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In this paper, we deal with the mean value theorem for tempered fractional operators, some properties with monotone functions and the Taylor theorem for Caputo tempered fractional derivative. Furthermore, we present a geometric interpretation of the Riemann-Liouville tempered fractional integral as a shadow on the wall. [ABSTRACT FROM AUTHOR]

In this work we generalize some results of the Riemann-Liouville fractional calculus for the tempered case, namely, we deal with some boundedness results of Riemann-Liouville tempered fractional integrals on continuous function space and Lebesgue spaces in bounded intervals and on the real line. Moreover, the limit behavior of the Riemann-Liouville tempered fractional integrals approaching to the Riemann-Liouville fractional integrals is considered. [ABSTRACT FROM AUTHOR]

In the present paper, we investigate the Hardy–Littlewood type and the integration by parts result for ψ$$ \psi $$–Riemann–Liouville fractional integrals. Also, we attack the integration by parts for the ψ$$ \psi $$–Riemann–Liouville and ψ$$ \psi $$–Hilfer fractional derivatives. To finish, we investigated Sobolev‐type inequalities involving the ψ$$ \psi $$–Riemann–Liouville and the ψ$$ \psi $$–Hilfer fractional derivatives in weighted space. [ABSTRACT FROM AUTHOR]

We are concerned with the following fractional Kirchhoff variational inequality: .... In this paper, by applying penalization techniques from Bensoussan and Lions (1978) combined with mountain pass theorem, we show the existence and concentration behavior of positive solution to the cited variational inequality. This result extend some results established by Alves, Barros and Torres [J. Math. Anal. Appl. 494 (2021)] to the fractional case. [ABSTRACT FROM AUTHOR]

In this paper we consider the existence of multiple solutions for the non-homogeneous fractional p-Laplacian equations of Schrödinger–Kirchhoff type M ⁢ ( ∫ ℝ n ∫ ℝ n | u ⁢ ( x ) - u ⁢ ( z ) | p | x - z | n + p ⁢ s ⁢ 𝑑 z ⁢ 𝑑 x ) ⁢ ( - Δ ) p s ⁢ u + V ⁢ ( x ) ⁢ | u | p - 2 ⁢ u = f ⁢ ( x , u ) + g ⁢ ( x ) M\Bigg{(}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(z)|^{p}}{|x-{% z}|^{n+ps}}\,dz\,dx\Bigg{)}(-\Delta)_{p}^{s}u+V(x)|u|^{p-2}u=f(x,u)+g(x) in ℝ n {\mathbb{R}^{n}} , where (-Δ ) p s )_{p}^{s} is the fractional p-Laplacian operator with 0¡s¡1¡p¡ ∞ \infty , ps¡n, f : ℝ n \mathbb{R}^{n} × \times ℝ \mathbb{R} → \to ℝ \mathbb{R} is a continuous function, V : ℝ n \mathbb{R}^{n} → \to ℝ + \mathbb{R}^{+} is a potential function and g : ℝ n \mathbb{R}^{n} → \to ℝ \mathbb{R} is a perturbation term. Assuming that the potential V is bounded from bellow, that f(x,t) satisfies the Ambrosetti–Rabinowitz condition and some other reasonable hypotheses, and that g(x) is sufficiently small in L p ′ L^{p^{\prime}} ( ℝ n \mathbb{R}^{n} ), we obtain some new criterion to guarantee that the equation above has at least two non-trivial solutions. [ABSTRACT FROM AUTHOR]

In this article, we are interested in the following non-linear Schrödinger equation with non-local regional diffusion(-Δ)α pn u + u = f (u) in Rn, where ϵ > 0, 0 < α < 1 and (-Δ)α pn is a variational version of the regional Laplacian, whose range of scope is a ball with radius p ϵ(x) = p(ϵx) > 0, where p is a continuous function.We give general conditions on p and f which assure the existence and multiplicity of solution for the cited problem. [ABSTRACT FROM AUTHOR]

In this article, we are interested in the non-linear Schrödinger equation with non-local regional diffussion wherefis a super-linear sub-critical function,is a variational version of the regional Laplacian, whose range of scope is a ball with radius. We study the existence of a ground state solution, furthermore we prove that the ground state level is achieved by a radially symmetric solution. The proof is carried out by using variational methods jointly with rearrangement arguments. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the nonlinear Schrödinger equation with non-local regional diffusion [ABSTRACT FROM AUTHOR]

In this paper we consider the existence of solution for the following class of fractional elliptic problem (0.1) (− Δ) s u + u = Q (x) | u | p − 1 u in R N ∖ Ω N s u (x) = 0 in Ω , where s ∈ (0 , 1) , N > 2 s , Ω ⊂ R N is a bounded set with smooth boundary, (− Δ) s denotes the fractional Laplacian operator and N s is the nonlocal operator that describes the Neumann boundary condition, which is given by N s u (x) = C N , s ∫ R N ∖ Ω u (x) − u (y) | x − y | N + 2 s d y , x ∈ Ω. [ABSTRACT FROM AUTHOR]

This work concerns with the existence of solutions for the following class of nonlocal elliptic problems { (− Δ) s u + u = | u | p − 2 u in Ω , u ≥ 0 in Ω and u ≢ 0 , u = 0 R N ∖ Ω , involving the fractional Laplacian operator (− Δ) s , where s ∈ (0 , 1) , N > 2 s , Ω ⊂ R N is an exterior domain with (non-empty) smooth boundary ∂Ω and p ∈ (2 , 2 s ⁎). The main technical approach is based on variational and topological methods. The variational analysis that we perform in this paper dealing with exterior domains is quite general and may be suitable for other goals too. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of heteroclinic solution for a class of nonlocal problems of the type (− Δ) α u + a (𝜖 x) V ′ (u) = 0 , x ∈ ℝ , lim x → − ∞ u (x) = − 1 and lim x → + ∞ u (x) = 1 , where α ∈ (1 / 2 , 1) , V , a : ℝ → ℝ are continuous functions verifying some technical conditions. For example a can be asymptotically periodic and potential V can be the Ginzburg–Landau potential, that is, V (t) = (t 2 − 1) 2 . [ABSTRACT FROM AUTHOR]

In this paper we study the existence of solution for a class of variational inequality in whole R N. The main tools used are the penalization method due to Bensoussan & Lions [8] and the Mountain Pass Theorem due to Ambrosetti & Rabinowitz [3]. [ABSTRACT FROM AUTHOR]