A fractional Laplacian problem with critical nonlinearity

1.   Introduction

In recent years, motivated by problems that appear in anomalous diffusions in plasmas, flames propagation and chemical reactions in liquids etc, fractional Laplacian problems have received extensive attention in the literature. Also, due to its rich multi-origins, fractional Laplacian operators turn out to have many different definitions. When studying fractional problems in the whole space ${\mathbb R}^{n}$, the fractional Laplacian $(-{\Delta})^{s}$ is usually defined via the Fourier transformation $(-{\Delta})^{s}f(x) = (|2\pi i\xi|^{2s}\hat{f}(\xi))^{\lor},$ see e.g. Frank and Lenzmann ^[12] and Di Nezza, Palatucci and Valdinoci ^[11]. Equivalently, this operator can be written as the difference quotient

for sufficiently regular $f$, where $c_{n, s}$ is a normalization constant, for a proof see e.g. ^[11]. An advantage of the difference quotient type definition is that it can be directly extended to define the so-called fractional $p$-Laplacian operator

and the so called regional fractional Laplacian operator

in case the problem under consideration is restricted to a bounded domain ${\Omega}\subset{\mathbb R}^{n}$, see e.g. Chen ^[10] and the references therein. It is beyond the size of this paper to present all the other definitions for fractional Laplacian operators in the literature. We refer the interested readers to the references mentioned above.

In this paper, we are concerned with the fractional Laplacian operator $(-{\Delta})^{s}$ on a bounded domain defined via the spectral of the Laplacian operator $-{\Delta}$. This type of fractional Laplacian operator was first defined by Cabré and Tan ^[7] for $s = 1/2$ and then extended by Brändle, Colorado, de Pablo and Sánchez ^[4] to the whole range $s\in (0, 1)$. This type of definition can be viewed as a discrete version of Fourier transformation for functions on a bounded domain. More precisely, let $s\in(0, 1)$ and ${\Omega}\subset{\mathbb R}^{n}$ be a bounded smooth domain. Let $\{{\lambda}_{k}, {\varphi}_{k}\}_{k\ge1}$ be the corresponding eigenvalues and eigenfunctions of $-{\Delta}$ on ${\Omega}$ with Dirichlet boundary value ${\varphi}_{k}|_{{\partial}{\Omega}} = 0$ and normalization $\|{\varphi}_{k}\|_{L^{2}({\Omega})} = 1$. Then, for $u = \sum_{k\ge1}c_{k}{\varphi}_{k}$ satisfying $\sum_{k\ge1}c_{k}^{2}{\lambda}_{k}^{2s} < {\infty}$, we define

Therefore, a natural function space of $(-{\Delta})^{s}$ in the sense of functional analysis is given by

such that

is well defined for every $u = \sum c_{k}{\varphi}_{k}, v = \sum d_{k}{\varphi}_{k}\in H_{0}^{s}({\Omega})$. Observe that a remarkable feature of the operator $(-{\Delta})^{s}$ is that it is nonlocal, in the sense that for any point $x\in{\Omega}$, the value of $(-{\Delta})^{s}u$ at point $x$ can be obtained only if one knows the distribution of $u$ in the whole domain, so that it is possible derive all of the coefficients $c_{k}\equiv\int_{{\Omega}}u{\varphi}_{k} {{\rm{d}}} x$. For a systematical study on this type of fractional Laplacian operators, we refer to e.g. Cabré and Tan ^[7] for the case $s = 1/2$ and Brändle et al. ^[4] for the general case $s\in(0, 1)$.

Now our problem under consideration can be stated as below. Fix $s\in(0, 1)$ and consider the following fractional Laplacian problem

where $2_{s}^{\ast} = 2n/(n-2s)$, $2\le q < 2_{s}^{\ast}$ and $k\in(0, {\infty})$ is an arbitrary positive number. A positive function $u\in H_{0}^{s}({\Omega})$ is called a (weak) solution of problem (1.3) if for every $v\in H_{0}^{s}({\Omega})$, there holds

Precise assumptions on the coefficient function $g$ will be given soon. Our aim is to prove that under appropriate conditions on the function $g$ and the parameters $q, n$, there does exist at least one positive solution to the above problem.

Problem (1.3) is not totally new. Problems of type (1.3) have been extensively studied in the literature with different leading operators. In the local case (i.e., $s = 1$) with $k = 0$, problem of type (1.3) dates back to the famous work Brézis and Nirenberg ^[6], where positive solutions to the critical growth problem was obtained:

In this case, the function $g$ is the constant function ${\lambda}$. Since then, there are numerous extensions and important variants which have been studied. In ^[1], Ambrosetti, Brezis and Cerami considered the problem combined with concave and convex nonlinearities

where $0 < q < 1 < p$ and ${\lambda} > 0$. They proved among many other results that when ${\lambda}$ is sufficiently small, this problem has a positive solution, see [1, Theorem 2.1] for more results. Quite recently, this problem was extended by Barrios, Colorado, Servadei, and Soria ^[3] to the nonlocal setting

where $0 < s < 1$, $0 < q < p = \frac{n+2s}{n-2s}$ and ${\lambda} > 0$, and $(-{\Delta})^{s}u$ is defined via the difference quotient (1.1). Note that in this problem the boundary condition is in fact assumed on the complement of ${\Omega}$. They obtained similar results [3, Theorem 1.1, 1.2] as that of [1, Theorem 2.1]. In ^[15] Servadei and Valdinoci considered the even more general integrodifferential problem of Brézis and Nirenberg type

where $s\in(0, 1)$ and ${\mathcal L}_{K}$ is a nonlocal operator with kernel function $K$ such that $(-{\Delta})^{s}$ plays a model case. Under various natural assumptions they proved several existence results to this problem, see [15, Theorem 1–3]. In particular, as an application, they obtained a positive solution in the model case ${\mathcal L}_{K} = (-{\Delta})^{s}$ and $f\equiv0$, see [15, Theorem 4]. Similar to ^[3], but with the leading operator $(-{\Delta})^{s}$ defined in the spectral way (1.2), Barrios, Colorado, de Pablo and Sánchez ^[2] considered the problem

with $0 < q < p = \frac{n+2s}{n-2s}$. Then also obtained existence and nonexistence results under various assumptions on ${\lambda}$, see [2, Theorem 1.1, 1.2, 1.3] for details. For the case $p < \frac{n+2s}{n-2s}$, see Br$\ddot{a}$ndle, Colorado, de Pablo and Sánchez ^[4].

In the case $g\not\equiv\rm{constant}$, Gazzola ^[13] studied this type of problem with $p$-Laplacian operator $-{\Delta}_{p}$ as the leading term,

where $1 < p < n$, $p^{*} = np/(n-p)$ and positive solutions were obtained among other results, see [13, Theorem 2]. Inspired by the work of Gazzola ^[13] and Cabré and Tan ^[7], Wang studied problem (1.3) with $s = 1/2$ and obtained a positive solution to her problem. We mention that Li and Xiang ^[14] extended Wang's work ^[17] to the setting of regional fractional Laplacion problems.

Inspired by Wang's work ^[17], in this paper we study problem (1.3) with $s$ belonging to the full range $(0, 1)$. Throughout we assume that $2\le q < 2_{s}^{\ast}$ and $g$ satisfies

(G1) $g\in C({\Omega})$ is a nonnegative nontrivial function, and

(G2) $g\in L^{2n/(2n-(n-2s)q)}({\Omega})$. In addition, if $q = 2$, then $\|g\|_{L^{n/2s}({\Omega})} < S$, where $S$ is defined as in (1.6) in the below.

Our main result reads as follows.

Theorem 1.1. Assume that $s\in (0, 1)$ and $g$ satisfies (G1) (G2). Then problem (1.3) admits a positive solution for every $k\in(0, {\infty})$, provided either $q\ge 2\mathit{\;{\rm{and}}\;}n > 2s(1+2/q)$, or $q = 2$ and $n = 4s$.

It is clear that the nonlocality of $(-{\Delta})^{s}$ makes it difficult to deal with problem (1.3) directly. Our strategy is to use a localization method. In the global case ${\Omega} = {\mathbb R}^n$, a localization principle for fractional Laplacian problem was first systematically developed by Caffarelli and Silvestre ^[9]. In our local case, a localization principle was also developed by Cabré and Tan ^[7] for the case $s = 1/2$ and by Brändle et al. ^[4] for the general case $s\in (0, 1)$. To state the localization principle, we need to introduce some notations. Denote

with coordinates $(x, y)\in{\mathcal C}$ for $x\in{\Omega}$ and $y > 0$. According to Brändle et al. ^[4], for every $u\in H_{0}^{s}({\Omega})$, there exists a unique $2s$-harmonic extension $\bar{u}$ which is equal to $u$ on ${\Omega}\times\{y = 0\}$ in the sense of trace with $\bar{u} = 0$ on the lateral boundary ${\partial}_L \mathcal{C}$, and satisfies the equation

Following Brändle et al. ^[4], introduce the function space

where ${\kappa}_{s} = 2^{1-2s}{\Gamma}(1-s)/{\Gamma}(s)$ is a normalization constant such that the $2s$-harmonic extension $\bar{u}$ of $u$ satisfies $\|\bar{u}\|_{X_{0}^{2s}({\mathcal C})} = \|u\|_{H_{0}^{s}({\Omega})}$. Then, problem (1.3) is equivalent to

where

That is, $u\in H_{0}^{s}({\Omega})$ is a solution to problem (1.3) if and only if $\bar{u}\in X_{0}^{2s}({\mathcal C})$ is a solution to problem (1.4) which means that

holds for all $\Phi\in X_{0}^{2s}({\mathcal C})$. This explains the localization method.

So, to prove Theorem 1.1, we turn to the equivalent problem (1.4). Note that problem (1.4) is variational with the energy functional ${\mathcal J}:X_{0}^{2s}({\mathcal C})\to{\mathbb R}$ being given by

for $\bar{u}\in X_{0}^{2s}({\mathcal C})$. To find a critical point of ${\mathcal J}$, we will use the mountain pass theorem (see e.g. Struwe [16, Chapter 6]).

Before ending the introduction, we record the following inequality (see [4, Theorem 2.1]) for later use: there exists $C = C(n, s) > 0$ such that, for every $w\in X^{2s}({\mathbb R}_{+}^{n+1})$, the closure of $C_{0}^{{\infty}}(\overline{{\mathbb R}_{+}^{n+1}})$ under the seminorm $\|u\|_{X^{2s}({\mathbb R}_{+}^{n+1})}^{2} = \int_{{\mathbb R}_{+}^{n+1}}{\kappa}_{s}y^{1-2s}|{\nabla} w(x, y)|^{2} {{\rm{d}}} x {{\rm{d}}} y$, there holds

Let

be the best constant for inequality (1.5). It is known [4, Theorem 2.1] that $S$ is attained by the functions

for all ${\epsilon} > 0$ and $x_{0}\in{\mathbb R}^{n}$, where $P_{y}^{1-2s}(x) = k_{1-2s}y^{-n}(1+|x|/y)^{-(n+2s)/2}$ is the so-called $s$-Poisson kernel and

The constant $c_{n} > 0$ is chosen such that

Our notations are standard. We use $C$ to denote positive constants that are different from line to line. For simplicity, we will use $\|\cdot\|$ to denote the norm $\|\cdot\|_{X_{0}^{2s}({\mathcal C})}$ throughout.

2.   Proof of the main result

This section is devoted to the proof of Theorem 1.1. Our method is to use the well known mountain pass theorem, see e.g. Struwe [16, Theorem 6.1]. The first lemma points out that

Lemma 2.1. $\mathcal{J}$ satisfies the mountain pass geometry.

The proof is standard, and we omit the details. We shall also need

Lemma 2.2. Under the assumptions $(G_{1})$ and $(G_{2})$, there hold

and

for every $(u_{m})\subset X_{0}^{2s}({\mathcal C})$ that converges weakly to $u\in X_{0}^{2s}({\mathcal C})$.

Lemma 2.2 is a consequence of Vitali's convergence theorem. We omit the details, see also Wang [17, Lemma 2.1] and Gazzola [13, Lemma 1].

We say that $(u_{m})\in X_{0}^{2s}({\mathcal C})$ is a $(PS)_c$ sequence, if $\mathcal{J}(u_{m})\rightarrow c\in {\mathbb R}$ and $\mathcal{J}'(u_{m})\rightarrow 0$ as $m\rightarrow \infty$. Say that the functional $\mathcal{J}$ satisfies $(PS)_c$ compactness condition if $(u_{m})\in X_{0}^{2s}({\mathcal C})$ is a $(PS)_c$ sequence, then it contains a convergent subsequence in $X_{0}^{2s}({\mathcal C})$. Next lemma says that this happens when $c$ is below a critical value, as that was observed by Brezis and Nirenberg ^[6].

Lemma 2.3. $\mathcal{J}$ satisfies the $(PS)_{c}$ condition provided $c < \frac {s}{n} S^{\frac{n}{2s}}$.

Proof. Let $(u_{m})\subset X_{0}^{2s}({\mathcal C})$ be a $(PS)_c$ sequence. First we claim that $(u_{m})$ is a bounded sequence in $X_{0}^{2s}({\mathcal C})$. This will follow from a simple combination of the $(PS)_c$ assumption. Let $\beta\geq2$. Then

Taking $\beta = q$ when $q > 2$ yields

Then the assumption $g\geq0$ implies that

When $q = 2$, take $2_{s}^{*} > \beta > 2$. Then using the assumption $(G_{2})$ and Hölder's inequality gives

Now the boundedness of $(u_{m})\subset X_{0}^{2s}({\mathcal C})$ follows from (2.1) and (2.2).

Next we prove that $(u_{m})$ has a convergent subsequence in $X_{0}^{2s}({\mathcal C})$. Since we have proved the boundedness of $(u_{m})$ in $X_{0}^{2s}({\mathcal C})$, we may assume without loss of generality that, up to a subsequence,

for some $u\in X_{0}^{2s}({\mathcal C}).$ Put $v_{m} = u_{m}-u$. It suffices to prove that $\|v_{m}\|\rightarrow0$.

It is standard to find that $\mathcal{J}'(u) = 0.$ To examine the sequence closer, note that

Using Lemma 2.2 we find that

By Lemma 2 of Brézis and Lieb ^[5],

As a result, we obtain the first decomposition

Similarly, we also have the second decomposition

where we used used the assumption $\mathcal{J}'(u_{m})\to 0$ in the first equality and the fact $\mathcal{J}'(u) = 0$ in the last line.

To continue, note that $\|v_{m}\|\leq C(\|u_{m}\|+\|u\|)$ is a bounded sequence. So we may assume $\|v_{m}\|^{2}\rightarrow b\geq0$. Then the above second decomposition gives

Recall the trace inequality (1.5). This implies $Sb^{\frac{2}{2_{s}^{*}}}\leq b$. As a consequence, we infer that either $b = 0$ or $b\geq S^{\frac{n}{2s}}$ holds.

We have to exclude the case $b\geq S^{\frac{n}{2s}}$. To this end, note that $\mathcal{J}'(u) = 0$ implies $\mathcal{J}(u)\geq0$. Thus the first decomposition leads to

If $b\geq S^{\frac{n}{2s}}$, then taking limit in this inequality gives $c\geq \frac{s}{n}S^{\frac{n}{2s}}$, which contradicts with our assumption $c < \frac{s}{n}S^{\frac{n}{2s}}$. Hence $b = 0$. That is $v_{m}\rightarrow 0$ in $X_{0}^{2s}({\mathcal C})$. The proof is finished.

To proceed, we may assume that $0\in {\Omega}$ and $g(0) > 0$ with no loss of generality. By the assumption $(G{1})$ we can assume $B_{r}(0)\subset \{x\in\Omega:g(x) > g(0)\}$ for some $r > 0$ sufficiently small. For simplicity, write $u_{{\epsilon}} = u_{{\epsilon}, 0}, U_{{\epsilon}} = U_{{\epsilon}, 0}$, where $u_{{\epsilon}, x_{0}}, U_{{\epsilon}, x_{0}}$ are defined as in (1.7). Let $0 < \rho < r$ and take a cut-off function $\eta\in C^{\infty}(\overline{\mathcal{C}})$ such that $0\leq\eta\leq1$ and $\eta(x, y)\equiv1$ for $|x| < \frac{\rho}{2}$ and $y\geq0$, and $\eta(x, y)\equiv0$ for $|x|\geq\rho$. These auxiliary functions and parameters are used to construct a special path in $X_{0}^{2s}({\mathcal C})$ starting from the origin such that the following lemma holds.

Lemma 2.4. For ${\epsilon} > 0$ sufficiently small there holds

Proof. Write $v_{{\epsilon}} = \eta U_{{\epsilon}}$ and $\gamma(t) = tv_{{\epsilon}}$ for $t\geq0$. Suffices to show that $\max_{t > 0}\mathcal{J}(\gamma(t)) < \frac{s}{n}S^{\frac{n}{2s}}$ for ${\epsilon}$ sufficiently small. By a direct computation we have

Since the second term on the right hand side is nonnegative, there holds

Also note that $v_{\epsilon}$ is a bounded function. So for $t$ sufficiently small, we have $(tv_{\epsilon}-k)_+\equiv 0$. Hence for $t$ sufficiently small,

So $\mathcal{J}(\gamma(t))$ achieves a positive maximum on $(0, {\infty})$.

Let $t_{{\epsilon}}$ be such that $\mathcal{J}(\gamma(t_{{\epsilon}})) = \max_{t > 0}\mathcal{J}(\gamma(t))$. We claim that for ${\epsilon}$ sufficiently small, there exist constants $C_{1}, C_{2}$ independent of ${\epsilon}$ such that

Notice that

where $\overline{t}_{{\epsilon}} = ({2_{s}^{*}\|v_{{\epsilon}}\|^{2}}/ {2\int_{\Omega}v_{{\epsilon}}^{{2_{s}^{*}}}})^{(n-2s)/{4s}}$. We infer that $\mathcal{J}(\gamma(t))\leq0$ holds for all $t\geq \overline{t}_{{\epsilon}}$, which implies

for some $C_2 > 0$ independent of ${\epsilon}$. On the other hand, by Lemma 2.1, $t_{{\epsilon}}$ cannot converge to zero. Therefore there exists $C_{1} > 0$ such that

for ${\epsilon}$ sufficiently small. The claim is proved.

To further compute the maximum $\mathcal{J}(\gamma(t_{{\epsilon}}))$, according to ^[2], there hold

for ${\epsilon}$ sufficiently small. Thus

where we used in the last line the fact that $\frac{t^{2}-1}{2}-\frac{t^{2_{s}^{*}}-1}{2_{s}^{*}}\le 0$ for all $t\ge 0$. Therefore we derive the estimate

We need to estimate $\int_{{\Omega}}g(t_{{\epsilon}}v_{{\epsilon}}-k)_{+}^{q}$. For ${\epsilon}$ sufficiently small, we always have

In the case $q\ge 2$ and $n > 2s(1+2/q)$, there holds

for some $C > 0$ independent of ${\epsilon}$, where we have used the estimate (2.3) for $t_{{\epsilon}}$. In the case $q = 2, n = 4s$, note that on the set $\{{\epsilon} < |x| < {\epsilon}^{3/4}\}$, there holds

for ${\epsilon}$ sufficiently small. Hence,

for some $C > 0$ independent of ${\epsilon}$. Thus, there exists $C > 0$ such that

Combining (2.4) and (2.5) yields

In the case $q\ge 2\;{\rm{and}}\;n > 2s(1+2/q)$, it holds $\frac{n-2s}{2}q-2s > 0$. Therefore, in both cases we can deduce that

provided ${\epsilon}$ is sufficiently small. The proof is complete.

Now we are in the position to prove Theorem 1.1.

Proof of Theorem 1.1. First choose $e = t_0 \eta U_{\epsilon}$, where ${\epsilon}$ and $\eta$ are chosen as in Lemma 2.4 and $t_0$ is sufficiently large such that ${\mathcal J}(t_0 \eta U_{\epsilon}) < 0$, and then let

and

By Lemma 2.4, we have $c_0 < \frac sn S^{n/2s}$. Since $\mathcal{J}$ satisfies the geometry of mountain pass, there exists a sequence $(u_{m})\subset X_{0}^{2s}({\mathcal C})$ satisfying ${\mathcal J}(u_{m})\to c_{0}$ and $\mathcal{J}^{\prime}(u_{m})\to0$ as $m\to{\infty}$. Therefore, Lemma 2.3 implies that problem (1.4) admits a nonnegative nontrivial solution. Finally, a maximum principle of Cabré and Sire [8, Remark 4.2] implies that the solution is positive in ${\Omega}$. The proof is complete.

Acknowledgments

This work is supported by the Project of Hubei University of Arts and Science (No. XK2021023). The first author would like to thank Miao Chen for many useful help and suggestions. She would also like to thank the second author for pointing out the problems. Both authors would like to thank the anonymous referees for valuable comments that greatly improve this manuscript.

Conflict of interest

All authors declare no conflict of interest in this paper.