[Paper Review] Existence, multiplicity and concentration for a class of fractional $p\&q$ Laplacian problems in $\mathbb{R}^{N}$

Claudianor O. Alves, Vincenzo Ambrosio+1 more|arXiv (Cornell University)|Jan 30, 2019

Nonlinear Partial Differential Equations45 references11 citations

TL;DR

This paper establishes the existence, multiplicity, and concentration of nontrivial nonnegative solutions for a class of fractional $p\&q$ Laplacian equations in $\mathbb{R}^N$ with a small parameter $\varepsilon > 0$. Using variational methods, Nehari manifold techniques, and Ljusternik-Schnirelmann category theory without assuming the Ambrosetti-Rabinowitz condition, the authors prove that solutions concentrate around points where the potential $V(x)$ achieves its minimum $V_0$, and the number of solutions is bounded below by the category of the set $\{x \in \mathbb{R}^N : V(x) = V_0\}$, with uniform boundedness of solutions established via a Moser iteration argument.

ABSTRACT

In this work we consider the following class of fractional $p\&q$ Laplacian problems \begin{equation*} (-\Delta)_{p}^{s}u+ (-\Delta)_{q}^{s}u + V(\varepsilon x) (|u|^{p-2}u + |u|^{q-2}u)= f(u) \mbox{ in } \mathbb{R}^{N}, \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $1< p<q<\frac{N}{s}$, $(-\Delta)^{s}_{t}$, with $t\in \{p,q\}$, is the fractional $t$-Laplacian operator, $V:\mathbb{R}^{N} ightarrow \mathbb{R}$ is a continuous potential and $f:\mathbb{R} ightarrow \mathbb{R}$ is a $\mathcal{C}^{1}$-function with subcritical growth. Applying minimax theorems and the Ljusternik-Schnirelmann theory, we investigate the existence, multiplicity and concentration of nontrivial solutions provided that $\varepsilon$ is sufficiently small.

Motivation & Objective

• To establish the existence of nontrivial nonnegative solutions for a class of fractional $p\&q$ Laplacian equations in $\mathbb{R}^N$ with a small parameter $\varepsilon > 0$.
• To investigate the multiplicity of solutions using the Ljusternik-Schnirelmann category theory without requiring the Ambrosetti-Rabinowitz condition.
• To analyze the concentration behavior of solutions as $\varepsilon \to 0$, showing they concentrate around points minimizing the potential $V(x)$.
• To prove the uniform boundedness of all solutions via a variant of the Moser iteration method.

Proposed method

• Variational methods are employed to study the energy functional associated with the fractional $p\&q$ Laplacian problem.
• The Nehari manifold is used to handle the nonlinearity and to establish the existence of a ground state solution.
• Ljusternik-Schnirelmann category theory is applied to the Nehari manifold to derive multiplicity results.
• A homotopy argument involving the deformation retraction $\beta_\varepsilon \circ \Phi_\varepsilon$ is used to relate the category of the solution set to that of the minimum set of the potential $V(x)$.
• A modified Moser iteration scheme is applied to the weak solution to prove its essential boundedness, establishing $L^\infty$-regularity.
• The proof of the Palais-Smale condition on the Nehari manifold is established without the Ambrosetti-Rabinowitz condition, a key technical novelty.

Experimental results

Research questions

• RQ1 Under what conditions does the fractional $p\&q$ Laplacian problem in $\mathbb{R}^N$ admit at least one nontrivial nonnegative solution for small $\varepsilon > 0$?
• RQ2 How many nontrivial nonnegative solutions does the problem admit, and how does this number relate to the topology of the set $\{x \in \mathbb{R}^N : V(x) = V_0\}$?
• RQ3 Where do the solutions concentrate as $\varepsilon \to 0$, and what is the precise asymptotic behavior of their mass distribution?
• RQ4 Can the boundedness of solutions be established without assuming the Ambrosetti-Rabinowitz condition, and if so, how?

Key findings

• For sufficiently small $\varepsilon > 0$, problem (1.1) admits a nonnegative ground state solution $u_\varepsilon$ that concentrates around a point $x_0 \in \mathbb{R}^N$ where $V(x_0) = V_0$.
• For any $\delta > 0$, there exists $\varepsilon_\delta > 0$ such that for all $\varepsilon \in (0, \varepsilon_\delta)$, the problem has at least $\text{cat}_{M_\delta}(M)$ nontrivial nonnegative solutions, where $M = \{x : V(x) = V_0\}$ and $M_\delta = \{x : \text{dist}(x, M) \leq \delta\}$.
• The concentration is quantified in integral form: for each sequence $\varepsilon_n \to 0$, the solution $u_{\varepsilon_n}$ satisfies $\int_{B_{\varepsilon_n \bar{R}}(y)} f(u_{\varepsilon_n}) u_{\varepsilon_n} \, dx \geq C \varepsilon_n^N$ and $\int_{\mathbb{R}^N \setminus B_{\varepsilon_n \bar{R}}(y)} f(u_{\varepsilon_n}) u_{\varepsilon_n} \, dx < \varepsilon_n^N \delta$ for large $n$, with $C > 0$ and $\bar{R} > 0$.
• All solutions to problem (1.1) are uniformly bounded in $L^\infty(\mathbb{R}^N)$, with $\|u\|_{L^\infty} \leq K$ for some $K > 0$ independent of $\varepsilon$, established via a Moser iteration argument.