Existence of solutions for a perturbed problem with logarithmic potential in $\mathbb{R}^2$

1.   Introduction

The Newton kernel $\Phi_N \colon \mathbb{R}^N \to \mathbb{R}$ is defined by

The non-local partial differential equation

where $a$ is a positive function, was proposed in the study of quantum physics of electrons in a ionic crystal (the so-called Pekar polaron model) for $N = 3$. The same equation can also be seen as a coupling of quantum physics with Newtonian gravitation: indeed, the system

in the unknown $\psi \colon \mathbb{R}^N \times \mathbb{R} \to \mathbb{R}$, $\psi = \psi(x, t)$, reduces to the single equation

via the ansatz $\psi(x, t) = \mathrm{e}^{-\mathrm{i} \lambda t} u(x)$ with $\lambda \in \mathbb{R}$ and $a(x) = E(x)+\lambda$. E.H. Lieb proved in ^[11] that (1.1) possesses, in dimension $N = 3$, a unique ground state solution which is positive and radially symmetric. E. Lenzmann proved in ^[10] that this solution is also non-degenerate. The analysis of (1.1) in dimension $N = 3$ is heavily based on the algebraic properties of the kernel $\Phi_3$, in particular its homogeneity. Lieb's proof of existence carries over to $N = 4$ and $N = 5$, while no solution with finite energy can exist in dimension $N \geq 6$, see ^[6].

In this note we consider (1.1) in the plane, i.e., when $N = 2$. The appearance of the logarithm in $\Phi_2$ changes drastically the setting of the problem, which has been an open field of study for several years. One of the main obstructions to a straightforward analysis in the planar case is the lack of positivity of the kernel $\Phi_2$.

Some preliminary numerical results contained in ^[9] encouraged Ph. Choquard, J. Stubbe and M. Vuffray to prove the existence of a unique positive radially symmetric solution by applying a shooting method, see ^[7]. But only in very recent years have variational methods been used to solve (1.1) for $N = 2$: the formal definition of a Euler functional

is not consistent with the metric structure of the Sobolev space $H^1(\mathbb{R}^2)$.

Stubbe proposed in ^[14] a variational setting for (1.1) in dimension two within the closed subspace

endowed with the norm

Although this space permits to use variational methods, several difficulties arise from the logarithmic term.

Using this functional approach, S. Cingolani and T. Weth (see ^[8]) proved some existence results for (1.1) under either a periodicity assumption on the potential $a$, or the action of a suitable group of transformations. Uniqueness and monotonicity of positive solutions are also proved.

Later on, D. Bonheure, S. Cingolani and J. Van Schaftingen (see ^[6]) proved that the positive solution $u$ of (1.1) with $a > 0$ is non-degenerate, in the sense that the only solutions of the linearized equation associated to (1.1) are the (linear combinations of) the two partial derivatives of $u$.

Motivated by these results, we consider the following perturbed equation, based on (1.1):

The quantity $\varepsilon$ plays the rôle of a "small" perturbation, and the function $h$ is a "weight" for the local nonlinearity $|u|^{p-1}u$. We refer to the next Sections for the precise assumptions we make.

We will face the problem of constructing solutions to (1.2) by means of a general technique in Critical Point Theory, introduced by A. Ambrosetti and M. Badiale in ^[1,2,3]. We refer to ^[4] for a presentation in book form. For the reader's convenience, we summarize here the main ideas of this method.

Suppose we are given a (real) Hilbert space $X$ and a functional $I_ \varepsilon \in C^2(X)$ of the form

Here $I_0 \in C^2 (X)$ is the so-called unperturbed functional, while $\varepsilon \in \mathbb{R}$ is a (small) perturbation parameter. We will suppose that there exists a (smooth) manifold $Z$ of dimension $d < \infty$, such that every $z \in Z$ is a critical point of $I_0$.

Letting $W = \left(T_z Z\right)^\perp$ for $z \in Z$, we look for solutions to the equation $I_ \varepsilon'(u) = 0$ of the form $u = z+w$, where $z \in Z$ and $w \in W$. We can split the equation $I_ \varepsilon'(u) = 0$ into two equations by means of the orthogonal projection $P \colon X \to W$:

We will assume that the following conditions hold:

(ND) for all $z \in Z$, we have $T_z Z = \ker I_0''(z)$;

(Fr) for all $z \in Z$, we have that the linear operator $I_0''(z)$ is Fredholm with index zero.

Remark 1.1. The condition (ND) can be seen as a non-degeneracy assumption, since it is always true that $T_z Z \subset \ker I_0''(z)$, by definition of $Z$.

It is possible to show that the first equation of system (1.3) can be (uniquely) solved with respect to $w = w(\varepsilon, z)$, with $z \in Z$ and $\varepsilon$ sufficiently small. The main result of this perturbation technique can be summarized in the following statement.

Theorem 1.2 (^[4]). Suppose that the function $\Phi_ \varepsilon \colon Z \to \mathbb{R}$ defined by $\Phi_ \varepsilon(z) = I_ \varepsilon(z+w(\varepsilon, z))$ possesses, for $| \varepsilon|$ sufficiently small, a critical point $z_ \varepsilon \in Z$. Then $u_ \varepsilon = z_ \varepsilon + w(\varepsilon, z_ \varepsilon)$ is a critical point of $I_ \varepsilon = I_0+ \varepsilon G$.

As it should be clear, the perturbation method of Ambrosetti and Badiale leans on the effect of the function $h$, which breaks the invariance of $I_0$ under translations. As such, the existence of a critical point of the function $\Phi_ \varepsilon$ depends crucially on the behavior of $h$.

We split our existence results into two categories. The first one assumes that the weight function $h$ is not only bounded, but also sufficiently integrable over $\mathbb{R}^2$; because of this, we can consider this results as a local existence result.

Theorem 1.3. Let $p > 1$ and $h \in L^{\infty}(\mathbb{R}^2) \cap L^q(\mathbb{R}^2)$ for some $q > 1$. Moreover, suppose that

(h$_1$) $\int_{ \mathbb{R}^2}h(x)|z_0|^{p+1}\, dx \neq 0$.

Then Eq. (1.2) has a solution provided $| \varepsilon|$ is small enough.

It is possibile to drop the integrability condition on $h$, at the cost of a more delicate analysis of the implicit function $w = w(\varepsilon, z)$ that describes $\Phi_ \varepsilon$. We have the following global result.

Theorem 1.4. Let $p > 2$ and suppose that $h$ satisfies

(h$_2$) $h \in L^{\infty}(\mathbb{R}^2)$ and $\lim_{|x| \to \infty} h(x) = 0$.

Then for all $| \varepsilon|$ small, Eq. (1.2) has a solution.

We highlight that our results differ from those appearing in the literature for several reasons. First of all, the non-degeneracy property appearing in Proposition 2.5 can be used as a basis for further investigation. Moreover, the right-hand side of Eq. (1.2) may (and indeed must) depend on $x$; no symmetry requirement, like radial symmetry, is needed in our proofs.

The paper is organized as follows. In Section 2 we first give the precise assumptions for our problem, then we recall some known results (classical and not) and we prove some properties for energy functional, such as regularity. In Section 3 we present the proof of the main theorems.

2.   Preliminary results

Consider the equation

with $a > 0$, $h \in L^{\infty}(\mathbb{R}^2)$ and $p > 1$. We introduce the function space

endowed with the norm

where

The norm $\|\cdot \|_X$ is associated naturally to an inner product. Let

be the energy functional associated to the equation, where

and

We define (see ^[8,14]) the symmetric bilinear forms

and

since for all $r > 0$ we have

Remark 2.1. The definitions above are restricted to measurable functions $u$, $v\colon \mathbb{R}^2 \to \mathbb{R}$ such that the corresponding double integral is well defined in the Lebesgue sense.

In order to find estimates for $B_1$ and $B_2$ we recall a classical result of Measure Theory.

Theorem 2.2 (Hardy-Littlewood-Sobolev's inequality ^[12]). Let $p > 1$, $q > 1$ and $0 < \lambda < N$ with $\frac{1}{p} + \frac{\lambda}{N} + \frac{1}{q} = 2.$ If $f \in L^p(\mathbb{R}^N)$ and $g \in L^q(\mathbb{R}^N)$, then there exists a sharp constant $C(N, \lambda, p)$, independent of $f$ and $g$, such that

The sharp constant satisfies

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

In this case there is equality in (2.4) if and only if $g \equiv cf$ with $c$ constant and

for some $A \in \mathbb{R}$, $\gamma \in \mathbb{R} \setminus \{0\}$ and $\alpha \in \mathbb{R}^N.$

We note that, since

we have by Schwarz's inequality

for $u$, $v$, $w$, $z \in X$. Next, since $0 \le \log(1+r) \le r$ for all $r > 0$, we have by Hardy-Littlewood-Sobolev's inequality

for $u, v \in L^\frac{4}{3}(\mathbb{R}^2)$, for some constant $C > 0$. In particular, from (2.5) we have

for all $u \in X$ and from (2.6) we have

for all $u \in L^\frac{8}{3}(\mathbb{R}^2)$.

Proposition 2.3. The functional $I_{ \varepsilon}$ is of class $C^2(X)$.

Proof. The proof is similar to [8,Lemma 2.2], so we just sketch the main ideas. Recalling (2.7), (2.8), the assumption $h \in L^{\infty}(\mathbb{R}^2)$ and the fact that $X$ is compactly embedded into $L^s(\mathbb{R}^2)$, $s \in [2, +\infty)$ (see ^[5,6,8] for a proof) we have

The first Gâteaux derivative of $I_{ \varepsilon}$ along $v$ is

We add and subtract $\int\limits_{ \mathbb{R}^2} u(x)v(x)\log(1+|x|) \, dx$ to recover the scalar product of $X$, so we obtain

Now,

The second Gâteaux derivative of $I_{ \varepsilon}(u)$ along $(v, w)$ is

In this case, we add and subtract $\int_{ \mathbb{R}^2} v(x)w(x)\log(1+|x|) \, dx$, hence

Finally,

It is now standard to conclude that the first and the second Gâteaux derivatives are continuous (with respect to $u \in X$), so that $I_ \varepsilon \in C^2(X)$.

Critical points of the unperturbed functional $I'_0(u) = 0$ are solutions of the equation

which admits for every $a > 0$ a unique — up to translations — radially symmetric solution $u \in X$ (see ^[8], Theorem 1.3).

Since (2.10) is invariant under translations, we can consider $z_{\xi}(x) = u(x-\xi)$, with $\xi \in \mathbb{R}^2$. The manifold

is therefore a critical manifold for $I_0$. We want to show that the manifold $Z$ satisfies the properties (ND) and (Fr).

The (ND) property follows from the following theorem, proved in ^[6].

Theorem 2.4. If $a > 0$ and $u \in X$ is a radial solution of (2.10) then there exists $\mu \in (0, \infty)$ such that

with

To prove that (Fr) holds, we actually show that $I''_0(z_{\xi})$ is a compact perturbation of the identity operator.

Proposition 2.5. $I''_0(z_{\xi}) = L-K$, where $L$ is a continuous invertible operator and $K$ is a continuous linear compact operator in $X$.

Proof. We recall that

We add and subtract $\int_{ \mathbb{R}^2} \log(1+|x|)v(x)w(x) \, dx$, hence

and so

The second derivative is therefore the linear operator defined by

where

Following the proof in ^[13], Lemma 15, let $\{v_n\}_n$, $\{w_n\}_n$ be two sequences in $X$ such that $\|v_n\| \leq 1$, $\|w_n\| \leq 1$, $v_n \rightharpoonup v_0$ and $w_n \rightharpoonup w_0$. Without loss of generality we can assume $v_0 = w_0 = 0$, so that

From the compact embedding of $X$ in $L^s(\mathbb{R}^2)$ for $s \geq 2$, we can say that

in $L^s(\mathbb{R}^2)$, for every $s \geq 2$. We compute, using Theorem 2.4,

Now, by Hölder's inequality,

thanks to Theorem 2.4 and (2.11).

Similarly,

This proves that the linear operator

is compact. At this point, we should notice that the linear operator

is not invertible on $X$. To overcome this difficulty, we set

and rewrite $\mathcal{L}(z_\xi)$ as follows:

Since

by [8,Proposition 2.3], the multiplication operator

is compact. We may conclude that the functional $I''_0(z_{\xi})$ is of the form $L - K$ where

is a linear, continuous, invertible operator and $K$ is a linear, continuous, compact operator. A different proof, based on a direct computation, appears in [?].

3.   Existence results for equation (1.2)

3.1.   Proof of Theorem 1.3

Since the properties (ND) and (Fr) hold we can say that, for $| \varepsilon|$ small, the reduced functional has the following form:

with $c_0 = I_0(z_{\xi})$.

Let $\Gamma \colon \mathbb{R}^2 \to \mathbb{R}$ be the function defined as

Lemma 3.1. Suppose that $h \in L^{\infty}(\mathbb{R}^2) \cap L^q(\mathbb{R}^2)$ for some $q > 1$. Then

Proof. By the Hölder inequality,

and since $z_{\xi}$ decays to zero as $|\xi| \to \infty$ we have

The proof is completed.

Thanks to the previous Lemma we can now prove the existence of local solutions for (2.1).

Proof of Theorem 1.3. The hypothesis of the previous Lemma are satisfied, so we have that $\Gamma(\xi)$ goes to $0$ as $|\xi| \to \infty$. From (h$_1$) follows that $\Gamma(0) = -\frac{1}{p+1}\int_{ \mathbb{R}^2}h(x)|z_0|^{p+1} \, dx \neq 0$. Then $\Gamma$ is not identically zero and follows that $\Gamma$ has a maximum or a minimum on $\mathbb{R}^2$ and the existence of a solution follows from Theorem 2.16 in ^[4].

3.2.   Proof of Theorem 1.4

As before, we call $P = P_{\xi}:X \to W_{\xi}$ the orthogonal projection onto $W_{\xi} = \left(T_{z_{\xi}}Z\right)^{\perp}$, $\widetilde{W_{\xi}}: = \langle z_{\xi} \rangle \oplus \left(T_{z_{\xi}}Z\right)$ and $R_{\xi}(w) = I'_0(z_{\xi}+w) - I''_0(z_{\xi})[w]$.

Remark 3.2. By the variational characterization of the Mountain-Pass solution $u$ as in [4,Remark 4.2], the spectrum of $PI''_0(u)$ has exactly one negative simple eigenvalue with eigenspace spanned by $u$ itself. Moreover, $\lambda = 0$ is an eigenvalue with multiplicity $N$ and eigenspace spanned by $D_iu$, $i = 1, \dotsc, N$ and there exists $\kappa > 0$ such that

and hence the rest of the spectrum is positive.

We prove the following

Theorem 3.3. (i) There is $C > 0$ such that $\|\left(PI''_0(z_{\xi})\right)^{-1}\|_{L(W_{\xi}, W_{\xi})} \leq C$, for every $\xi \in \mathbb{R}^2$,

(ii) $R_{\xi}(w) = o(\|w\|)$, uniformly with respect to $\xi \in \mathbb{R}^2$.

Proof. Since $z_{\xi}$ is a Mountain-Pass solution, Remark 3.2 holds, hence it suffices to show that there exists $\kappa > 0$ such that

For any fixed $\xi \in \mathbb{R}^2$, say $\xi = 0$, $PI''_0(z_0) = PI''_0(u)$ is invertible and there exists $\kappa > 0$ such that

Let $v^{\xi}(x) = v(x+\xi)$, then

and thanks to the change of variables $x = t+\xi$ and $y = s+\xi$ we obtain

Moreover, $v^{\xi} \perp \widetilde{W}$ whenever $v \perp \widetilde{W_{\xi}}$, hence

and so $(i)$ si true.

To prove $(ii)$ we observe that

After some computations we obtain

and since $z_{\xi}$ is critical point we finally have

By (2.7), (2.8), (2.4), Hölder's inequality we have

by the compact embedding of $X$ into $L^s(\mathbb{R}^2)$, $s \geq 2$

and finally

Hence,

and this goes to $0$ as $\|w\|_X \to 0$ uniformly with respect to $\xi \in \mathbb{R}^2$.

This Lemma allows us to use Lemma 2.21 in ^[4], so there exists $\varepsilon_0 > 0$ such that for all $| \varepsilon| \leq \varepsilon_0$ and all $\xi \in \mathbb{R}^2$ the auxiliary equation $PI'_ \varepsilon\left(z_{\xi} + w\right) = 0$ has a unique solution $w_{ \varepsilon}(z_{\xi})$ with

uniformly with respect to $\xi \in \mathbb{R}^2$.

We now prove

Lemma 3.4. There exists $\varepsilon_1 > 0$ such that for all $| \varepsilon| \leq \varepsilon_1$, the following result holds:

Proof. We first show two preliminaries results:

(a) $w_{\xi}$ weakly converges in $X$ to some $w_{ \varepsilon, \infty} \in X$ as $|\xi| \to \infty$. Moreover, the weak limit $w_{ \varepsilon, \infty}$ is a weak solution of

(b) $w_{ \varepsilon, \infty} = 0$.

As a consequence of (3.2) we have that $w_{ \varepsilon}(z_{\xi})$ weakly converges in $X$ to some $w_{ \varepsilon, \infty} \in X$, as $|\xi| \to \infty$. Recall that $w_{ \varepsilon}(z_{\xi})$ is a solution of the auxialiary equation $PI'_{ \varepsilon}\left(z_{\xi}+w_{\xi}(z_{\xi})\right) = 0$, namely

where

and $D_i$ denotes the partial derivative with respect to $x_i$.

Let $v$ be any test function, then

Following the computations in the proof of Proposition 2.5, in particular (2.12), and by Theorem 2.4 we can say that

as $|\xi| \to \infty$.

Now, we need to pass to the limit in

In order to do that, we show that

We split the integral as

where $\rho > 0$. Using Hölder's inequality with $p = k+1$, so $p' = \frac{p}{p-1-k}$, we obtain

and this goes to $0$ as $\rho$ goes to $\infty$. On the other hand,

and since $v$ is a test function this integral goes to $0$ as $\rho$ goes to $\infty$.

Hence, (3.4) holds and then

Moreover,

as $|\xi| \to \infty$ and again by Theorem 2.4,

as $|\xi| \to \infty$.

Finally, we obtain

thus $w_{ \varepsilon, \infty}$ is a weak solution of (3.3), namely (a) holds.

By (3.2) we have that $\lim\limits_{| \varepsilon| \to 0} w_{ \varepsilon, \infty} = 0$. Since the unique solution $w \in X$ of

with small norm is $w = 0$. To show that, we need to prove that there exists a constant $C > 0$ such that $\|w\|_X \geq C$.

Consider the first Gâteaux derivative, computed in (2.9), evaluated at $w$ along $w$, namely

thus

and thus

Now, observe that from (2.3) we have

hence

and finally

Moreover,

By (3.5) we obtain

where $\eta = \max\{p+1, 4\} > 2$ since $p > 1$; recalling that $\|w\|^{\eta-2}_{H^1(\mathbb{R}^2)} \leq C_4\|w\|^{\eta-2}_{X}$ we obtain

From this we infer that $w_{ \varepsilon, \infty} = 0$, provided that $| \varepsilon| \ll 1$. Hence (b) is true.

We recall that $w_{ \varepsilon, \xi}$ satisfies

where

and

From (3.6) and Theorem 3.3 it follows

We infer that

and this goes to 0 as $|\xi| \to \infty$ by the compact embedding of $X$ in $L^s(\mathbb{R}^2)$ for all $s \geq 2$, (a) and (b) proved above.

Then, by (3.1) we find that

thus,

Inserting the above inequality in (3.7) we obtain that

and passing to the limit we find that

Since $w_{ \varepsilon, \xi}$ is small in $X$ as $| \varepsilon| \to 0$ we conclude that

Now we are ready to prove the existence of a global solution for the problem (2.1).

Proof of Theorem 1.4. Consider the reduced function $\Phi_{ \varepsilon}(\xi) = I_{ \varepsilon}\left(z_{\xi} + w_{ \varepsilon, \xi}\right)$, namely

From $I_0(z_{\xi}) = \frac{1}{2}\|z_{\xi}\|^2_{H^1(\mathbb{R}^2)} - \frac{1}{8\pi}\int_{ \mathbb{R}^2 \times \mathbb{R}^2}\log\frac{1}{|x-y|}z_{\xi}^2(x)z_{\xi}^2(y) \, dx \, dy$ and setting $c_0 = I_0(z_{\xi})$ we have that

Moreover, $-\Delta z_{\xi} + az_{\xi} - \frac{1}{2\pi}\left[\log\frac{1}{|\cdot|}*z_{\xi}^2\right]z_{\xi} = 0$ implies

Substituting into (3.8) we obtain

thus

and after a short computation

Now, by repating the same arguments used in Proposition 2.5 all the integrals over $\mathbb{R}^2 \times \mathbb{R}^2$ converge to $0$.

Moreover, by Lemma 3.4 we have $\|w_{ \varepsilon, \xi}\|^2_{H^1(\mathbb{R}^2)} \to 0$ as $|\xi| \to +\infty$ and by Minkowski's inequality, Proposition 2.4, Lemma 3.4 and (h$_\mathbf{2}$):

Hence,

This means that either $\Phi_{ \varepsilon}$ is constant, or it has a maximum or minimum. In any case $\Phi_{ \varepsilon}$ has a critical point and we can apply Theorem 2.23 in ^[4] to find a solution for problem (2.1).

Acknowledgements

The authors are grateful to the anonymous referee for their comments and suggestions about the first version of the paper.

Conflict of interest

The authors declare no conflict of interest.