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We focus on the following fractional (p, q)-Choquard problem: (−Δ)psu+(−Δ)qsu+V(εx)(|u|p−2u+|u|q−2u)=1|x|μ*F(u)f(u) in RN,u∈Ws,p(RN)∩Ws,q(RN),u>0 in RN, $\begin{cases}{\left(-{\Delta}\right)}_{p}^{s}u+{\left(-{\Delta}\right)}_{q}^{s}u+V\left(\varepsilon x\right)\left(\vert u{\vert }^{p-2}u+\vert u{\vert }^{q-2}u\right)=\left(\frac{1}{\vert x{\vert }^{\mu }}{\ast}F\left(u\right)\right)f\left(u\right) \,\text{in}\,{\mathbb{R}}^{N},\quad \hfill \\ u\in {W}^{s,p}\left({\mathbb{R}}^{N}\right)\cap {W}^{s,q}\left({\mathbb{R}}^{N}\right), u{ >}0\,\text{in}\,{\mathbb{R}}^{N},\quad \hfill \end{cases}$ where ɛ > 0 is a small parameter, 0 < s < 1, 1

In this paper, we deal with the following class of (p 1 , p 2) -Laplacian problems: - Δ p 1 u - Δ p 2 u = g (u) in R N , u ∈ W 1 , p 1 (R N) ∩ W 1 , p 2 (R N) , where N ≥ 2 , 1 < p 1 < p 2 ≤ N , Δ p i is the p i -Laplacian operator, for i = 1 , 2 , and g : R → R is a Berestycki-Lions type nonlinearity. Using appropriate variational arguments, we obtain the existence of a ground state solution. In particular, we provide three different approaches to deduce this result. Finally, we prove the existence of infinitely many radially symmetric solutions. Our results improve and complement those that have appeared in the literature for this class of problems. Furthermore, the arguments performed throughout the paper are rather flexible and can be also applied to study other p-Laplacian and (p 1 , p 2) -Laplacian equations with general nonlinearities. [ABSTRACT FROM AUTHOR]

We consider the following class of fractional Schrödinger equations:

In this paper, we investigate the existence of multiple solutions for the following two fractional problems:

In this work we study the following fractional scalar field equation:

We study the multiplicity and concentration of complex-valued solutions for a fractional magnetic Schrödinger equation involving a scalar continuous electric potential satisfying a local condition and a continuous nonlinearity with subcritical growth. The main results are obtained by applying a penalization technique, generalized Nehari manifold method and Ljusternik–Schnirelman theory. We also prove a Kato's inequality for the fractional magnetic Laplacian which we believe to be useful in the study of other fractional magnetic problems. [ABSTRACT FROM AUTHOR]

We study the existence and concentration of positive solutions for the following class of fractional p-Kirchhoff type problems: $$ \left\{\begin{array}{@{}ll} \left(\varepsilon^{sp}a+\varepsilon^{2sp-3}b \,[u]_{s, p}^{p}\right)(-\Delta)_{p}^{s}u+V(x)u^{p-1}=f(u) & \text{in}\ \mathbb{R}^{3},\\ \noalign{ u\in W^{s, p}(\mathbb{R}^{3}), \quad u>0 & \text{in}\ \mathbb{R}^{3}, \end{array}\right.$$ where ɛ is a small positive parameter, a and b are positive constants, s ∈ (0, 1) and p ∈ (1, ∞) are such that $sp \in (\frac {3}{2}, 3)$ , $(-\Delta)^{s}_{p}$ is the fractional p-Laplacian operator, f: ℝ → ℝ is a superlinear continuous function with subcritical growth and V: ℝ3 → ℝ is a continuous potential having a local minimum. We also prove a multiplicity result and relate the number of positive solutions with the topology of the set where the potential V attains its minimum values. Finally, we obtain an existence result when f(u) = uq−1 + γur−1, where γ > 0 is sufficiently small, and the powers q and r satisfy 2p < q < p*s ⩽ r. The main results are obtained by using some appropriate variational arguments. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of fractional Schrödinger-Poisson type equations with magnetic field of the type $$\varepsilon^{2s}(-\Delta)_{A/\varepsilon}^{s}u + V(x)u + {\rm e}^{-2t}(\vert x \vert^{2t-3} \ast \vert u\vert ^{2})u = f(\vert u \vert^{2})u \quad \hbox{in} \ \open{R}^{3},$$ where ε > 0 is a parameter, s , t ∈ (0, 1) are such that 2 s +2 t >3, A :ℝ3 → ℝ3 is a smooth magnetic potential, (−Δ) A s is the fractional magnetic Laplacian, V :ℝ3 → ℝ is a continuous electric potential and f :ℝ → ℝ is a C 1 subcritical nonlinear term. Using variational methods, we obtain the existence, multiplicity and concentration of nontrivial solutions for e > 0 small enough. [ABSTRACT FROM AUTHOR]

We consider the following class of fractional parametric problems (−ΔDir)su=f(x,u)+tφ1+hinΩ,u=0on∂Ω,where Ω⊂RN is a smooth bounded domain, s∈(0,1), N>2s, (−ΔDir)s is the fractional Dirichlet Laplacian, f:Ω¯×R→R is a locally Lipschitz nonlinearity having linear or superlinear growth and satisfying Ambrosetti–Prodi type assumptions, t∈R, φ1 is the first eigenfunction of the Laplacian with homogenous boundary conditions, and h:Ω→R is a bounded function. Using variational methods, we prove that there exists a t0∈R such that the above problem admits at least two distinct solutions for any t≤t0. We also discuss the existence of solutions for a fractional periodic Ambrosetti–Prodi type problem. [ABSTRACT FROM AUTHOR]

We investigate the existence, multiplicity and concentration of nontrivial solutions for the following fractional magnetic Kirchhoff equation with critical growth: a ε 2 s + b ε 4 s - 3 [ u ] A / ε 2 (- Δ) A / ε s u + V (x) u = f (| u | 2 ) u + | u | 2 s ∗ - 2 u in R 3 , where ε is a small positive parameter, a , b > 0 are fixed constants, s ∈ (3 4 , 1) , 2 s ∗ = 6 3 - 2 s is the fractional critical exponent, (- Δ) A s is the fractional magnetic Laplacian, A : R 3 → R 3 is a smooth magnetic potential, V : R 3 → R is a positive continuous potential verifying the global condition due to Rabinowitz (Z Angew Math Phys 43:270–291, 1992), and f : R → R is a C 1 subcritical nonlinearity. Due to the presence of the magnetic field and the critical growth of the nonlinearity, several difficulties arise in the study of our problem and a careful analysis will be needed. The main results presented here are established by using minimax methods, concentration compactness principle of Lions (Ann Inst H Poincaré Anal Non Linéaire 1(2):109–145, 1984), a fractional Kato's type inequality and the Ljusternik–Schnirelmann theory of critical points. [ABSTRACT FROM AUTHOR]

We consider some nonlinear fractional Schrödinger equations with magnetic field and involving continuous nonlinearities having subcritical, critical or supercritical growth. Under a local condition on the potential, we use minimax methods to investigate the existence and concentration of nontrivial weak solutions. [ABSTRACT FROM AUTHOR]

We study the existence and multiplicity of periodic weak solutions for a non-local equation involving an odd subcritical nonlinearity which is asymptotically linear at infinity. We investigate such problem by applying the pseudo-index theory developed by Bartolo, Benci and Fortunato [11] after transforming the problem to a degenerate elliptic problem in a half-cylinder with a Neumann boundary condition, via a Caffarelli-Silvestre type extension in periodic setting. The periodic nonlocal case, considered here, presents, respect to the cases studied in the literature, some new additional difficulties and a careful analysis of the fractional spaces involved is necessary. [ABSTRACT FROM AUTHOR]

In this paper we discuss the existence and non-existence of weak solutions to parametric fractional equations involving the square root of the Laplacian A1/2 in a smooth bounded domain Ω ⊂ Rn (n ≥ 2) and with zero Dirichlet boundary conditions. Namely, our simple model is the following equation ... The existence of at least two non-trivial L∞-bounded weak solutions is established for large value of the parameter λ, requiring that the nonlinear term f is continuous, superlinear at zero and sublinear at infinity. Our approach is based on variational arguments and a suitable variant of the Caffarelli-Silvestre extension method. [ABSTRACT FROM AUTHOR]

In this paper we focus our attention on the following nonlinear fractional Schrödinger equation with magnetic field ε 2 s ( − Δ ) A / ε s u + V ( x ) u = f ( | u | 2 ) u in R N , where ε > 0 is a parameter, s ∈ ( 0 , 1 ) , N ≥ 3 , ( − Δ ) A s is the fractional magnetic Laplacian, V : R N → R and A : R N → R N are continuous potentials and f : R N → R is a subcritical nonlinearity. By applying variational methods and Ljusternick–Schnirelmann theory, we prove existence and multiplicity of solutions for ε small. [ABSTRACT FROM AUTHOR]

By using a generalization of the Struwe–Jeanjean monotonicity trick we prove the existence of a non-trivial weak solution for the following problem whereis a smooth bounded open set in,,,,is the fractionalp-Laplace operator andis a continuous function withand there areandsuch thatwhenever. [ABSTRACT FROM PUBLISHER]