[Paper Review] Existence and symmetry result for Fractional p-Laplacian in $\mathbb{R}^{n}$

César E. Torres Ledesma|arXiv (Cornell University)|Dec 10, 2014

Nonlinear Partial Differential Equations13 references3 citations

TL;DR

This paper establishes the existence of a nontrivial weak solution to a fractional p-Laplacian equation in $\mathbb{R}^n$ with a potential and subcritical, superlinear nonlinearity, using the mountain pass theorem under the Cerami condition. It further proves that this solution is radially symmetric via symmetric decreasing rearrangement and the Polya-Szegö inequality for fractional Sobolev spaces.

ABSTRACT

In this article we are interested in the following fractional $p$-Laplacian equation in $\mathbb{R}^n$ \begin{eqnarray*} &(-Δ)_{p}^αu + V(x)u^{p-2}u = f(x,u) \mbox{ in } \mathbb{R}^{n}, \end{eqnarray*} where $p\geq 2$, $0< s < 1$, $n\geq 2$ and subcritical p-superlinear nonlinearity. By using mountain pass theorem with Cerami condition we prove the existence of nontrivial solution. Furthermore, we show that this solution is radially simmetry.

Motivation & Objective

• To establish the existence of nontrivial weak solutions for a class of fractional p-Laplacian equations with a potential term and subcritical, superlinear nonlinearity in $\mathbb{R}^n$.
• To analyze the symmetry properties of the obtained solution, specifically proving radial symmetry.
• To extend variational methods, particularly the mountain pass theorem with the Cerami condition, to the nonlocal, nonlinear setting of the fractional p-Laplacian.
• To apply symmetric decreasing rearrangement techniques to show that the energy functional of the solution is minimized by its rearranged, radially symmetric version.
• To verify that the energy functional satisfies the necessary compactness and geometric conditions for critical point theory in the fractional Sobolev space $X^s$.

Proposed method

• Formalize the problem in the fractional Sobolev space $X^s = \{u \in W^{s,p}(\mathbb{R}^n) : \int_{\mathbb{R}^n} V(x)|u|^p dx < \infty\}$, equipped with the norm $\|u\|_{X^s} = \left( [u]_{s,p}^p + \|V^{1/p}u\|_p^p \right)^{1/p}$.
• Define the energy functional $I(u) = \frac{1}{p}\|u\|_{X^s}^p - \int_{\mathbb{R}^n} F(x,u) dx$, where $F(x,t) = \int_0^t f(x,s) ds$, and show it is $C^1$ under the given assumptions.
• Verify that $I$ satisfies the mountain pass geometry: $I(0) = 0$, there exist $\rho, \alpha > 0$ such that $I(u) \geq \alpha$ for $\|u\|_{X^s} = \rho$, and $I(e) < 0$ for some $e$ with $\|e\|_{X^s} > \rho$.
• Apply the Cerami condition to ensure that any $(C)_c$ sequence has a convergent subsequence, enabling the use of the mountain pass theorem.
• Use symmetric decreasing rearrangement $u^*$ to show that $I(u^*) \leq I(u)$ for all $u \in X^s$, leveraging the fractional Polya-Szegö inequality and the radial symmetry of $V$.
• Employ Ekeland's variational principle on the rearranged path to extract a Cerami sequence converging to a critical point, which is shown to be radially symmetric.

Experimental results

Research questions

• RQ1 Under what conditions does the fractional p-Laplacian equation $(-\Delta)_p^s u + V(x)u^{p-2}u = f(x,u)$ in $\mathbb{R}^n$ admit a nontrivial weak solution?
• RQ2 Can the mountain pass theorem with the Cerami condition be successfully applied to the energy functional associated with the fractional p-Laplacian in an unbounded domain?
• RQ3 Does the solution obtained via variational methods inherit radial symmetry under the given assumptions on $f$ and $V$?
• RQ4 How do symmetric decreasing rearrangements affect the energy functional in the context of fractional Sobolev spaces?
• RQ5 What role does the potential $V(x)$, assumed to be positive and radially increasing, play in ensuring the symmetry of the solution?

Key findings

• The energy functional $I$ associated with the fractional p-Laplacian equation satisfies the mountain pass geometry and the Cerami condition, ensuring the existence of a nontrivial critical point.
• A nontrivial weak solution exists for the equation $(-\Delta)_p^s u + V(x)u^{p-2}u = f(x,u)$ in $\mathbb{R}^n$ under the assumptions $(V)$, $(f_1)$–$(f_3)$, as guaranteed by the mountain pass theorem.
• The solution obtained via the mountain pass procedure is radially symmetric, as shown by applying symmetric decreasing rearrangement and the fractional Polya-Szegö inequality.
• The rearranged functional satisfies $I(u^*) \leq I(u)$, and equality holds only if $u$ is already radially symmetric and non-increasing, implying the minimizer must be symmetric.
• The Cerami sequence extracted via Ekeland's principle converges strongly in $X^s$ to a critical point $w$ with $I(w) = c$ and $I'(w) = 0$, confirming the existence of a nontrivial solution.
• The radial symmetry of the solution is preserved due to the radial structure of $V(x)$ and the fact that the rearrangement reduces the energy, implying the minimizer must be symmetric.