Lebesgue and Sobolev spaces with variable exponents

A critical Kirchhoff‐type problem driven by a p (·)‐fractional Laplace operator with variable s (·) ‐order

This paper is concerned with the existence and multiplicity of solutions for the fractional variable order Choquard type problem (−Δ)p(⋅)s(⋅)u(x)+(−Δ)q(⋅)s(⋅)u(x)=λ|u(x)|β(x)−2u(x)+∫ΩF(y,u(y))|x−y|...

Existence and multiplicity results for p(⋅)&q(⋅) fractional Choquard problems with variable order

Abstract We present the theory of a new fractional Sobolev space in complete manifolds with variable exponent. As a result, we investigate some of our new space’s qualitative properties, such as completeness, reflexivity, separability, and density. We also show that continuous and compact embedding results are valid. We apply the conclusions of this study to the variational analysis of a class of fractional $p(z, \cdot )$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>,</mml:mo><mml:mo>⋅</mml:mo><mml:mo>)</mml:mo></mml:math> -Laplacian problems involving potentials with vanishing behavior at infinity as an application. 

/pdf/on-a-new-fractional-sobolev-space-with-variable-exponent-on-12he61bj.pdf

On a new fractional Sobolev space with variable exponent on complete manifolds

This paper deals with Strauss and Lions-type theorems for fractional Sobolev spaces with variable exponent $$W^{s,p(.),{\tilde{p}}(.,.)} (\Omega )$$
 , when p and $${\tilde{p}}$$
 satisfy some conditions. As application, we study the existence of solutions for a class of Kirchhoff–Choquard problem in $${\mathbb {R}}^N$$
 .

Strauss and Lions Type Theorems for the Fractional Sobolev Spaces with Variable Exponent and Applications to Nonlocal Kirchhoff–Choquard Problem

In this article, we study the existence/multiplicity results for the variable order nonlocal Choquard problem with variable exponents (−Δ)p(⋅)s(⋅)u(x)=λ|u(x)|α(x)−2u(x)+∫ΩF(y,u(y))|x−y|μ(x,y)dyf(x,...

Variable order nonlocal Choquard problem with variable exponents

In this article, we study the existence/multiplicity results for the following variable order nonlocal Choquard problem with variable exponents (-\Delta)_{p(\cdot)}^{s(\cdot)}u(x)&=\lambda|u(x)|^{\alpha(x)-2}u(x)+ \left(\DD\int_\Omega\frac{F(y,u(y))}{|x-y|^{\mu(x,y)}}dy\right)f(x,u(x)), x\in \Omega, u(x)&=0, x\in \mathbb R^N\setminus\Omega, where $\Omega\subset\mathbb R^N$ is a smooth and bounded domain, $N\geq 2$, $p,s,\mu$ and $\alpha$ are continuous functions on $\mathbb R^N\times\mathbb R^N$ and $f(x,t)$ is Carathedory function. Under suitable assumption on $s,p,\mu,\alpha$ 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.

In this article we study the existence of weak solution, existence of ground state solution using Nehari manifold and existence of infinitely many solutions using Fountain theorem and Dual fountain theorem for a class of doubly nonlocal Kirchhoff-Choquard type equations involving the variable-order fractional $p(\cdot)-$ Laplacian operator. Here the nonlinearity does not satisfy the well known Ambrosetti-Rabinowitz type condition.

On a class of Kirchhoff-Choquard equations involving variable-order fractional $p(\cdot)-$ Laplacian and without Ambrosetti-Rabinowitz type condition

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} (-\Delta)_{p(\cdot)}^{s(\cdot)}u(x)&=\lambda|u(x)|^{\alpha(x)-2}u(x)+\left(\DD\int_\Omega\frac{F(y,u(y))}{|x-y|^{\mu(x,y)}}dy\right)f(x,u(x)),\\ &~\hspace{6cm} x\in \Omega, \\ u(x)&=0 ,\hspace{20mm} x\in \Omega^c:=\mathbb R^N\setminus\Omega, \end{array} \end{equation*} where $\Om\subset\mathbb R^N$ is a smooth and bounded domain, $N\geq 2$, $p,s,\mu$ and $\alpha$ are continuous functions on $\mathbb R^N\times\mathbb R^N$ and $f(x,t)$ is continuous function with $F(x,t):=\displaystyle\int_{0}^{t} f(x,s)ds$. Under suitable assumption on $s,p,\mu,\alpha$ 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.

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

. In this article, we consider the following modiﬁed quasilinear critical Kirchhoﬀ-Schr¨odinger problem involving Stein-Weiss type nonlinearity: where λ > 0 is a parameter, N = 0 < µ < N , 0 < 2 β + µ < N , 2 ≤ q < 2 p ∗ . Here p ∗ = NpN − p is the Sobolev critical exponent and p ∗ β,µ := p 2 (2 N − 2 β − µ ) N − 2 is the critical exponent with respect to the doubly weighted Hardy-Littlewood-Sobolev inequality (also called Stein- Weiss type inequality). Then by establishing a concentration-compactness argument for our problem, we show the existence of inﬁnitely many nontrivial solutions to the equations with respect to the parameter λ by using Krasnoselskii’s genus theory, symmetric mountain pass theorem and Z 2 - symmetric version of mountain pass theorem for diﬀerent range of q . We further show that these solutions belong to L ∞ ( R N ).

/pdf/multiplicity-results-for-p-kirchhoff-modified-schrodinger-3oxzgmia.pdf

Reshmi Biswas

Papers

Variable order nonlocal Choquard problem with variable exponents

Variable order nonlocal Choquard problem with variable exponents

On a class of Kirchhoff-Choquard equations involving variable-order fractional $p(\cdot)-$ Laplacian and without Ambrosetti-Rabinowitz type condition

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

Multiplicity results for $p$-Kirchhoff modified Schrödinger equations with Stein-Weiss type critical nonlinearity in $\mathbb R^N$