Ground states for planar axially Schrödinger–Newton system with an exponential critical growth

<jats:p>In this paper, we study the following planar Schrödinger–Newton system: <jats:disp-formula><jats:alternatives><jats:tex-math>$$ \left \{ \textstyle\begin{array}{l} -\Delta u+ V(x)u+\lambda\phi u= f(x,u)\quad \textrm{in } \mathbb{R}^{2},\\ \Delta\phi=u^{2}\quad\textrm{in } \mathbb{R}^{2}, \end{array}\displaystyle \right . $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>{</mml:mo><mml:mtable><mml:mtr><mml:mtd><mml:mo>−</mml:mo><mml:mi>Δ</mml:mi><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:mi>V</mml:mi><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:mi>λ</mml:mi><mml:mi>ϕ</mml:mi><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>u</mml:mi><mml:mo>)</mml:mo><mml:mspace /><mml:mtext>in </mml:mtext><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mi>Δ</mml:mi><mml:mi>ϕ</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>u</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mspace /><mml:mtext>in </mml:mtext><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></jats:alternatives></jats:disp-formula> where <jats:italic>V</jats:italic>, <jats:italic>f</jats:italic> are axially symmetric about <jats:italic>x</jats:italic>, <jats:italic>V</jats:italic> is positive, and <jats:italic>f</jats:italic> is super-linear at zero and exponential critical at infinity. Using a weaker condition <jats:disp-formula><jats:alternatives><jats:tex-math>$$ \biggl[\frac{f(x,u)}{u^{3}}-\frac{f(x,tu)}{(tu)^{3}} \biggr]\operatorname {sign}(1-t)+ \theta V(x)\frac{ \vert 1-t^{2} \vert }{(tu)^{2}}\geq0,\quad \forall x\in \mathbb{R}^{2}, t>0, u\neq0 $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>[</mml:mo><mml:mfrac><mml:mrow><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>u</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mi>u</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:mfrac><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mi>u</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mi>u</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:mfrac><mml:mo>]</mml:mo></mml:mrow><mml:mo>sign</mml:mo><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>+</mml:mo><mml:mi>θ</mml:mi><mml:mi>V</mml:mi><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo><mml:mfrac><mml:mrow><mml:mo>|</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msup><mml:mi>t</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>|</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mi>u</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mo>≥</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mspace /><mml:mi>∀</mml:mi><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>u</mml:mi><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math></jats:alternatives></jats:disp-formula> with <jats:inline-formula><jats:alternatives><jats:tex-math>$\theta\in[0,1)$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>θ</mml:mi><mml:mo>∈</mml:mo><mml:mo>[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:math></jats:alternatives></jats:inline-formula> instead of the Nehari type monotonic condition on <jats:inline-formula><jats:alternatives><jats:tex-math>$\frac{f(x,u)}{|u|^{3}}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mfrac><mml:mrow><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>u</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo>|</mml:mo><mml:mi>u</mml:mi><mml:mo>|</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:mfrac></mml:math></jats:alternatives></jats:inline-formula>, we obtain a ground state solution of the above problem via variational methods.</jats:p>


Introduction and main results
In the present paper, we are concerned with the wave solutions of the Schrödinger-Newton system where ψ : R d × R → C is the wave function, W (x) is a real external potential, λ > 0 is a parameter. Problems of the type (1.1) arise in many problems from physics. We refer the readers to [15], therein (1.1) appears in a quantum mechanical context in the case d ≤ 3.
A standing wave solution of (1.1) is a solution of the form ψ(x, t) = e -iEt u(x) and its existence reduces (1.1) to the system where V (x) = W (x) -E, g(x, e -iEt u) = f (x, u)e -iEt . For the case d = 3, it is called the Schrödinger-Poisson system and it has been well studied. For the existence, multiplicity, and concentration, we refer the readers to [2,3,9,10,13,20] and the references therein. For Kirchhoff type equations involving subcritical and critical growth in three dimensions, please see [19] and the references therein. We also quote the paper [12] for Hardy-Schrödinger-Kirchhoff systems. However, much less is known about the case d = 2. For φ = u 2 , in R 2 , one has Substituting it into (1.2), we obtain the integro-differential equation For simplicity, throughout this paper, let λ = 2π . The approach for d = 3 cannot be easily adapted to d = 2 since which is the functional associated with the third term in (1.4), is sign-changing, and is neither bounded from above nor from below on H 1 (R 2 ). This difficulty has been overcome recently in [7] or [16]. For by introducing the following subspace of H 1 (R 2 ) Stubbe considered the L 2 -constraint minimization problem and proved that (1.6) admits a ground state. Soon afterwards, in [8], Cingolani and Weth processed successfully the two dimensional Schrödinger-Newton equations with nonlinear term |u| p-2 u, p ≥ 4. Du and Weth [11] provided some results about p > 2 and p ≥ 3. The key tool is Pohozaev type identity (see [11,Lemma 2.4]). Chen, Shi, and Tang [4] used the same idea to obtain a ground state but they could deal with the general nonlinearity f (u). Simultaneously, Chen and Tang [5] investigated the existence of an axially symmetric Nehari type ground state and nontrivial solution for where V , f is axially symmetric about x. Please see [6,17] for further results about two dimensional Schrödinger-Newton equations with the axially symmetric assumptions. Recently, when V (x) = 1, Alves and Figueiredo [1] proved that (1.4) admits a positive ground state, where f is a continuous function with the exponential critical growth.
In this paper, motivated by the papers [1] and [5], we shall study the existence of ground state solutions of planar problem (1.1) with an exponential critical growth. In order to state our main result, we assume that Here we also give an example which satisfies (f 1 )-(f 5 ): where K ∈ (R 2 , R) is axially symmetric and inf x∈R 2 K(x) > 0, V satisfies (V 1 ) and (V 2 ). But it does not satisfy the Nehari type monotonic condition Now we state our main result as follows.
, where m is the least energy (it will be defined in (2.22)), θ is from (f 3 ), (1.7) possesses a ground state solution.
m ) is used to prove the minimizing sequence of m is bounded, and please see Lemma 3.3. Up to now, we have not been able to remove it.
The paper is organized as follows. Section 2 is to establish the variational setting and to give some preliminaries. Section 3 is to prove the existence of ground states. Throughout the paper, we always assume that (V 1 ), (V 2 ) and (f 1 )-(f 5 ) hold and make use of the following notations: • C, C i (i = 0, 1, 2, . . .) for positive constants (possibly different from line to line).

Variational setting and preliminaries
In this section, we begin our study by establishing the variational setting for (1.7). Let H 1 (R 2 ) be the usual fractional Sobolev space with the usual norm By (V 1 ) and (f 1 ), similar to [5], let E be defined as According to [1, Lemma 2.1], we have the following.
We formally formulate problem (1.7) in a variational way as follows: For simplicity of notations, denote Similar to [8], using ln(r) = ln(1 + r) -ln(1 + 1 r ), ∀r > 0, it holds that We give the following proposition which is used to estimate the nonlinearity.

Lemma 2.6
For any u ∈ E, there exists unique t > 0 such that tu ∈ N .
Since u ∈ N , by Lemma 2.4, one has Up to this stage, preparations have been made. We point out that we can define m without using the condition α ∈ (0, π (1-θ) m ). In the next section, taking full advantage of the condition α ∈ (0, π (1-θ) m ), we shall prove the existence of ground state solutions of (1.7).

Existence of ground states
In this section, with the additional condition α ∈ (0, π (1-θ) m ), we are devoted to showing that m is achieved and the minimizer is a ground state solution of equation (1.7). Lemma 3.1 There exists C > 0 such that u ≥ C for all u ∈ N ; furthermore, m > 0.
Proof Assume by contradiction that there is {u n } ⊂ N such that u n → 0. Obviously, u n 2 + 4 I 1 (u n ), u n = 4 I 2 (u n ), u n + R 2 f (x, u n )u n dx.
In view of (f 1 ) -(f 3 ), combining Hölder's inequality, it follows that With Proposition 2.2 in hand, using the Sobolev embedding, it leads to By direct calculation, it holds that 4 I 2 (u n ), u n ≤ C u n 4 8 3 = o n (1).
Therefore, we obtain That is, Noting that u n → 0, using Proposition 2.2 again, we get which is ridiculous. Combining with (2.21), we have m > 0.
Next, we give the following lemma which shall be used later.
Proof Similar to (2.21), { u n } is bounded. Similar to (2.5), {I 2 (u n )} is bounded. Next, we want to estimate the {I 1 (u n )}. Note that For the second term on the right, using Hölder's inequality with s > 1 and s ≈ 1, it holds that Taking into account α ∈ (0, π (1-θ) m ), jointly with for n large enough, we obtain αs u n 2 < 4π . So, by Proposition 2.2, we get Since u n 2 + I 1 (u n ) = I 2 (u n ) + R 2 f (x, u n )u n dx, (3.7) which yields that {I 1 (u n )} is bounded. And it follows from Lemma 3.2 that {u n } is bounded in E.
Next, we claim that there are R, η > 0 such that lim inf n→∞ B R (y n ) |u n | 2 dx ≥ η.

Lemma 3.4 m is achieved and the minimizer is a weak solution of (1.7).
Proof Now, we can assume that u n u 0 = 0 in E, u n → u 0 in L t (R 2 ) for all t ∈ [2, ∞) and u n (x) → u 0 (x) a.e. in R 2 . By a standard argument, one can deduce that I (u 0 ) = 0. Obviously, we have f (x, u 0 )u 0 dx + o n (1). (3.12) Here, we only check 3.12 since (3.11) is similar. We have already known that f (x, u n )u n ≤ ε|u n | 2 + C(ε)|u n | p exp α u n 2 u n u n 2 -1 . (3.13) Noting that α ∈ (0, π (1-θ) m ) and (3.5), we obtain that α u n 2 < 4π for n large enough. By Proposition 2.2, there exists C > 0 independent of n such that