·))− f(·,u(·))|₂→0 asn→+_∞ and (2.10) is proved.

By exploiting the arguments carried out in [5,6], we get that the pair(u,φ)∈E_V×D^1,2(_R³) is a solution of problem (P_g) (resp. (P₀)) if and only ifu∈EV is a critical point of Ig(resp. I0) andφ=φ_u.

Finally, we prove that dϕ_F is compact from E_V to E_V⁰ . Let {u_n}be a sequence in E_V such thatun *uinEV. By Proposition2.1, for alli=1, . . . ,rit isun→uin L_W^wi

i(_R³)namely Z

R³W_i(x)|un−u|^widx→0 as n→+_∞. (2.12) Fixedi=1, . . . ,rand taken α_i = _w²

i ∈ (0, 2), by Hölder’s inequality we get Z

R³(W_i(x))²|u_n−u|^2(wi−1)dx

=

Z

R³(W_i(x))^αi(W_i(x))^2−αi|un−u|^2(wi−1)dx

≤ Z

R³(W_i(x))^{µi α}_µⁱ

i Z

R³

(W_i(x))^2−αi|u_n−u|^2(wi−1)(

µi αi)⁰

dx ¹

(^µ_αⁱ i)0

=|W_i|^α_µⁱ

i|u_n−u|

1 (^µi αi)0 wi,Wi, hence, by (2.12) it follows that

Z

R³(W_i(x))²|un−u|^2(wi−1)dx→0 asn→+_∞.

Then, arguing as in the proof of the continuity of dϕ_F, as soon as u_n * u in E_V we get f(·,un(·)) → f(·,u(·)) in L²(_R³)so dϕ_F(un) → dϕ_F(u)in E⁰_V as n → +_∞and we conclude thatdϕ_F(u)is compact fromE_V to E_V⁰ .

Finally it is standard to prove thatϕ_g∈ C¹(E_V,R)with derivative dϕ_g(u))[ζ] =

Z

R³g(x)ζdx for everyu,ζ ∈ E_V and the proof is completed.

Now, in order to prove in next Section3 the existence result by minimization arguments, we will exploit the following generalized version of the Weierstrass theorem.

Theorem 2.4. Let(X,k · k)be a reflexive Banach space and M ⊆ X be a weakly closed subset of X.

Suppose that the functional I : M → _R is coercive and (sequentially) weak lower semi-continuous on M.

Then, I is bounded from below on M and

there exists u₀ ∈ M such that I(u₀) =_min

u∈MI(u).