Sign-changing and signed solutions for fractional Laplacian equations with critical or supercritical nonlinearity

1.   Introduction

This paper is devoted to the study of the existence of sign-changing and signed solutions for the following nonlocal elliptic equations:

where $\mathcal{L}_K$ is the integro-differential operator defined as follows:

here

is a function with the properties that:

$(K_1)$ $mK\in{L^1}(\mathbb{R}^n),$ where

$(K_2)$ There exist $\gamma > 0$ and $s\in (0, 1)$ such that

for any $x\in \mathbb{R}^n\setminus\{0\}.$

A typical model for $K$ is given by the singular kernel

which coincides with the fractional Laplace operator $-(-\triangle)^s$ of the following fractional Laplacian equations

where

In problems (1.1) and (1.2), the set $\Omega\subset \mathbb{R}^n$ is an open bounded with Lipschitz boundary, $n > 2s, s\in (0, 1), \lambda$ is a positive real parameter, $p\geq 2^*$ and

is the fractional critical Sobolev exponent. The nonlinear term $f$ satisfies the following conditions:

$(A_{1})\ f\in C(\bar{\Omega}\times\mathbb{R}, \mathbb{R})$, there exist $C > 0$ and $2 < q < 2^*$ such that

$(A_{2})$ $\lim\limits_{t\rightarrow 0}f(x, t) = 0$ uniformly in $x\in \bar{\Omega};$

$(A_{3})$ $\frac{f(x, t)}{t}$ is increasing in $|t| > 0$ for $a.e.\ x\in\Omega.$

The operator $(-\triangle)^s$ can be seen as the infinitesimal generators of Lévy stable diffusion Processes; see ^[1] and the references therein. This operator arises in several areas, such as biology, chemistry, physics and finance (see ^[2,3,4]). It is easy to see that the integro-differential operator $\mathcal{L}_K$ is a generalization of the fractional Laplace operator $-(-\triangle)^s$ (see ^[5,6,7]). Moreover, the interest in non-local integro-differential problems (1.1) goes beyond the mathematical curiosity. They have impressive applications in different fields, such as the thin obstacle problem, portfolio optimization, pricing of financial instruments, phase transitions, stratified materials, statistical mechanics, fluid flow, anomalous diffusion, crystal dislocation, deblurring and denoising of images, and so on, see ^[8,9,10]. In the past few years, a great deal of attention has been devoted to nonlocal operators of elliptic type, both for their interesting theoretical structure and in view of concrete applications, see ^[11,12] and the references therein. By the minimax method, topological degree theory, or constrained minimization method, many authors obtained the existence results of nontrivial solutions, positive solutions or sign-changing solutions of some nonlinear elliptic equations, see ^[13,14,15]. To show their results, the authors always assumed the nonlinearity $f(x, t)$ involves subcritical or critical growth and/or $f(x, t)$ satisfies Ambrosetti-Rabinowitz condition. However, the existence of nontrivial solutions, positive solutions, negative solutions and sign-changing solutions for the nonlocal elliptic problem (1.1) with $p\leq 2^*$ has been investigated by using the variational method, fixed-point index theory, and critical point theorems, see ^[16,17,18]. There are only a few results about the existence and multiplicity of solutions for (1.1) with $p > 2^*$. Fortunately, Li et al. ^[19] investigated the following fractional Schr$\ddot{o}$dinger equation with electromagnetic fields and critical or supercritical nonlinearity:

where $(-\triangle)_A^s$ is the fractional magnetic operator with

and $\lambda$ is a positive real parameter. When the nonlinearity $f$ satisfies the Ambrosetti-Rabinowitz condition, they obtained the existence of a nontrivial solution for the above equation via truncation argument and the mountain pass theorem.

Motivated by the above works, the main purpose of this paper is to study the existence of sign-changing and signed solutions of (1.1) under the conditions $(K_1)$, $(K_2)$ and $(A_1)$–$(A_3)$. To the best of our knowledge, there are no papers about the existence of sign-changing and signed solutions for (1.1) and (1.2) with supercritical growth.

To state our main result, we define the sets $X$ and $X_0$ as

and

where $u|_\Omega$ represents the restriction to $\Omega$ of function

We note that $X$ and $X_0$ are non-empty, since

(see ^[20]). We endows $X$ with the norm defined by

where

(see ^[21]). Moreover, we can take the function

as a norm on $X_0,$ which is equivalent to the usual one defined in (1.3) (see ^[22]). Also, $(X_0, \|\cdot\|)$ is a Hilbert space with a scalar product given by

Lemma 1.1. The embedding $X_0\hookrightarrow L^\nu(\mathbb{R}^n)$ is continuous if $\nu\in [1, 2^*]$ and compact if $\nu\in [1, 2^*),$ where $u\in L^\nu(R^n)$ means $u = 0\ a.e.$ in $\mathbb{R}^n\backslash\Omega.$

It is well known that there is the best fractional critical Sobolev constant, such that

Observing that the energy functional of (1.1) is given by

To the best of our knowledge, the Sobolev embedding theorems no longer hold when $p > 2^*$. On the one hand, it causes the second integral in $\mathcal{J}$ to be divergent, which makes the functional $\mathcal{J}$ cannot be well defined on $X_0$. On the other hand, it leads to the lack of compactness in studying problem (1.1). Hence, we cannot directly use variational methods to prove the existence of sign-changing and signed solutions.To overcome these difficulties, we use a new method, which came from the papers ^[19,23]. The main idea of this method is to reduce the supercritical problem into a subcritical one. In comparison with previous works, this paper has several new features. First, we consider the more general nonlinear term without Ambrosetti-Rabinowitz condition. Second, the nonlinear term involves supcritical growth. Finally, the existence of a sign-changing solution and a signed solution is obtained by combining an appropriate truncation argument with a constrained minimization method and the Moser iteration method. The results in this paper generalize and improve the results in ^[24,25,26]]. There have been no previous studies considering the existence of sign-changing and signed solutions for problems (1.1) and (1.2) involving supcritical growth to the best of our knowledge.

The main result of this paper is the following:

Theorem 1.1. Suppose that $(K_1)$, $(K_2),$ and $(A_1)$–$(A_3)$ are satisfied. Then there exists $\lambda_{*} > 0$ for any $\lambda \in(0, \lambda_*]$, problem (1.1) admits a sign-changing solution and a signed solution.

Remark 1.1. Comparing with ^[4,25,27], we prove the existence of sign-changing solutions of (1.1) without the Ambrosetti-Rabinowitz condition. The results can be regarded as the complementary work of ^[4,25,27]. Moreover, comparing with ^[2,4,27], we consider the supercritical fractional Laplace equations. Our results are new. Therefore, the results of this paper can enrich the results in the previous papers.

Theorem 1.2. Suppose that $(A_1)$–$(A_3)$ are satisfied. Then there exists $\lambda_{**} > 0$, such that, for any $\lambda \in(0, \lambda_{**}]$, the problem (1.2) admits a sign-changing solution and a signed solution.

This paper is organized as follows: In Section 2, we will prove the existence of sign-changing and signed solutions for the truncation problem of (1.1). Section 3 is devoted to completing the proof of Theorems 1.1 and 1.2.

2.   Preliminaries

In this section, we give a truncation argument in order to overcome the lack of compactness in studying critical and supercritical growth. Let $M > 0$ be a constant. For each $\lambda > 0$ and $M > 0$ fixed, we investigate the existence of sign-changing and signed solutions for the following truncation problem:

where

To investigate (2.1), we define the energy functional

by

where

By $(A_1)$ and the standard argument, it is easy to obtain that $I_\lambda\in{C^1(X_{0}, \mathbb{R})}$ and

where $u, v\in X_{0}.$

Let

for any

we have

and

Obviously, the critical points of $I_\lambda$ are equivalent to the weak solutions of problem (2.1). Furthermore, if $u\in X_{0}$ is a solutions of (2.1) and $u^{\pm}\neq 0$ in $\Omega,$ then $u$ is called a sign-changing solution of (2.1). If $u\in X_{0}$ is a solution of (2.1) and $u > 0$ (or $u < 0$) in $\Omega,$ then $u$ is called a signed solution of (2.1).

Next, we consider the minimization problems:

where

and

Theorem 2.1. Suppose that $(K_1)$, $(K_2)$ and $(A_1)$–$(A_3)$ are satisfied. Then, for each $\lambda > 0, M > 0$, problem (2.1) admits a sign-changing solution $u_1\in \mathcal{M}$ and a signed solution $u_2\in \mathcal{N}$. Furthermore,

In the following, we shall give some properties for $\mathcal{M}$ and $\mathcal{N}.$ By $(A_1)$ and $(A_2)$, we easily see that for any $\varepsilon > 0$, there exists $C_\varepsilon > 0$ such that

for all $t\in \mathbb{R}$ and $2 < q < 2^*$. By $(A_1)$–$(A_3)$, we easily deduce that

and

First, we show that the sets $\mathcal{M}$ and $\mathcal{N}$ are nonempty in $X_0$, and then we seek critical points of $I_\lambda$ by constraint minimizations on $\mathcal{M}$ and $\mathcal{N}$.

Lemma 2.1. Suppose that $(K_1)$, $(K_2)$ and $(A_1)$–$(A_3)$ hold.

(1) If $u\in X_0$ with $u^\pm\neq 0$, then there exists a unique pair $(\alpha_u, \beta_u)\in \mathbb{R}_+\times\mathbb{R}_+$ such that $\alpha_uu^++\beta_uu^-\in \mathcal{M}$.

(2) If $u\in X_0 \setminus \{0\}$, then there exists a unique number $t_u > 0$ such that $t_uu\in \mathcal{N}$ and

Proof. (1) For fixed $u\in X_0$ with

we claim the existence of $\alpha_u$ and $\beta_u$.

Set

By (2.6) and $q\in (2, 2^*)$, we can find that

for a sufficiently small $\alpha > 0$ and

for a sufficiently large $\beta > 0$. Therefore, there exist $0 < r < R$ such that

Taking into account (2.10)–(2.12), we deduce

and

Therefore, there exists some point $(\alpha_u, \ \beta_u)$ with

such that

by Miranda's theorem. Thus

Next, we prove the uniqueness of the pair $(\alpha_u, \beta_u)$.

Case 1. $u\in \mathcal{M}$.

Assume $u\in \mathcal{M}$, we have

We obtain

that is

and

Now we prove that there exists a unique pair

such that

If there exists another pair $(\tilde{\alpha}_u, \tilde{\beta}_u)$ such that

then we obtain

and

Assume that $0 < \tilde{\alpha}_u\leq\tilde{\beta}_u$, by using (2.15), we deduce

Multiply the above inequality by $\tilde{\alpha}_u^{-2}$, we obtain

Putting together (2.17) and (2.13), we have

Since $\varphi(t)$ is increasing in $t > 0$, combining (2.7) and (2.18), we obtain

Similarly, by (2.16), it results

which implies $\tilde{\beta}_u\leq 1.$ Then, combining

we have

Case 2. $u\notin \mathcal{M}$.

(1) Assume $u\notin \mathcal{M}$, then there exists a pair $(\alpha_u, \beta_u)$ such that

If there exists another pair $(\hat{\alpha}_u, \hat{\beta}_u)$ such that

Set

and

we have

Since $w\in\mathcal{M}$, we have

So, there exists a unique pair $(\alpha_u, \beta_u)$ such that

(2) For $t > 0,$ let

By (2.6) and Lemma 1.1, for $\varepsilon > 0$ sufficiently small we have

where

Since $q > 2$, we obtain that $h(t) > 0$ for $t > 0$ small. From the Eq (2.9), we easily get that $h(t)\rightarrow -\infty$ as $t\rightarrow +\infty$. Hence $h$ has a positive maximum at

Therefore, $h^\prime (t_u) = 0$ and $t_uu\in \mathcal{N}.$ Obviously, $h^\prime (t) = 0$ is equivalent to

From (2.7), the right side of (2.19) is increasing for $t > 0.$ As a consequence, there exists a unique number $t_u > 0$ such that (2.19) holds. The uniqueness of $t_u$ is proved, and

□

Lemma 2.2. Suppose that $(K_1)$, $(K_2)$, and $(A_1)$–$(A_3)$ hold.

(1) If

for fixed $u\in X_0$ with $u^\pm\neq 0$, then there exists a unique pair

such that

(2) If

for fixed $u\in X_0 \setminus \{0\}$, then there exists a unique number $t_u\in (0, 1]$ such that

Proof. We only prove Lemma 2.2 (1); the proof of Lemma 2.3 (2) is analogous.□

For fixed $u\in X_0$ with $u^\pm\neq 0$, by Lemma 2.1, we obtain that there exist a unique pair $(\alpha_u, \ \beta_u)$ such that

Assume that $\alpha_u \geq \beta_u > 0$. In addition,

Since

it holds

Therefore (2.20) and (2.21) lead to

By (2.7), we have $\alpha_u \leq 1.$ Thus, $0 < \beta_u\leq\alpha_u\leq 1$.

Lemma 2.3. For fixed $u\in X_0$ with $u^\pm\neq 0$, then $(\alpha_u, \beta_u)$ obtained in Lemma 2.2 is the unique maximum point of the function

where

Proof. By Lemma 2.1, it yields that $(\alpha_u, \beta_u)$ is the unique critical point of $\Theta$ in $\mathbb{R}^+\times \mathbb{R}^+$. By (2.9), we can see that

uniformly as

then we can prove that there is no maximum point on the boundary of $(\mathbb{R}^+, \mathbb{R}^+)$. If we suppose that there exists $\bar{\beta}\geq 0$ such that $(0, \bar{\beta})$ is a maximum point of $\Theta$. Since

is an increasing function of $\alpha$ for $\alpha$ sufficiently small, the pair $(0, \bar{\beta})$ cannot be a maximum point of $\Theta$ in $\mathbb{R}^+\times \mathbb{R}^+$. □

Lemma 2.4. Suppose that $(K_1)$, $(K_2)$ and $(A_1)$–$(A_3)$ hold, then

(1)

and

(2) $\ m_1 > 0$ and $m_2 > 0$ can be achieved respectively.

Proof. (1) By Lemmas 2.1 and 2.3, it is easy to see that

and

(2) For $u\in \mathcal{M}$, we obtain

By (2.6), for any $\varepsilon > 0$ sufficiently small, we have

Taking

then for sufficiently small $\rho > 0$ where

we can know

For $u\in\mathcal{M}$, there exists $t > 0$ such that $tu\in S_\rho$. From Lemmas 2.1 and 2.3, we obtain

Therefore,

Let

be such that

then we claim that $\{u_n\}$ is bounded. By contradiction, we may suppose that $\|u_n\|\rightarrow \infty$ as $n\rightarrow \infty$. Let

then $\omega_n\in X_0$ and $\|\omega_n\| = 1$. Passing to a subsequence if necessary, we may assume that there exists $\omega\in X_0$ such that $\omega_n \rightharpoonup \omega \ \text{in}\ X_0, \ \omega_n \rightarrow \omega \ \text{in}\ L^r(\mathbb{R}^n)$, where $2\leq r < 2^*, \ \omega_n \rightarrow \omega \ \text{a.e. in }\ \mathbb{R}^n.$

If $\omega\neq 0$, then $|\Omega_{\neq}| > 0$, where

In view of

So

Noting that

we have

consequently,

Therefore, by Fatou's lemma and (2.9), we have

which is a contradiction.

If $\omega\equiv 0$, then $\omega_n\rightarrow 0$ in $L^r(\mathbb{R}^n)$. So,

So, by Lemma 2.1, we have

Taking

it is a contradiction. Thus, $\{u_n\}$ is bounded in $X_0$. By Lemma 1.1, up to a subsequence, we can assume that

In addition, $(A_1), (A_2)$, and Lemma 1.1 lead to

Since $u_n\in\mathcal{M}$, then

that is

and

Thanks to (2.6), (2.24), and (2.25), we have

Choose

Thus, there exists a constant $\delta > 0$ such that

By applying (2.24) and (2.25) again, we deduce that

Since $\{u_n\}$ is bounded, by Lemma 1.1, there is

such that

Picking

we have

By (2.22) and (2.26), we have

Thus,

By Lemma 2.1, there exists

such that

Next, we aim to prove that

Putting together (2.22), (2.24) and Fatou's lemma, we deduce

By (2.27) and Lemma 2.1, we have

In the similar way, we can obtain

By (2.7), it follows that

Then we have

Therefore, we have that

This completes the proof. □

The proof for $m_2$ is analogous.

Proof. From Lemma 2.4, we get that

Similar to the discussion of the last step of Theorem 1.2 in ^[22], we can obtain that

is a critical point of $I_\lambda$ on $X_0$ and $u_1$ is a sign-changing solution of (2.1). Similarly, we can obtain $u_2\in \mathcal{N}$ is a nontrivial solution of (2.1) and

Thanks to $u_1^\pm\neq 0$, by Lemma 2.2, there exists a unique number $\alpha_{u_1^+} > 0$ such that $\alpha_{u_1^+}u_1^+ \in \mathcal{N}.$ Similarly, there is a unique number $\beta_{u_1^-} > 0$ such that $\beta_{ u_1^-}u_1^- \in \mathcal{N}.$ Therefore, by (2.3), (2.4), and Lemma 2.5, we have

that is, $0 < m_2 < m_1.$ It follows that $m_2 > 0$ cannot be achieved by a sign-changing function; thus, $u_2\in \mathcal{N}$ is a signed solution of (2.1). □

3.   Proof of main results

In this section, we devote ourselves to completing the proof of Theorems 1.1 and 1.2. From the truncation argument in Section 2, we can see that if the solutions $u_1$ and $u_2$ of (2.1) satisfy

Then $u_1\in X_0$ is a sign-changing solution of (1.1), and $u_2\in X_0$ is a signed solution of (1.1). For convenience, for each $\lambda > 0, M > 0$ fixed, we let

Lemma 3.1. Let $u_1$ and $u_2$ be a sign-changing solution and a signed solution of problem (2.1), respectively; then there exists a constant $K > 0$ independent of $\lambda, M > 0$ such that

Proof. From (2.8), we have

And as

Together with (3.1)–(3.3), we have

where

By (3.4), we have

So, there exists a constant $K_1 > 0$ independent of $\lambda, M > 0$ such that

Similarly, we obtain

So, there exists a constant $K_2 > 0$ independent of $\lambda, M > 0$ such that

Taking

then

This completes the proof. □

Lemma 3.2. Let $u_1$ and $u_2$ be a sign-changing solution and a signed solution of problem (2.1), respectively, then there exists a constant $B > 0$ independent on $\lambda$ and $M$ such that

We only prove Lemma 3.2 for $u_1$, the proof for $u_2$ is analogous.

Proof. For $L > 0$ and $\beta > 1,$ set

where

It is easy to obtain that

and

Recall that

It is easy to see that

and

Choose $\zeta(u_1)$ as a test function in (2.3), combining (3.5) and (3.6), we conclude

which gives

By (2.6), for any $\varepsilon > 0$, there exists $C_\varepsilon > 0$ such that

Let

by (3.7) and (3.8) and H$\ddot{o}$lder's inequality, it holds that

where

It is obvious that $2t\in(2, 2^*)$.

Together with Lemma 1.1 and (1.6), we have

Therefore, by (3.9) and (3.10), we obtain

where $C_7 > 0$. Letting $L\rightarrow \infty$,

Thus

Let

then $\alpha > 1$. Taking $\beta = \alpha$, we have

Taking $\beta = \alpha^2$, we have

Therefore, we have

Taking $\beta = \alpha^i, i\in \mathbb{N}$, we have

Letting $i\rightarrow \infty$, we can know that

Finally, by (3.10) and Lemma 3.1, there exists $C_9 > 0$ such that

Therefore, it follows from (3.11) and

there exists a constant $B > 0$ independent on $\lambda$ and $M$, such that

This completes the proof. □

Proof of Theorem 1.1. By Lemma 3.2, there exists a positive constant $B$ independent on $\lambda$ and $M$ such that

Thus, for large $M > 0,$ we can choose small $\lambda_* > 0$ such that

for all $\lambda\in(0, \lambda_*].$ By Theorem 2.1, problem (1.1) admits a sign-changing solution and a signed solution for $\lambda\in(0, \lambda_*]$.

This completes the proof. □

Proof of Theorem 1.2. We take

then it is obvious that $K(x)$ satisfies the conditions $(K_1), (K_2)$ and problem (1.1) turns into problem (1.2). By using [4,Lemma 5], we can obtain that

Thus, the assertion of Theorem 1.2 follows from Theorem 1.1. □

4.   Conclusions

In this study, we have investigated the existence of sign-changing and signed solutions for nonlinear elliptic equations driven by nonlocal integro-differential operators with critical or supercritical nonlinearity. The main idea of this paper is to reduce the supercritical problem into a subcritical one. In comparison with previous works, this paper has several new features. First, we consider the more general nonlinear term without Ambrosetti-Rabinowitz condition. Second, the nonlinear term involves supcritical growth. Finally, the existence of a sign-changing solution and a signed solution is obtained by combining an appropriate truncation argument with a constrained minimization method and the Moser iteration method. In the future, our work will focus on the existence of normalized solutions to the nonlinear elliptic equations driven by nonlocal integro-differential operators with critical or supercritical nonlinearity.

Use of Generative-AI tools declaration

The authors declares they have not used artificial intelligence (AI) tools in the creation of this article.

Acknowledgments

Supported by the National Natural Science Foundation of China (61803236), Natural Science Foundation of Shandong Province (ZR2018MA022).

Conflict of interest

All authors declare that there are no conflicts of interest in this paper.