Direct Method of Scaling Spheres for the Laplacian and Fractional Laplacian Equations with Hardy-Hénon Type Nonlinearity

In this paper we consider the partial differential equation

where $n>\alpha$, $0<\alpha\leq 2$, and the Hardy-Hénon type nonlinearity $f:(\mathbb{R}^{n}\backslash\{0\})\times[0,+\infty)\to[0,+\infty)$. For $\alpha=2$, $(-\Delta)^{\frac{\alpha}{2}}$ is the classical Laplacians operator, that is, $-\Delta=-\sum_{i=1}^{n}\frac{\partial^{2}}{\partial x_{i}^{2}}$. For $\alpha$ taking any real number between 0 and 2, $(-\Delta)^{\frac{\alpha}{2}}$ is a nonlocal differential operator defined by

where P.V. is the Cauchy principal value. The integral on the right-hand side of (1.2) is well defined for $u\in C_{loc}^{[\alpha],\{\alpha\}+\epsilon}(\mathbb{R}^{n})\cap\mathcal{L}_{\alpha}(\mathbb{R}^{n})$, where $\epsilon>0$ is arbitrary small, $[\alpha]$ denotes the integer part of $\alpha$, $\{\alpha\}=\alpha-[\alpha]$,

For more details, see [46, 13].

For $\alpha=2$, we assume the solution $u\in C^{2}(\mathbb{R}^{n}\backslash\{0\})\cap C(\mathbb{R}^{n})$. For $0<\alpha<2$, we assume the solution $u\in C_{loc}^{[\alpha],\{\alpha\}+\epsilon}(\mathbb{R}^{n}\backslash\{0\})\cap\mathcal{L}_{\alpha}(\mathbb{R}^{n})$.

We need the following assumptions about the Hardy-Hénon type nonlinearity $f(x,u)$.

1. $(f_{0})$

   The nonlinearity $f$ has subcritical growth, i.e.

   $$f(t^{-\frac{2}{n-\alpha}}x,tu(x))/{t^{\frac{n+\alpha}{n-\alpha}}}>f(x,u(x))\;\;\;\;\text{for any $t>1$ and $(x,u)\in(\mathbb{R}^{n}\backslash\{0\})\times\mathbb{R}^{n}_{+}$}.$$

2. $(f_{1})$

   The nonlinearity $f$ is nondecreasing about $u$ in $(\mathbb{R}^{n}\backslash\{0\})\times\mathbb{R}^{n}_{+}$.

3. $(f_{2})$

   There exists a $\sigma<\alpha$ such that $|x|^{\sigma}f(x,u)$ is locally Lipschitz on $u$ in $\mathbb{R}^{n}\times[0,+\infty)$.

4. $(f_{3})$

   There exists a cone $\mathcal{C}\subset\mathbb{R}^{n}$ with vertex at 0, constants $C>0$, $-\alpha<a<+\infty$, $0<p<p_{c}(a)$ such that

   $$f(x,u)\geq C|x|^{a}u^{p}(x)\;\;\;\;\text{in $\mathcal{C}\times[0,+\infty)$.}$$

The Liouville type theorems of the Lane-Emden equations, Hardy-Hénon equations and equations with general nonlinearities have been extensively studied (see [35, 52, 33, 34, 31, 49, 5, 6, 7, 53, 17, 29]). In the study of Liouville-type theorems concerning the nonexistence of non-trivial solutions, the most commonly employed approach is the method of moving planes. This method has been extensively explored in prior works such as [3, 1, 8, 9, 32] and further developed in recent research contributions, including [39, 10, 15, 11, 12, 9, 20]. However, the method of moving planes has limitations. For example, it requires the solution to be bounded or integrable. In [23], Dai and Qin introduced the method of scaling spheres, which offers significant advantages. Notably, in the fractional case, it does not necessitate any boundedness or integrability of the solution, in contrast to the method of moving planes. This variation can potentially provide novel perspectives for exploring Liouville-type theorems. Another distinct feature is that the potential term in nonlinearity does not need to be strictly decreasing after the Kelvin transform, as required by the method of moving planes. This extension greatly broadens the range of the exponent $p$, from $1<p<\frac{p+a+\alpha}{n-\alpha}$ to $0<p<\frac{p+2a+\alpha}{n-\alpha}$. To be precise, by the direct method of scaling spheres, Dai and Qin expanded the range of $p$ in the following ways (only relevant parts of the results are listed here):

1. 1.

   For nonnegative solution $u$ of the equation

   $$(-\Delta)^{\frac{\alpha}{2}}u(x)=|x|^{a}u^{p}(x),\;\;\;\;x\in\mathbb{R}^{n},$$

   where $0<\alpha\leq 2$, $n>\alpha$, $-\alpha<a<+\infty$, $0<p<\frac{p+2a+\alpha}{n-\alpha}$, $u$ can only be trivial.

2. 2.

   For a non-negative solution $u$ of the equation

   $$\begin{cases}(-\Delta)^{\frac{\alpha}{2}}u(x)=f(x,u),\;\;\;\;&x\in\mathbb{R}^{n}_{+},\\ u(x)=0,\;\;\;\;&x\in(\mathbb{R}^{n}_{+})^{c},\end{cases}$$

   where $0<\alpha\leq 2$, $n>\alpha$, and Hardy-Hénon type nonlinearity $f$ is subcritical and satisfies assumptions $(f_{1})$, $(f_{2})$ and $(f_{3})$, $u$ can only be trivial.

Dai and Qin also indicated that the problems addressable with the direct method of scaling spheres are not limited to the ones mentioned above. In fact, they explicitly point out that method of scaling spheres can also be employed to equation

with Hardy-Hénon type nonlinearity $f$. By further developing their idea and applying the method of scaling spheres, we derive the following Liouville-type theorem.

To establish Theorem 1.2, we begin by giving the equivalence (Theorem 2.1) between (1.1) and the corresponding integral equation

Then we establish the narrow region principle (Theorem 3.2), and use the (direct) method of scaling spheres.

In particular, $f(x,u)=|x|^{a}u^{p}$ with $a>-\alpha$ satisfies all the assumptions in Theorem 1.2. Therefore, this result covers the Liouville-type theorem presented in [23] for the equation $(-\Delta)^{\frac{\alpha}{2}}u(x)=|x|^{a}u^{p}$. In [47], Yu applied the method of moving planes in the integral form to get the Liouville-type theorem of the integral equation

where $f(u)$ is continuous and non-decreasing, and $f(t)/t^{\frac{n+\alpha}{n-\alpha}}$ is non-decreasing. Also in [14], Chen, Li and Zhang applied the direct method of moving spheres to get the Liouville-type theorem of the partial differential equation

where $f(u)$ is locally bounded and $f(t)/t^{\frac{n+\alpha}{n-\alpha}}$ is non-decreasing. Distinguishing our research from previous works where the nonlinearity $f(u)$ depends solely on the solution $u$, we midly extend the scope by considering the spatial dependence in the nonlinearity $f(x,u(x))$. For additional research on fractional Laplacians with non-linearity, see [22, 23, 44, 28, 27, 43, 26, 45, 38, 37, 36, 48]. For further studies using the method of scaling spheres, see [41, 40, 42, 25, 21, 19, 30, 24, 18, 4]

As mentioned in [23], when dealing with fractional Laplacian, it is often more convenient to apply the Hardy-Littlewood-Sobolev inequality and the method of scaling spheres in integral form. We hope to apply the latter approach to the higher-order fractional Laplacian equations with general nonlinearity and get the Liouville-type theorem.

The paper is organized as follows. In Section 2 we prove the equivalence between PDE (1.1) and IE (1.4). In Section 3 we leverage contradiction arguments, the narrow region principle (Theorem 3.2) and the direct method of scaling spheres to prove Theorem 1.2.

We first show the equivalence between (1.1) and the integral equation i.e. the solution of the PDE also solves the IE (1.4). The idea comes from [51, 23].

One can check that if $u$ solves the IE then it solves the PDE. Thus we only check that any solution of the PDE is also a solution of IE. The idea of proof basically comes from [23], Theorem 2.1.

Moreover, we can derive a lower bound estimate of $u$ from the equivalent IE. To this end, we can see that for any $|x|\geq 1$,

Next we will apply the method of scaling spheres to give the following lower bounds for positive $u$, which will lead a contraction to IE (1.4).