Existence And Uniqueness Of Some PDE

May 14, 2025 by ADMIN 37 views

**Existence and Uniqueness of Solutions to a Class of Partial Differential Equations**

Introduction

Partial Differential Equations (PDEs) are a fundamental tool in modeling various phenomena in physics, engineering, and other fields. The existence and uniqueness of solutions to PDEs are crucial in understanding the behavior of these phenomena. In this article, we will discuss the existence and uniqueness of solutions to a specific class of PDEs, which is given by:

\begin{align} \begin{cases} (-\Delta)^\alpha u(x) + \frac{f(x)}{u^{\beta}(x)} = c \quad & \text{for $x \in \mathbb{R}^n$ } \ u(x) \to 0 \quad & \text{as $|x| \to \infty$ } \end{cases} \end{align}

where $(-\Delta)^\alpha$ is the fractional Laplacian operator, $f(x)$ is a given function, $c$ is a constant, and $\alpha$ and $\beta$ are positive real numbers.

Preliminaries

Before we proceed to the main discussion, let us recall some basic concepts and results in functional analysis and partial differential equations.

Fractional Laplacian Operator

The fractional Laplacian operator $(-\Delta)^\alpha$ is defined as:

(-\Delta)^\alpha u(x) = \int_{\mathbb{R}^n} \frac{u(x) - u(y)}{|x-y|^{n+2\alpha}} dy

for $u \in L^1_{loc}(\mathbb{R}^n)$ .

Weak Solutions

A weak solution to the PDE is a function $u \in L^1_{loc}(\mathbb{R}^n)$ that satisfies the following equation:

\int_{\mathbb{R}^n} (-\Delta)^\alpha u(x) \phi(x) dx + \int_{\mathbb{R}^n} \frac{f(x)}{u^{\beta}(x)} \phi(x) dx = c \int_{\mathbb{R}^n} \phi(x) dx

for all $\phi \in C^\infty_c(\mathbb{R}^n)$ .

Existence of Solutions

To show the existence of solutions to the PDE, we will use the following theorem:

Theorem 1 (Mountain Pass Theorem).: Let $X$ be a Banach space and $I \in C^1(X, \mathbb{R})$ be a functional such that $I(0) = 0$ and $I$ satisfies the Palais-Smale condition. Suppose that there exist $r > 0$ and $\delta > 0$ such that $I(u) \geq \delta$ for all $u \in X$ with $\|u\| = r$ . Then, there exists $u \in X$ such that $I(u) = \inf_{v \in X} \max_{t \in [0,1]} I(tv)$ . Moreover, if $I$ is even and $I(u) \geq 0$ for all $u \in X$ , then the solution $u$ is non-negative.

We will apply this theorem to the functional:

I(u) = \frac{1}{2} \int_{\mathbb{R}^n} (-\Delta)^\alpha u(x) u(x) dx + \frac{1}{\beta} \int_{\mathbb{R}^n} \frac{f(x)}{u^{\beta}(x)} u(x) dx - c \int_{\mathbb{R}^n} u(x) dx

which is defined on the space $X = L^1_{loc}(\mathbb{R}^n)$ .

Proof of Existence

To show that the functional $I$ satisfies the Palais-Smale condition, we need to prove that any sequence $\{u_n\} \subset X$ such that $I(u_n) \to \infty$ as $n \to \infty$ has a convergent subsequence.

Let $\{u_n\} \subset X$ be a sequence such that $I(u_n) \to \infty$ as $n \to \infty$ . We need to show that there exists a subsequence $\{u_{n_k}\}$ such that $u_{n_k} \to u$ in $X$ as $k \to \infty$ .

Since $I(u_n) \to \infty$ as $n \to \infty$ , we have that:

\int_{\mathbb{R}^n} (-\Delta)^\alpha u_n(x) u_n(x) dx + \int_{\mathbb{R}^n} \frac{f(x)}{u_n^{\beta}(x)} u_n(x) dx - c \int_{\mathbb{R}^n} u_n(x) dx \to \infty

as $n \to \infty$ . This implies that:

\int_{\mathbb{R}^n} (-\Delta)^\alpha u_n(x) u_n(x) dx \to \infty

as $n \to \infty$ . Since the fractional Laplacian operator is a bounded operator, we have that:

\|u_n\|_{L^2(\mathbb{R}^n)} \to \infty

as $n \to \infty$ . This implies that:

\frac{1}{\|u_n\|_{L^2(\mathbb{R}^n)}} \int_{\mathbb{R}^n} \frac{f(x)}{u_n^{\beta}(x)} u_n(x) dx \to 0

as $n \to \infty$ . Therefore, we have that:

\int_{\mathbb{R}^n} (-\Delta)^\alpha u_n(x) u_n(x) dx \to \infty

as $n \to \infty$ .

Since the fractional Laplacian operator is a bounded operator, we have that:

\|u_n\|_{L^2(\mathbb{R}^n)} \to \infty

as $n \to \infty$ . This implies that:

\frac{1}{\|u_n\|_{L^2(\mathbb{R}^n)}} \int_{\mathbb{R}^n} (-\Delta)^\alpha u_n(x) u_n(x) dx \to 0

as $n \to \infty$ . Therefore, we have that:

\int_{\mathbb{R}^n} \frac{f(x)}{u_n^{\beta}(x)} u_n(x) dx \to 0

as $n \to \infty$ .

Since $f(x)$ is a bounded function, we have that:

\int_{\mathbb{R}^n} \frac{f(x)}{u_n^{\beta}(x)} u_n(x) dx \leq \|f\|_{L^\infty(\mathbb{R}^n)} \int_{\mathbb{R}^n} \frac{1}{u_n^{\beta}(x)} u_n(x) dx

as $n \to \infty$ . This implies that:

\int_{\mathbb{R}^n} \frac{1}{u_n^{\beta}(x)} u_n(x) dx \to 0

as $n \to \infty$ .

Since $u_n(x) \to 0$ as $|x| \to \infty$ , we have that:

\int_{\mathbb{R}^n} \frac{1}{u_n^{\beta}(x)} u_n(x) dx \leq \int_{\mathbb{R}^n} \frac{1}{u_n^{\beta}(x)} |u_n(x)| dx

as $n \to \infty$ . This implies that:

\int_{\mathbb{R}^n} \frac{1}{u_n^{\beta}(x)} |u_n(x)| dx \to 0

as $n \to \infty$ .

Since $u_n(x) \to 0$ as $|x| \to \infty$ , we have that:

\int_{\mathbb{R}^n} \frac{1}{u_n^{\beta}(x)} |u_n(x)| dx \leq \int_{\mathbb{R}^n} \frac{1}{|u_n(x)|^{\beta-1}} dx

as $n \to \infty$ . This implies that:

\int_{\mathbb{R}^n} \frac{1}{|u_n(x)|^{\beta-1}} dx \to 0

as $n \to \infty$ .

Since $\beta > 1$ , we have that:

\int_{\mathbb{R}^n} \frac{1}{|u_n(x)|^{\beta-1}} dx \leq \int_{\mathbb<br/> **Q&A: Existence and Uniqueness of Solutions to a Class of Partial Differential Equations** =====================================================================================

Q: What is the main goal of this article?

A: The main goal of this article is to discuss the existence and uniqueness of solutions to a specific class of partial differential equations (PDEs), which is given by:

\begin{align} \begin{cases} (-\Delta)^\alpha u(x) + \frac{f(x)}{u^{\beta}(x)} = c \quad & \text{for $x \in \mathbb{R}^n$ } \ u(x) \to 0 \quad & \text{as $|x| \to \infty$ } \end{cases} \end{align}

Q: What is the fractional Laplacian operator?

A: The fractional Laplacian operator $(-\Delta)^\alpha$ is defined as: