[Paper Review] On superlinear Fractional p-Laplacian in R^n

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

Nonlinear Differential Equations Analysis3 citations

TL;DR

This paper investigates the existence of nontrivial solutions to a fractional p-Laplacian equation with a subcritical p-superlinear nonlinearity in R^n, where p ≥ 2 and 0 < s < 1. By applying the mountain pass theorem under the Cerami condition, the authors establish the existence of at least one nontrivial weak solution under suitable assumptions on the potential V(x) and the nonlinearity f(x,u).

ABSTRACT

In this article we are interested in the following fractional $p$-Laplacian equation in $\mathbb{R}^n$ $(-\Delta)_{p}^{\alpha}u$ + $V(x)u^{p-2}u = f(x,u) {in} \mathbb{R}^{n}$ where $p\geq 2$, $0< s < 1$, $n\geq 2$ and subcritical p-superlinear nonlinearity. By using mountain pass theorem with Cerami condition and existence result is obtained.

Motivation & Objective

• To study the existence of nontrivial solutions to a fractional p-Laplacian equation with subcritical p-superlinear nonlinearity in R^n.
• To analyze the behavior of solutions under the influence of a potential V(x) and a nonlinear term f(x,u).
• To establish existence results using variational methods under appropriate growth and regularity conditions.
• To extend the applicability of the mountain pass theorem in the context of fractional p-Laplacian operators with superlinear growth.

Proposed method

• Formulate the fractional p-Laplacian equation using the nonlocal operator (−Δ)p^s with s ∈ (0,1) and p ≥ 2.
• Define the energy functional associated with the equation in a suitable Sobolev-type space, ensuring the functional is well-defined and continuously differentiable.
• Apply the mountain pass theorem to the energy functional under the Cerami condition to guarantee the existence of a critical point.
• Verify that the Cerami condition holds under the given assumptions on f(x,u) and V(x), ensuring the existence of a bounded Palais-Smale sequence.
• Use the subcritical and p-superlinear growth of f(x,u) to ensure the functional satisfies the geometric conditions required by the mountain pass theorem.
• Establish the existence of at least one nontrivial weak solution via the linking structure and compactness arguments.

Experimental results

Research questions

• RQ1 Under what conditions does the fractional p-Laplacian equation with a potential V(x) and p-superlinear nonlinearity f(x,u) admit a nontrivial weak solution in R^n?
• RQ2 How does the interplay between the nonlocal operator (−Δ)p^s and the potential V(x) affect the existence of solutions?
• RQ3 Can the mountain pass theorem be effectively applied to this class of fractional p-Laplacian problems under the Cerami condition?
• RQ4 What growth and regularity assumptions on f(x,u) ensure the existence of a nontrivial solution in the subcritical regime?
• RQ5 What role does the parameter s ∈ (0,1) play in the solvability of the equation when p ≥ 2?

Key findings

• The paper establishes the existence of at least one nontrivial weak solution to the fractional p-Laplacian equation in R^n under the given conditions.
• The solution is obtained via the mountain pass theorem, which guarantees a critical point of the associated energy functional.
• The Cerami condition is verified for the energy functional, ensuring the existence of a bounded Palais-Smale sequence.
• The nonlinearity f(x,u) is assumed to be subcritical and p-superlinear, which is essential for satisfying the mountain pass geometry.
• The potential V(x) is assumed to be measurable and bounded from below, ensuring the functional is well-defined and coercive in part.
• The result holds for all n ≥ 2, p ≥ 2, and s ∈ (0,1), extending the applicability of variational methods to this class of nonlocal equations.