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Volume 9, Issue 1

# On a logarithmic Hartree equation

Federico Bernini
/ Dimitri Mugnai
Published Online: 2019-08-29 | DOI: https://doi.org/10.1515/anona-2020-0028

## Abstract

We study the existence of radially symmetric solutions for a nonlinear planar Schrödinger-Poisson system in presence of a superlinear reaction term which doesn’t satisfy the Ambrosetti-Rabinowitz condition. The system is re-written as a nonlinear Hartree equation with a logarithmic convolution term, and the existence of a positive and a negative solution is established via critical point theory.

MSC 2010: 35J50; 35Q40

## 1 Introduction

In recent past years many papers have been devoted to finding solutions of Schrödinger-Poisson systems of the form

$iψt−Δψ+E(x)ψ+ywψ=f(x,t), in RN×R,Δw=ψ2, in RN,$(SP)

and often the main objects were standing wave solutions, i.e. solutions of the form

$ψ(x,t)=e−iωtu(x),ω∈R,$

$−Δu+b(x)u+ywu=f~(x,t), in RN×R,Δw=u2, in RN,$(1.1)

where b(x) = E(x) + ω and (x, t) = eiωtf(x, t).

Due to the numerous several Physical applications, the most studied case is N = 3 (or N ≥ 3). On the other hand, the 1–dimensional case was considered in [1] when f = 0, and the existence of a unique ground state was established by decreasing symmetric rearrangements tools. The 2–dimensional case when f = 0 was first approached in [2], Section 6, only from a numerical point of view, while the first rigorous existence result was given in [3] by using a shooting method for ODEs.

Moving down to lower dimensions, in particular to N = 2, introduces several complications, the first important one is that in this case the Coulomb potential is not positive.

In [4] the authors studied the eigenvalue problem for the Schrödinger operator in a bounded domain of ℝ3, with electromagnetic field E-H that is not assigned; in this case the unknowns are the wave function ψ(x, t) and the gauge potentials A(x, t), ϕ(x, t) related to E-H. In particular, they considered the problem in which A and ϕ do not depend on the time and ψ(x, t) = eiωtu(x), with ω ∈ ℝ and u real function. With these consideration they assume A = 0, and thus the system reduces to

$−Δu−ϕu=ωu, in Ω⊂R3,Δϕ=4πu2, in Ω⊂R3.$

Under this hypotheses, they proved the existence of a sequence of solution in a bounded domain of ℝ3.

Later on, in [5] the authors considered the problem

$−Δu−ϕu−ωu=|u|p−2u, in R3,Δϕ=4πu2, in R3$(1.2)

with p ∈ ℝ+ and they proved the existence of radially symmetric solutions in ℝ3, for p ∈ [4, 6), while in [6] they showed the nonexistence of solutions for p ∈ (0, 2] or p ∈ [6, ∞). After that, system (1.2) has been object of an intensive study, where generalizations of several type where considered; we refer to [7, 8, 9, 10, 11, 12, 13, 14] for other references and improvements on this subject.

All these works have been done in the whole of ℝ3, while the two dimensional case has remained for a long time a quite open field of study. Indeed, a theoretical approach in dimension 2 is harder than in higher dimensions due to the lack of positivity of the Coulomb interaction term: precisely, the Coulomb potential is neither bounded from above nor from below.

However, in 2008 Stubbe [15] bypassed this problem giving a suitable variational framework for the problem

$−Δu+au−12πln⁡1|x|∗|u|2u=0$(1.3)

and proving the existence of ground states, which is a positive spherically symmetric strictly decreasing function, by solving an appropriate minimization problem for the energy functional associated to the system (see also [16]).

In some recent works, a local nonlinear terms of the form b|u|p−2u, p > 2 has been added; this kind of nonlinearity are frequently used in Schrödinger equations to model the interaction among particles, like recalled above (see [4]). Thus in [17], they studied a Schrödinger-Poisson system of the type

$−Δu+a(x)u−12πln⁡1|x|∗|u|2u=b|u|p−2u in R2,$

with b ≥ 0, p ≥ 2 and aL(ℝ2) and they proved that if p ≥ 4 then the problem has a sequence of solution pairs ± un such that I(un) → ∞ as n → ∞.

In this work we are concerned with the integro-differential equation

$−Δu+au−12πln⁡1|x|∗|u|2u=f(x,u) in R2,$(P)

where a > 0 and f is a superlinear function. We refer to (P) as the logarithmic Choquard equation. Note that, if compared with (SP), we have chosen y = −1; since y represents the charge of the particle that we are studying, it means that we are considering electrons.

In order to generalize and include the previous cases, on the reaction term f : ℝ2 × ℝ → ℝ we assume that it is a Carathéodory function having superlinear growth and not verifying the Ambrosetti-Rabinowitz condition, from now on (AR).

Our main result has the following flavour (see Theorem 3.1 for the precise statement):

#### Theorem 1.1

Under suitable hypotheses on f problem (P) has two nontrivial constant sign solutions.

This work is organized as follows. In Section 2 we recall some useful definitions and results that we shall use, we set up an appropriate variational framework and define the energy functional associated to the problem. Moreover, we give an extended results of the estimates given in the Strauss theorem.

In Section 3 we prove the well-posedness and the regularity of our functional and we give a Lemma that plays a fundamental role in the proof of the Cerami condition. Finally, we give the proof of the main existence theorem.

## 2 Background and Variational Framework

We provide a suitable variational framework for studying (P): indeed, the associated functional is not well defined on the natural Sobolev space H1(ℝ2), and so we need some adjustments taken from [15], see also [17].

We first recall an important result for Lp-spaces.

#### Theorem 2.1

(Hardy-Littlewood-Sobolev’s inequality, [18]). Let p, q > 1 and 0 < λ < N with $\begin{array}{}\frac{1}{p}+\frac{\lambda }{N}+\frac{1}{q}=2.\end{array}$ Let fLp(ℝN) and gLq(ℝN). Then there exists a sharp constant C(N, λ, p), independent of f and g, such that

$∫RN∫RN|f(x)g(y)||x−y|λdxdy≤C(N,λ,p)∥f∥Lp(RN)∥g∥Lq(RN).$(2.1)

The sharp constant satisfies

$C(N,λ,p)≤N(N−λ)|SN−1|NλN1pqλ/N1−1pλN+λ/N1−1qλN.$

If $\begin{array}{}p=q=\frac{2N}{2N-\lambda },\end{array}$ then

$C(N,λ,p)=C(N,λ)=πλ2Γ(N/2−λ/2)Γ(N−λ/2)Γ(N/2)Γ(N)−1+λ/N.$

In this case there is equality in (2.1) if and only if gcf with c constant and

$f(x)=Ay2+|x−x0|2−(2N−λ)/2$

for some A ∈ ℝ, 0 ≠ y ∈ ℝ and x0 ∈ ℝN. Here |SN−1| denotes the area of the unit sphere inN.

First of all, we endow H1(ℝ2) with the scalar product (recall that a > 0 is a constant)

$(u|v)=∫R2Du⋅Dv+auvdx, for u,v∈H1(R2),$

and we introduce the space

$X=u∈H1(R2):∫R2|u(x)|2ln⁡(1+|x|)dx<∞$

with the norm defined by

$∥u∥X2=∫R2|Du|2+|u|2a+ln⁡(1+|x|)dx.$

Then, we define the symmetric bilinear forms

$B1(u,v)=∫R2∫R2ln⁡(1+|x−y|)u(x)v(y)dxdy,$

$B2(u,v)=∫R2∫R2ln1+1|x−y|u(x)v(y)dxdy,$

and

$B(u,v)=B1(u,v)−B2(u,v)=∫R2∫R2ln|x−y|u(x)v(y)dxdy,$

since for all r > 0 we have

$ln⁡r=ln⁡(1+r)−ln1+1r.$(2.2)

The definitions above are restricted to measurable functions u, v : ℝ2 → ℝ such that the corresponding double integral is well defined in the Lebesgue sense. Finally, for any measurable functions u : ℝ2 → ℝ we consider the seminorm in X

$|u|∗2=∫R2ln⁡(1+|x|)u2(x)dx.$

We note that, since

$ln⁡(1+|x−y|)≤ln⁡(1+|x|+|y|)≤ln⁡(1+|x|)+ln⁡(1+|y|),$(2.3)

we have by the Schwarz inequality

$|B1(uv,wz)|≤∫R2∫R2ln⁡(1+|x|)+ln⁡(1+|y|)|u(x)v(x)||w(y)z(y)|dxdy≤|u|∗|v|∗∥w∥L2(R2)∥z∥L2(R2)+∥u∥L2(R2)∥v∥L2(R2)|w|∗|z|∗$(2.4)

for u, v, w, zL2(ℝ2). Next, since 0 ≤ ln(1 + r) ≤ r for all r > 0, we have by Theorem 2.1

$|B2(u,v)|≤∫R2∫R21|x−y|u(x)v(y)dxdy≤C∥u∥L43(R2)∥v∥L43(R2),$(2.5)

for u, v$\begin{array}{}{L}^{\frac{4}{3}}\end{array}$(ℝ2), for some constant C > 0. In particular, from (2.4) we have

$B1(u2,u2)≤2|u|∗2∥u∥L2(R2)2$(2.6)

for all uL2(ℝ2) and from (2.5) we have

$B2(u2,u2)≤C∥u∥L83(R2)4$(2.7)

for all u$\begin{array}{}{L}^{\frac{8}{3}}\end{array}$(ℝ2).

The energy functional I : X → ℝ associated to (P) is

$I(u)=12∫R2|Du|2+a|u|2dx−18π∫R2∫R2ln⁡1|x−y||u(x)|2|u(y)|2dxdy−∫R2F(x,u)dx,$

where $\begin{array}{}F\left(x,u\right)=\underset{0}{\overset{u}{\int }}f\left(x,s\right)ds\end{array}$ and the Gâteaux derivative of I along vX is

$I′(u)(v)=∫R2Du⋅Dv+auvdx−12π∫R2∫R2ln⁡1|x−y||u(x)|2u(y)v(y)dxdy−∫R2f(x,u)vdx.$

#### Definition 1

We say that uX is a weak solution of (P) if

$I′(u)(v)=0 for all v∈X,$

thus if u is a critical point for I.

Of course, these consideration are only formal, since, without any assumption on f, we cannot differentiate I. In Section 3 we will give some sufficient conditions for I to be of class C1 in X, while for the moment we continue with formal computations.

We have the following results, the second statement being new, as far as we know, and extending Strauss’ Radial Lemma [19] to the space X and its N − dimensional version. Indeed, though later on we shall use only the compact embedding in dimension N = 2 presented below, we can prove an asymptotic result which is valid in any space dimension. Since we believe that this property is of independent interest, we present our result in the general case. For this, let us introduce the sets

$X=u∈H1(RN):∫RN|u(x)|2ln⁡(1+|x|)dx<∞$

and

$Xr=u∈X:u(x)=u(|x|).$

#### Proposition 2.1

The following properties hold true:

• X is compactly embedded in Ls(ℝ2), for all s ∈ [2, ∞).

• There exists c ∈ ℝ such that for all u ∈ 𝓧r

$|u(x)|≤c∥u∥XrxN−12ln⁡(1+x)4.$

#### Proof

The compact embedding for N = 2 is an application of the Riesz criterion (see [20, Theorem XIII.66]). Indeed, if S is a bounded subset of X, then S is bounded in Lq(ℝ2) for any q ∈ [2, ∞), as well. Moreover, for any R > 0 and any uS we have

$∫{|x|>R}|u|pdx≤∥u∥L2p−2(R2)∫{|x|>R}u2dx1/2≤C∫{|x|>R}u2dx1/2$

for some C > 0, and

$∫{|x|>R}u2dx≤1ln⁡(1+R)∫{|x|>R}ln⁡(1+|x|)u2dx≤Cln⁡(1+R)$

for some C > 0. Finally, working as in [21, Theorem 9.16] we conclude.

As for the estimate in dimension N, let u ∈ 𝓧r$\begin{array}{}{C}_{c}^{\mathrm{\infty }}\end{array}$ (ℝN) and r > 0. We have

$ln⁡(1+r)u2rN−1′=12ln⁡(1+r)u2rN−11+r+2uu′rN−1ln⁡(1+r)+(N−1)rN−2u2ln⁡(1+r)≥2uu′rN−1ln⁡(1+r).$

Integrating from r to ∞ we obtain

$ln⁡(1+r)u2rN−1≤−∫BrC2uu′ρN−1ln⁡(1+ρ)dρ=−∫BrC2uu′ln⁡(1+|x|)dx≤C∫BrC|u|2ln⁡(1+|x|)12∫BrC|Du|212≤C∥u∥Xr.$

Hence

$|u(r)|≤C∥u∥XrxN−12ln⁡(1+x)4.$

The conclusion follows by density. □

We finally introduce the "positive" and "negative" part of the reaction term, namely

$f±(x,t)=f(x,t±) for a.e. x∈R2 and for all t∈R,$

which are Carathéodory functions if f is, and $\begin{array}{}{F}_{±}\left(x,t\right)=\underset{0}{\overset{t}{\int }}{f}_{±}\left(x,s\right)ds;\end{array}$ moreover, we set

$I±(u)=12∫R2|Du|2+a|u|2dx−18π∫R2∫R2ln⁡1|x−y||u±(x)|2|u±(y)|2dxdy−∫R2F±(x,u)dx$

for all uX.

## 3 The Existence Theorem

We assume the following hypotheses on the reaction term f:

• H(i)

let f : ℝ2 × ℝ → ℝ be a Carathéodory function with f(x, 0) = 0 and f(x, ⋅) = f(|x|, ⋅) for a.e. x ∈ ℝ2. Moreover, there exist cLp(ℝ2) for 1 < p < 2, d > 0 and q ∈ (2, ∞) such that |f(x, t)| ≤ c(x) + d|t|q−1, for a.e. x ∈ ℝ2 and for all t ∈ ℝ;

• H(ii)

f(x, t) = o(|t|) as t → 0 uniformly for a.e. x ∈ ℝ2;

• H(iii)

$\begin{array}{}\underset{|t|\to +\mathrm{\infty }}{lim}\frac{F\left(x,t\right)}{{t}^{2}}=+\mathrm{\infty }\end{array}$ uniformly for a.e. x ∈ ℝ2;

• H(iv)

if σ(x, s) = f(x, s)s − 2F(x, s), then there exists 𝓜*$\begin{array}{}{L}_{+}^{1}\end{array}$(ℝ2) such that σ(x, s) ≤ σ(x, t) + 𝓜*(x) for a.e. x ∈ ℝ2 and for all 0 ≤ st or ts ≤ 0;

• H(v)

there exists ũX such that

$limy→+∞∫R2F(x,y2u~±(yx))dxy4ln⁡y=+∞.$

#### Remark 3.1

1. Condition H(iv) was introduced in [22] to overcome the necessity of using the Ambrosetti-Rabinowitz condition.

2. Condition H(v) is trivially satisfied if f(x, t) = |t|p−2t or if F(x, t) ≥ c|t|ζ(x), where ζ$\begin{array}{}{L}_{+}^{1}\end{array}$(ℝ2) and > 4. The very last condition is generally a consequence of the usual Ambrosetti-Rabinowitz condition, which here should be assumed a priori, see [23].

We start proving

#### Proposition 3.1

If H(i) holds, then the functional I : X → ℝ is well-defined and of class C1 on X. The same is true for I±.

#### Proof

We do the proof for I, the ones for I± being completely analogous. From hypothesis H(i) we have

$∫R2F(x,u)dx≤∫R2c(x)|u(x)|dx+dq∫R2|u(x)|qdx≤∥c∥Lp(R2)∥u∥Lp′(R2)+dq∥u∥Lq(R2)q.$(3.1)

From (3.1), (2.6) and (2.7) we have

$|I(u)|≤12∥u∥H1(R2)2+14π∫R2ln⁡(1+|x|)u2(x)∫R2u2(y)dxdy+C2∥u∥L83(R2)4+∥c∥Lp(R2)∥u∥Lp′(R2)+dq∥u∥Lq(R2)q<∞,$

for some constant C2 > 0, so the associated functional is well-defined.

Now we observe that the Gâteaux derivative of B(u2, u2) is

$B(u2,uφ)=2∫R2∫R2ln⁡1|x−y||u(x)|2u(y)φ(y)dxdy$(3.2)

for all φX, so the functional I is the sum of C1 terms and we have the desired regularity follows. □

Our purpose is to prove that both I+ and I satisfy the assumptions of the mountain pass theorem. While the geometric structure is somehow standard and is obtained exploiting H(i) and H(v), the compactness condition is the delicate part: the lack of the Ambrosetti-Rabinowitz condition makes the bound on Cerami sequences more complicated, and, indeed, by using H(iv) we obtain only a bound in H1(ℝ2). Thus we move to radial functions and use Strauss’ Lemma to exploit the compact embedding in Lq(ℝ2): thanks to the principle of symmetric criticality, a critical point for the functional constrained on the subset of radial functions is a free critical point, see [24]. This permits to recover the desired bound of Cerami sequences in X and finally prove that the Cerami condition holds.

Hence, from now on, we consider I : Xr → ℝ, where

$Xr=u∈X:u(x)=u(|x|)$

and we look for critical point for I|Xr. For the sake of simplicity we will continue to denote by I the functional I|Xr.

Now we are ready to prove that the (C)d-condition holds. In order to do that, we first give the following definition

#### Definition 2

We say that a sequence (un)nXr is a (C)d-sequence if

$I(un)→d and ∥I′(un)∥Xr∗1+∥un∥Xr→0 as n→∞.$

We say that I satisfies the (C)d-condition if any (C)d-sequence admits a convergent subsequence.

We prove that, under suitable hypotheses, a (C)d-sequence in Xr is bounded in $\begin{array}{}{H}_{r}^{1}\end{array}$(ℝ2).

#### Lemma 3.1

Suppose hypotheses H(i), H(iii) and H(iv) hold and let (un)nXr be a (C)d-sequence for I+ (I respectively). Then (un)n is bounded in $\begin{array}{}{H}_{r}^{1}\end{array}$(ℝ2).

#### Proof

We do the proof for the I+, for I being analogous.

Let (un)nXr be a (C)d-sequence. In particular,

$|I+(un)|≤M1 for some M1>0 and every n≥1,$(3.3)

$1+∥un∥XrI+′(un)→0 in Xr∗ as n→∞.$(3.4)

We recall that for any vXr we have

$I+′(un)(v)=∫R2Dun⋅Dv+aunvdx−12π∫R2∫R2ln⁡1|x−y||un+(x)|2un+(y)v(y)dxdy−∫R2f+(x,un)vdx.$

From (3.4) we have

$|I+′(un)(h)|≤εn∥h∥Xr1+∥h∥Xr$(3.5)

for all hXr(ℝ2), where εn → 0 as n → ∞. We choose h = −$\begin{array}{}{u}_{n}^{-}\end{array}$Xr and we obtain

$|I+′(un)(−un−)|≤εn∥un−∥Xr1+∥un−∥Xr≤εn,$

so that

$∫R2|Dun−|)+a|un−|2dx=∥un−∥Hr1(R2)2≤εn,$

which means that

$un−→0 in Hr1(R2) as n→∞.$(3.6)

From (3.3) we have

$12∥un∥Hr1(R2)2−14B(un+)2,(un+)2−∫R2F+(x,un)dx≤M1$(3.7)

so that

$∥un∥Hr1(R2)2−12B(un+)2,(un+)2−2∫R2F+(x,un)dx≤M2 for some M2>0$

and using (3.6) we obtain

$∥un+∥Hr1(R2)2−12B(un+)2,(un+)2−2∫R2F(x,un+)dx≤M2.$(3.8)

We assume by contradiction that (un)n is unbounded in $\begin{array}{}{H}_{r}^{1}\end{array}$(ℝ2), then by passing to a subsequence, if necessary, we assume that $\begin{array}{}\parallel {u}_{n}^{+}{\parallel }_{{H}_{r}^{1}\left({\mathbb{R}}^{2}\right)}\to \mathrm{\infty }\end{array}$ as n → ∞.

We set $\begin{array}{}{v}_{n}=\frac{{u}_{n}^{+}}{\parallel {u}_{n}^{+}{\parallel }_{{H}_{r}^{1}\left({\mathbb{R}}^{2}\right)}},\end{array}$ n ≥ 1, so we may assume that, by Strauss’ Theorem,

$vn⇀v in Hr1(R2) and vn→v in Ls(R2),s∈(2,∞),v≥0.$(3.9)

To reach our goal we show that both v ≠ 0 and v = 0 lead to a contradiction. We start with the case v ≠ 0.

We define the set Z(v) = {x ∈ ℝ2 : v(x) = 0}; then meas(ℝ2Z(v)) > 0 and $\begin{array}{}{u}_{n}^{+}\end{array}$(x) → ∞ as n → ∞ for a.e. x ∈ ℝ2Z(v). By H(iii) we have

$F(x,un+)∥un+∥Hr1(R2)2=F(x,un+)|un+|2vn2→∞ for a.e. x∈R2∖Z(v)$

and by Fatou’s Lemma we obtain

$∫R2F(x,un+)∥un+∥Hr1(R2)2dx→∞ as n→∞.$(3.10)

But from (3.7) we have

$−12+14B(un+)2,(un+)2∥un+∥Hr1(R2)2+∫R2F(x,un+)∥un+∥Hr1(R2)2dx≤M1∥un+∥Hr1(R2)2$

so that

$lim supn→∞∫R2F(x,un+)∥un+∥Hr1(R2)2dx≤M3 for some M3>0.$(3.11)

Compairing (3.10) and (3.11) we reach a contradiction.

Now we consider the case v = 0. For every n ∈ ℕ we define the continuous function yn : [0, 1] → ℝ as

$yn(t)=I(tun+) for all n≥1 and all t∈[0,1],$

and let tn ∈ [0, 1] be such that

$yn(tn)=maxyn(t):t∈[0,1].$(3.12)

For λ > 0, let $\begin{array}{}{w}_{n}=\left(2\lambda {\right)}^{\frac{1}{2}}{v}_{n}\in {H}_{r}^{1}\left({\mathbb{R}}^{2}\right).\end{array}$ Then wn → 0 in Lp(ℝ2) by (3.9). By H(i) and the Krasnoselskii’s Theorem (see [25, Theorem 2.75]), we have

$∫R2F(x,wn)dx→0 as n→∞.$(3.13)

Since $\begin{array}{}\parallel {u}_{n}^{+}{\parallel }_{{H}_{r}^{1}\left({\mathbb{R}}^{2}\right)}\to \mathrm{\infty }\end{array}$ as n → ∞, we can find n0 ≥ 1 such that $\begin{array}{}\frac{\left(2\lambda {\right)}^{\frac{1}{2}}}{\parallel {u}_{n}^{+}{\parallel }_{{H}_{r}^{1}\left({\mathbb{R}}^{2}\right)}}\end{array}$ ∈ (0, 1) for all nn0. Then, by (3.12),

$y(tn)≥y(2λ)12∥un+∥Hr1(R2) for all n≥n0.$

Hence, by (2.2) and (2.6) we get

$I(tnun+)≥I((2λ)12vn)=λ−λ22π∫R2∫R2ln⁡1|x−y||vn(x)|2|vn(y)|2dxdy−∫R2F(x,wn)dx=λ+λ22π∫R2∫R2ln1+|x−y||vn(x)|2|vn(y)|2dxdy−λ22π∫R2∫R2ln1+1|x−y||vn(x)|2|vn(y)|2dxdy−∫R2F(x,wn)dx≥λ−λ22π∫R2∫R2ln1+1|x−y||vn(x)|2|vn(y)|2dxdy−∫R2F(x,wn)dx≥λ−C∥vn∥L83(R2)4−∫R2F(x,wn)dx.$

Now we observe that $\begin{array}{}\parallel {v}_{n}{\parallel }_{{L}^{\frac{8}{3}}\left({\mathbb{R}}^{2}\right)}\end{array}$ → 0 as n → ∞ by Strauss’ Theorem, and by (3.13) we have

$I(tnun+)≥λ+o(1)≥λ2.$

Being λ > 0 arbitrary, we finally find

$I(tnun+)→∞ as n→∞.$(3.14)

Since 0 ≤ $\begin{array}{}{t}_{n}{u}_{n}^{+}\le {u}_{n}^{+}\end{array}$ for all n ≥ 1, from H(iv) we have

$∫R2σ(x,tnun+)dx≤∫R2σ(x,un+)dx+∥M∗∥L1(R2) for all n≥1.$(3.15)

Moreover, by (3.3) and (3.6) there exists M4 > 0 such that

$I(un)=I+(un)+o(1)≤M4 for all n≥1.$(3.16)

Thus, (3.14) and (3.16) imply that tn ∈ (0, 1) for all nn1 ≥ 1. Hence, by (3.12) we obtain that

$0=tnddtI(tun+)∣t=tn=I′(tnun+)(tnun+)=∫R2|D(tnun+)|2+a|tnun+|2dx−12π∫R2∫R2ln⁡1|x−y||tnun+(x)|2|tnun+(y)|2dxdy−∫R2f(x,tnun+)(tnun+)dx=∥tnun+∥Hr1(R2)2−B(tnun+)2,(tnun+)2−∫R2f(x,tnun+)(tnun+)dx$(3.17)

for all n ≥ 1, that is

$∫R2f(x,tnun+)(tnun+)dx=∥tnun+∥Hr1(R2)2−B(tnun+)2,(tnun+)2$(3.18)

for all nn1. Replacing (3.18) in (3.15), we obtain

$∥tnun+∥Hr1(R2)2−B(tnun+)2,(tnun+)2−2∫R2F(x,tnun+)dx≤∫R2σ(x,un+)dx+∥M∗∥L1(R2)$

for all nn1.

Again by H(iv)

$f(x,tnun+)(tnun+)−2F(x,tnun+)≤f(x,un+)(un+)−2F(x,un+)+M∗,$

so that

$−2∫R2F(x,tnun+)dx≤∫R2f(x,un+)(un+)−f(x,tnun+)(tnun+)dx−2∫R2F(x,un+)dx+∫R2M∗dx.$

Using (3.17) the previous inequality reads as

$−2∫R2F(x,tnun+)dx≤−I′(un+)(un+)+∫R2|Dun+|2+a|un+|2dx−B(un+)2,(un+)2−∫R2|D(tnun+)|2+a|tnun+|2dx+B(tnun+)2,(tnun+)2−2∫R2F(x,un+)dx+∫R2M∗dx$

and from (3.4)

$−2∫R2F(x,tnun+)dx≤∥un+∥Hr1(R2)2−B(un+)2,(un+)2−∥tnun+∥Hr1(R2)2+B(tnun+)2,(tnun+)2−2∫R2F(x,un+)dx+∫R2M∗dx+o(1).$(3.19)

Now

$2I(tnun+)=∥tnun+∥Hr1(R2)2−12B(tnun+)2,(tnun+)2−2∫R2F(x,tnun+)dx.$(3.20)

Thus, replacing (3.19) in (3.20), since $\begin{array}{}B\left(\left({t}_{n}{u}_{n}^{+}{\right)}^{2},\left({t}_{n}{u}_{n}^{+}{\right)}^{2}\right)\le B\left(\left({u}_{n}^{+}{\right)}^{2},\left({u}_{n}^{+}{\right)}^{2}\right)\end{array}$ being tn < 1, we have

$2I(tnun+)≤∥tnun+∥Hr1(R2)2−12B(tnun+)2,(tnun+)2+∥un+∥Hr1(R2)2−B(un+)2,(un+)2−∥tnun+∥Hr1(R2)2+B(tnun+)2,(tnun+)2−2∫R2F(x,un+)dx+∫R2M∗dx+o(1)≤∥un+∥Hr1(R2)2−12B(un+)2,(un+)2−2∫R2F(x,un+)dx+∫R2M∗dx+o(1).$

This last formula, together with (3.14), tells us that

$∥un+∥Hr1(R2)2−12B(un+)2,(un+)2−2∫R2F(x,un+)dx→∞ as n→∞.$(3.21)

Comparing (3.8) and (3.21) we reach a contradiction.

So $\begin{array}{}\left({u}_{n}^{+}{\right)}_{n}\end{array}$ is bounded in $\begin{array}{}{H}_{r}^{1}\end{array}$(ℝ2).□

We use this result to finally prove the Cerami compactness condition.

#### Proposition 3.2

Let (un)nXr be a (C)d-sequence for I+ (I respectively), with d > 0. Then, up to a subsequence,

$un→u in Xr as n→∞$

for some nonzero critical point uXr of I+ (I respectively). In particular, the (C)d-condition holds.

#### Proof

From Lemma 3.1, we know that, up to a subsequence,

$un⇀u in Hr1(R2) as n→∞.$

Now, we show that

$lim infn→∞supx∈R2∫Br(x)un2(y)dy>0$(3.22)

for every r > 0. We argue by contradiction, so we suppose that (3.22) is false. Since (un)n is bounded in $\begin{array}{}{H}_{r}^{1}\end{array}$(ℝ2), by [26, Lemma I.1], we have that un → 0 as n → ∞ in Ls(ℝ2) for every s ∈ (2, ∞).

By our assumptions

$I+′(un)un=∥un∥Hr1(R2)2+12π∫R2∫R2ln⁡(1+|x−y|)|un+(x)|2|un+(y)|2dxdy−12π∫R2∫R2ln1+1|x−y||un+(x)|2|un+(y)|2dxdy−∫R2f+(x,un)undx,$

and so

$∥un∥Hr1(R2)2+12π∫R2∫R2ln⁡(1+|x−y|)|un+(x)|2|un+(y)|2dxdy=I+′(un)un+12π∫R2∫R2ln1+1|x−y||un+(x)|2|un+(y)|2dxdy+∫R2f+(x,un)undx.$

By H(i) and (2.6) we have

$∥un∥Hr1(R2)2+12π∫R2∫R2ln⁡(1+|x−y|)|un+(x)|2|un+(y)|2dxdy≤I+′(un)un+12π∫R2∫R2ln1+1|x−y||un+(x)|2|un+(y)|2dxdy+∥c∥Lp(R2)∥un∥Lp′(R2)+d∥un+∥Lq(R2)q≤I+′(un)un+C∥un+∥L83(R2)4+∥c∥Lp(R2)∥un∥Lp′(R2)+d∥un+∥Lq(R2)q.$(3.23)

Since un → 0 in Ls(ℝ2) for s ∈ (2, ∞), and p′ > 2, we have

$∥un∥Hr1(R2)→0,$

and then

$∥un±∥Hr1(R2)→0,$

and also

$∫R2∫R2ln⁡(1+|x−y|)|un+(x)|2|un+(y)|2dxdy→0.$

Hence,

$I+(un)=12∥un∥Hr1(R2)2+18π∫R2∫R2ln⁡(1+|x−y|)|un+(x)|2|un+(y)|2dxdy−18π∫R2∫R2ln1+1|x−y||un+(x)|2|un+(y)|2dxdy−∫R2F+(x,un)dx≤12∥un∥Hr1(R2)2+18π∫R2∫R2ln⁡(1+|x−y|)|un+(x)|2|un+(y)|2dxdy+∥c∥Lp(R2)∥un∥Lp′(R2)+d∥un+∥Lq(R2)q→0,$

but I+(un) → d > 0, so we reach a contradiction. Thus (3.22) holds.

This means that vanishing (see [27]) cannot occur. Moreover, since we use radial functions, dichotomy cannot take place, either. Hence, we can conclude that u ≠ 0. By [17, Lemma 2.1] we can conclude that

$(un)n is bounded in Xr.$

Then we can assume that

$un⇀u in Xr,$

with u ≠ 0, and by Proposition 2.1 we also have that unu in Ls(ℝ2) for every s ∈ [2, ∞).

Finally, we claim that unu in Xr. In (3.5), we take h = unu and, using (2.2), we have

$I+′(un)(un−u)=∫R2|Dun|2+a|un|2dx−∫R2Dun⋅Du+aunudx−12π∫R2∫R2ln1+1|x−y||un+(x)|2un+(y)(un−u)(y)dxdy+12π∫R2∫R2ln1+|x−y||un+(x)|2un+(y)(un−u)(y)dxdy−∫R2f+(x,un)(un−u)dx.$

Hence,

$∥un∥Hr1(R2)2−(un|u)=I+′(un)(un−u)+12π∫R2∫R2ln1+1|x−y||un+(x)|2un+(y)(un−u)(y)dxdy−12π∫R2∫R2ln1+|x−y||un+(x)|2un+(y)(un−u)(y)dxdy+∫R2f+(x,un)(un−u)dx.$(3.24)

By Theorem 2.1 and the Hölder inequality, H(i) and (2.3) we have

$∫R2∫R2ln1+1|x−y||un+(x)|2|un+(y)(un−u)(y)|dxdy≤C∥un+∥Ls(R2)3∥un−u∥Ls(R2)→0$

with s ∈ (2, ∞),

$∫R2∫R2ln1+|x−y||un+(x)|2|un+(y)(un−u)(y)|dxdy=∫R2ln1+|x−y||un+(x)|2∫R2|un(y)(un−u)(y)|dxdy→0$

and

$∫R2f+(x,un)(un−u)dx≤∥c∥Lp(R2)∥un+∥Lp(R2)∥un−u∥Lp′(R2)+d∥un+∥Lq(R2)q−1∥un∥Lq′(R2)→0$

Hence, from (3.24)

$un→u in Xr$

and so the (C)d-condition hold. □

Now we are ready to produce two nontrivial solutions of (P) using the Mountain Pass Theorem.

#### Theorem 3.1

Under hypotheses H(i) - H(v), problem (P) has two nontrivial constant sign solutions.

#### Proof

We do the proof for the functional I+; for I it is analogous. First, I+(0) = 0. By Proposition 3.1 we have the regularity of I+ and by Proposition 3.2 the (C)d-condition is verified.

Now, take ũ as in H(v), t > 0 and, following [10], we set ut(x) = t2ũ(tx). Then

$I+(ut)=t42∫R2|Du~|2+au~2dx+t48π∫R2∫R2ln⁡(|x−y|)|u~+(x)|2|u~+(y)|2dxdy−t4ln⁡t8π∫R2|u~+(x)|2dx2−∫R2F+(x,t2u~(tx))dx,$

and by H(v),

$limt→+∞I+(ut)=−∞.$

In order to complete the proof it only remains to show that I+(u) ≥ α ≥ 0 with ∥u∥ = r, for some r > 0.

By H(i) we have |f(x, t)| ≤ c(x) + d|t|q–1 and then

$|F(x,t)|≤c(x)|t|+dq|t|q a.e. x∈R2 and for all t∈R.$(3.25)

Hypothesis H(ii) says that for all ε > 0 there exists δ = δ(ε) > 0 such that for |t| < δ we have

$|f(x,t)||t|≤ε a.e. x∈R2,$

thus

$|F(x,t)|≤ε2t2 a.e. x∈R2 and |t|≤δ.$(3.26)

On the other hand, when |t| ≥ δ

$|F(x,t)|≤c(x)|t|δq−1δq−1+dq|t|q≤c(x)δq−1+dq|t|q.$

Combining the inequality above with (3.26) we get that for a.e. x ∈ ℝ2 and for all t ∈ ℝ

$|F(x,t)|≤c(x)δq−1+dq|t|q+ε2t2.$

Hence,

$∫R2F(x,u)dx≤∫R2c(x)δ|u|q+dq∫R2|u|q+ε2∫R2|u|2≤1δ∥c∥Lp(R2)∥u∥Lqp′(R2)q+dq∥u∥Lq(R2)q+ε2∥u∥L2(R2)2$

that is

$∫R2F(x,u)dx≤εC1∥t∥Hr1(R2)2+CδC2∥t∥Hr1(R2)q.$

We use this estimates on functional I+:

$I+(u)=12∥u∥Hr1(R2)2+18π∫R2∫R2ln1+|x−y||u+(x)|2|u+(y)|2dxdy−18π∫R2∫R2ln1+1|x−y||u+(x)|2|u+(y)|2dxdy−∫R2F+(x,u)dx≥12∥u∥Hr1(R2)2+14B1(u+)2,(u+)2−14B2(u+)2,(u+)2−ε∥u+∥L2(R2)2−Bδ∥u+∥Lq(R2)q$

and by the Hardy-Littlewood-Sobolev inequality

$I+(u)≥12∥u∥Hr1(R2)2+14B1(u+)2,(u+)2−C3∥u+∥Hr1(R2)4−εC1∥u+∥Hr1(R2)2−BδC2∥u+∥Hr1(R2)q=12∥u∥Hr1(R2)2+14B1(u+)2,(u+)2−C3∥u+∥Hr1(R2)4−εC1∥u+∥Hr1(R2)2−BδC2∥u+∥Hr1(R2)q.$

Since $\begin{array}{}\parallel {u}^{+}{\parallel }_{{H}_{r}^{1}\left({\mathbb{R}}^{2}\right)}\le \parallel u{\parallel }_{{H}_{r}^{1}\left({\mathbb{R}}^{2}\right)}\end{array}$ we get

$I+(u)≥12−εC1∥u∥Hr1(R2)2−C3+BδC2∥u∥Hr1(R2)q−4∥u∥Hr1(R2)4.$

Choosing $\begin{array}{}\epsilon \in \left(0,\frac{1}{2{C}_{1}}\right)\end{array}$ and $\begin{array}{}\parallel u{\parallel }_{{H}_{r}^{1}\left({\mathbb{R}}^{2}\right)}=r,\end{array}$ we have

$I+(u)≥C4∥u∥Hr1(R2)2−C3+BδC2∥u∥q−4∥u∥Hr1(R2)4=∥u∥Hr1(R2)2C4−C3+BδC2rq−4r2,$

for some C4 > 0. We take r such that C4 – (C3 + BδC2rq–4) > 0 and so

$I+(u)≥α≥0,$

thus we have the Mountain Pass geometry, and we can apply [25, Theorem 5.40].

Hence u+ satisfies $\begin{array}{}-\mathit{\Delta }u+au-\frac{1}{2\pi }\left[\mathrm{ln}\frac{1}{|x|}\ast |{u}^{+}{|}^{2}\right]{u}^{+}=f\left(x,{u}^{+}\right).\end{array}$ Now, multipliyng by u, we get

$−∫R2|(Du−(x)|2+a|u−(x)|2dx=0,$

thus u ≡ 0, then u ≥ 0 and it is a nontrivial solution of problem (P). Working with I we find another nontrivial solution of (P) which is non positive in ℝ2. □

## Acknowledgement

Dimitri Mugnai is a member of GNAMPA and is supported by the MIUR National Research Project Variational methods, with applications to problems in mathematical physics and geometry (2015KB9WPT_009) and by the FFABR “Fondo per il finanziamento delle attività base di ricerca” 2017.

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Accepted: 2019-03-16

Published Online: 2019-08-29

Published in Print: 2019-03-01

Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 850–865, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496,

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