[Paper Review] Multiplicity and uniform estimate for a class of variable order fractional $p(x)$-Laplacian problems with concave-convex nonlinearities

Reshmi Biswas, Sweta Tiwari|arXiv (Cornell University)|Oct 30, 2018

Nonlinear Partial Differential Equations10 references6 citations

TL;DR

This paper establishes existence and multiplicity of solutions for a variable order fractional $p(x)$-Laplacian problem with concave-convex nonlinearities and nonlocal Choquard-type terms. By developing a variable-exponent Hardy-Sobolev-Littlewood inequality and applying variational methods, it proves the existence of at least two nontrivial weak solutions under suitable conditions on the exponents and nonlinearity.

ABSTRACT

In this article, we study the existence/multiplicity results for the following variable order nonlocal Choquard problem with variable exponents \begin{equation*} \begin{array}{rl} (-Δ)_{p(\cdot)}^{s(\cdot)}u(x)&=λ|u(x)|^{α(x)-2}u(x)+\left(\DD\int_Ω\frac{F(y,u(y))}{|x-y|^{μ(x,y)}}dy ight)f(x,u(x)),\\ &~\hspace{6cm} x\in Ω, \\ u(x)&=0 ,\hspace{20mm} x\in Ω^c:=\mathbb R^N\setminusΩ, \end{array} \end{equation*} where $\Om\subset\mathbb R^N$ is a smooth and bounded domain, $N\geq 2$, $p,s,μ$ and $α$ are continuous functions on $\mathbb R^N imes\mathbb R^N$ and $f(x,t)$ is continuous function with $F(x,t):=\displaystyle\int_{0}^{t} f(x,s)ds$. Under suitable assumption on $s,p,μ,α$ and $f(x,t)$, first we study the analogous Hardy-Sobolev-Littlewood-type result for variable exponents suitable for the fractional Sobolev space with variable order and variable exponents. Then we give the existence/multiplicity results for the above equation.

Motivation & Objective

• To investigate the existence and multiplicity of weak solutions for a class of variable order fractional $p(x)$-Laplacian equations with concave-convex nonlinearities.
• To establish a variable-order and variable-exponent version of the Hardy-Sobolev-Littlewood inequality suitable for fractional Sobolev spaces.
• To analyze the interplay between variable exponents $p(x), s(x), \mu(x,y)$, and the nonlinearity $f(x,u)$ in determining solution multiplicity.
• To extend variational methods to nonlocal problems with variable-order operators and nonstandard growth conditions.
• To provide sufficient conditions on the functions $p, s, \mu, \alpha$, and $f$ ensuring the existence of at least two nontrivial weak solutions.

Proposed method

• Derive a generalized Hardy-Sobolev-Littlewood-type inequality for variable exponents and variable order fractional operators.
• Define the energy functional associated with the nonlocal Choquard-type equation in a suitable variable exponent fractional Sobolev space.
• Apply the mountain pass theorem and Ekeland’s variational principle to establish existence of at least one nontrivial weak solution.
• Use the Nehari manifold method to analyze the structure of the energy functional and prove the existence of multiple solutions.
• Employ compactness arguments and Sobolev embeddings in variable exponent spaces to ensure the Palais-Smale condition.
• Verify that the nonlinearity satisfies the necessary growth and continuity conditions to apply critical point theory.

Experimental results

Research questions

• RQ1 Under what conditions on $p(x), s(x), \mu(x,y), \alpha(x)$, and $f(x,u)$ does the variable order $p(x)$-Laplacian problem with nonlocal Choquard term admit at least one weak solution?
• RQ2 Can multiple weak solutions be guaranteed for this class of variable exponent and variable order problems under suitable structural assumptions?
• RQ3 How can the Hardy-Sobolev-Littlewood inequality be extended to the setting of variable-order and variable-exponent fractional integrals?
• RQ4 What role do the variable exponents and the nonlocal integral term play in the compactness and geometry of the associated energy functional?
• RQ5 How do the concave-convex nonlinearities influence the multiplicity of solutions in this nonlocal, nonstandard growth framework?

Key findings

• A new variable-order and variable-exponent Hardy-Sobolev-Littlewood inequality is established, enabling analysis in the relevant fractional Sobolev space framework.
• The existence of at least one nontrivial weak solution is proven using the mountain pass theorem under appropriate growth and coercivity conditions.
• The existence of at least two nontrivial weak solutions is established via the Nehari manifold method and the geometry of the energy functional.
• The nonlocal Choquard-type term is shown to be well-defined and continuous in the variable exponent fractional Sobolev space under the derived inequalities.
• The functional associated with the problem satisfies the Palais-Smale condition in the specified space, ensuring convergence of Palais-Smale sequences.
• The results are valid for smooth, bounded domains $\Omega \subset \mathbb{R}^N$ with $N \geq 2$, under continuity and boundedness assumptions on the variable exponents.