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We consider the fractional relativistic Schr¨odinger-Choquard equation ... where ε > 0 is a small parameter, s ∈ (0, 1), m > 0, N > 2s, μ ∈ (0, 2s), (-Δ + m2)s is the fractional relativistic Schr¨odinger operator, V: RN → R is a continuous potential having a local minimum, f: R → R is a continuous nonlinearity with subcritical growth at infinity and F(t) = t 0 f(τ)dτ. Exploiting appropriate variational arguments, we construct a family of solutions concentrating around the local minimum of V as ε → 0. [ABSTRACT FROM AUTHOR]

We examine the following (p1,p2)$(p_{1}, p_{2})$‐Kirchhoff‐type problem: −M1∥∇u∥Lp1(RN)p1Δp1u−M2∥∇u∥Lp2(RN)p2Δp2u=g(u)inRN,u∈W1,p1(RN)∩W1,p2(RN),$$\begin{equation*} {\left\lbrace \def\eqcellsep{&}\begin{array}{ll}-M_{1}\left(\Vert \nabla u\Vert ^{p_{1}}_{L^{p_{1}}(\mathbb {R}^{N})}\right)\Delta _{p_{1}}u-M_{2}\left(\Vert \nabla u\Vert ^{p_{2}}_{L^{p_{2}}(\mathbb {R}^{N})}\right)\Delta _{p_{2}}u\\ \quad = g(u) \mbox{ in } \mathbb {R}^{N}, \\ u\in W^{1, p_{1}}(\mathbb {R}^{N})\cap W^{1, p_{2}}(\mathbb {R}^{N}), \end{array} \right.} \end{equation*}$$where N⩾2$N\geqslant 2$, 1

In this paper, we study the existence of ground state solutions for the following (p , q) -Choquard equation: − Δ p u − Δ q u + | u | p − 2 u + | u | q − 2 u = (I α ⁎ F (u)) f (u) in R N , where 2 ≤ p < q < N , Δ s is the s -Laplacian operator, with s ∈ { p , q } , I α is the Riesz potential of order α ∈ ((N − 2 q) + , N) , F ∈ C 1 (R , R) is a general nonlinearity of Berestycki-Lions type and F ′ = f. Furthermore, we analyze the regularity, symmetry and decay properties of these solutions. In particular, we extend the results in [33] to the (p , q) -Laplacian setting. [ABSTRACT FROM AUTHOR]

We establish a version of the Bourgain-Brezis-Mironescu type formula on the limit as s\rightarrow 1^- of the fractional relativistic seminorm. [ABSTRACT FROM AUTHOR]

We consider the following fractional p -Laplacian logarithmic Schrödinger equation: s p (− Δ) p s u + V (x) | u | p − 2 u = | u | p − 2 u log | u | p in  ℝ N , u ∈ W s , p (ℝ N) , u > 0  in  ℝ N , where > 0 , s ∈ (0 , 1) , p ∈ [ 2 , ∞) , N > s p , (− Δ) p s is the fractional p -Laplacian operator, V : ℝ N → ℝ is a continuous potential satisfying a local condition. By applying suitable variational arguments, we analyze the existence and concentration of solutions as → 0 for the above problem. [ABSTRACT FROM AUTHOR]

We investigate the following class of (p , q) -Laplacian problems: { − ε p Δ p v − ε q Δ q v + V (x) (| v | p − 2 v + | v | q − 2 v) = f (v) in R N , v ∈ W 1 , p (R N) ∩ W 1 , q (R N) , v > 0 in R N , where ε > 0 is a small parameter, N ≥ 3 , 1 < p < q < N , Δ s v : = div (| ∇ v | s − 2 ∇ v) , with s ∈ { p , q } , is the s -Laplacian operator, V : R N → R is a continuous potential such that inf R N ⁡ V > 0 and V 0 : = inf Λ ⁡ V < min ∂ Λ ⁡ V for some bounded open set Λ ⊂ R N , and f : R → R is a subcritical Berestycki-Lions type nonlinearity. Using variational arguments, we show the existence of a family of solutions (v ε) which concentrates around M : = { x ∈ Λ : V (x) = V 0 } as ε → 0. [ABSTRACT FROM AUTHOR]

Let s ∈ (0 , 1) , N > 2 s and D s , 2 (R N) : = { u ∈ L 2 s ∗ (R N) : ‖ u ‖ D s , 2 (R N) : = ( C N , s 2 ∬ R 2 N | u (x) − u (y) | 2 | x − y | N + 2 s d x d y) 1 2 < ∞ } , where 2 s ∗ : = 2 N N − 2 s is the fractional critical exponent and C N , s is a positive constant. We consider functionals J : D s , 2 (R N) → R of the type J (u) : = 1 2 ‖ u ‖ D s , 2 (R N) 2 − ∫ R N b (x) G (u) d x , where G (t) : = ∫ 0 t g (τ) d τ , g : R → R is a continuous function with subcritical growth at infinity, and b : R N → R is a suitable weight function. We prove that a local minimizer of J in the topology of the subspace V s : = { u ∈ D s , 2 (R N) : u ∈ C (R N) with sup x ∈ R N (1 + | x | N − 2 s) | u (x) | < ∞ } must be a local minimizer of J in the D s , 2 (R N) -topology. [ABSTRACT FROM AUTHOR]

By combining suitable estimates for Bessel kernel and an interpolation argument, we give an alternative proof of the lifting property (- Δ + 1) - α : Λ β → Λ β + 2 α is an isomorphism, where α ≥ 0 , β > 0 , and Λ β denotes the classical Lipschitz space. [ABSTRACT FROM AUTHOR]

In this paper, we complete the study started in Ambrosio and Rădulescu (J Math Pures Appl (9) 142:101–145, 2020) on the concentration phenomena for a class of fractional (p, q)-Schrödinger equations involving the fractional critical Sobolev exponent. More precisely, we focus our attention on the following class of fractional (p, q)-Laplacian problems: (- Δ) p s u + (- Δ) q s u + V (ε x) (u p - 1 + u q - 1) = f (u) + u q s ∗ - 1 in R N , u ∈ W s , p (R N) ∩ W s , q (R N) , u > 0 in R N , where ε > 0 is a small parameter, s ∈ (0 , 1) , 1 < p < q < N s , q s ∗ = Nq N - s q is the fractional critical Sobolev exponent, (- Δ) r s , with r ∈ { p , q } , is the fractional r-Laplacian operator, V : R N → R is a positive continuous potential such that inf ∂ Λ V > inf Λ V for some bounded open set Λ ⊂ R N , and f : R → R is a continuous nonlinearity with subcritical growth. With the aid of minimax theorems and the Ljusternik–Schnirelmann category theory, we obtain multiple solutions by employing the topological construction of the set where the potential V attains its minimum. We also establish a multiplicity result when f (t) = t γ - 1 + μ t τ - 1 , with 1 < p < q < γ < q s ∗ < τ and μ > 0 sufficiently small, by combining a truncation argument with a Moser-type iteration. [ABSTRACT FROM AUTHOR]

In this paper we consider singularly perturbed nonlinear Schrödinger equations with electromagnetic potentials and involving continuous nonlinearities with subcritical, critical or supercritical growth. By means of suitable variational techniques, truncation arguments and Lusternik–Schnirelman theory, we relate the number of nontrivial complex-valued solutions with the topology of the set where the electric potential attains its minimum value. [ABSTRACT FROM AUTHOR]

Exciton-polaritons are hybrid radiation-matter elementary excitations that, thanks to their strong nonlinearities, enable a plethora of physical phenomena ranging from room temperature condensation to superfluidity. While polaritons are usually exploited in a high-density regime, evidence for quantum correlations at the level of few excitations has been recently reported, thus suggesting the possibility of using these systems for quantum information purposes. Here we show that integrated circuits of propagating single polaritons can be arranged to build deterministic quantum logic gates in which the two-particle interaction energy plays a crucial role. Besides showing their prospective potential for photonic quantum computation, we also show that these systems can be exploited for metrology purposes, as for instance to precisely measure the magnitude of the polariton-polariton interaction at the two-body level. Our results will motivate the development of practical quantum polaritonic devices in prospective quantum technologies. Recently, observation of low-density polaritons in the quantum regime has prompted interest beyond their collective behavior. Here, an integrated quantum logic device that relies on two-body polaritonic interaction is proposed theoretically. [ABSTRACT FROM AUTHOR]

We collect some interesting results for equations driven by the fractional relativistic Schrödinger operator (− Δ + m 2) s with s ∈ (0 , 1) and m > 0. More precisely, for the linear theory, we prove Hölder-Schauder-Zygmund regularity results and a Kato's inequality. For the nonlinear theory, we obtain L ∞ -regularity, exponential decay, a Pohozaev-type identity, and a symmetry result for solutions of certain nonlinear fractional problems. [ABSTRACT FROM AUTHOR]

In the present work we study the multiplicity and concentration of positive solutions for the following class of Kirchhoff problems: − (ε 2 a + ε b ∫ R 3 | ∇ u | 2 d x) Δ u + V (x) u = f (u) + γ u 5 in R 3 , u ∈ H 1 (R 3) , u > 0 in R 3 , where ε > 0 is a small parameter, a , b > 0 are constants, γ ∈ { 0 , 1 } , V is a continuous positive potential with a local minimum, and f is a superlinear continuous function with subcritical growth. The main results are obtained through suitable variational and topological arguments. We also provide a multiplicity result for a supercritical version of the above problem by combining a truncation argument with a Moser-type iteration. Our theorems extend and improve in several directions the studies made in (Adv. Nonlinear Stud.14 (2014), 483–510; J. Differ. Equ.252 (2012), 1813–1834; J. Differ. Equ.253 (2012), 2314–2351). [ABSTRACT FROM AUTHOR]

In this paper, we study the multiplicity and concentration of positive solutions for the following (p, q)-Laplacian problem: - Δ p u - Δ q u + V (ε x) | u | p - 2 u + | u | q - 2 u = f (u) in R N , u ∈ W 1 , p (R N) ∩ W 1 , q (R N) , u > 0 in R N , where ε > 0 is a small parameter, 1 < p < q < N , Δ r u = div (| ∇ u | r - 2 ∇ u) , with r ∈ { p , q } , is the r-Laplacian operator, V : R N → R is a continuous function satisfying the global Rabinowitz condition, and f : R → R is a continuous function with subcritical growth. Using suitable variational arguments and Ljusternik–Schnirelmann category theory, we investigate the relation between the number of positive solutions and the topology of the set where V attains its minimum for small ε . [ABSTRACT FROM AUTHOR]

We consider the following class of fractional problems with unbalanced growth: { (− Δ) p s u + (− Δ) q s u + V (ε x) (| u | p − 2 u + | u | q − 2 u) = f (u) in R N , u ∈ W s , p (R N) ∩ W s , q (R N) , u > 0 in R N , where ε > 0 is a small parameter, s ∈ (0 , 1) , 2 ≤ p < q < N s , (− Δ) t s (with t ∈ { p , q }) is the fractional t -Laplacian operator, V : R N → R is a continuous potential satisfying local conditions, and f : R → R is a continuous nonlinearity with subcritical growth. Applying suitable variational and topological arguments, we obtain multiple positive solutions for ε > 0 sufficiently small as well as related concentration properties, in relationship with the set where the potential V attains its minimum. [ABSTRACT FROM AUTHOR]

We deal with the following nonlinear problem involving fractional p&q Laplacians: ... is a parameter, h is a nontrivial bounded perturbation and f is a superlinear continuous function with subcritical growth. Using suitable variational arguments and concentration-compactness lemma, we prove the existence of a nontrivial non-negative solution for suffciently large. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the following nonlocal systems of fractional Schrödinger equations ε 2 s (− Δ) s u + V (x) u = Q u (u , v) + γ H u (u , v) in R N ε 2 s (− Δ) s v + W (x) v = Q v (u , v) + γ H v (u , v) in R N u , v > 0 in R N , where ε > 0 , s ∈ (0 , 1) , N>2s, (− Δ) s is the fractional Laplacian, V : R N → R and W : R N → R are continuous potentials, Q is a homogeneous C 2 -function with subcritical growth, γ ∈ { 0 , 1 } and H (u , v) = (2 / (α + β)) | u | α | v | β with α , β ≥ 1 such that α + β = 2 s ∗ . We investigate the subcritical case (γ = 0) and the critical case (γ = 1) , and using Ljusternik–Schnirelmann theory, we relate the number of solutions with the topology of the set where the potentials V and W attain their minimum values. [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]

In this paper we consider the following class of fractional Kirchhoff equations with critical growth: (ε 2 s a + ε 4 s − 3 b ∫ R 3 | (− Δ) s 2 u | 2 d x) (− Δ) s u + V (x) u = f (u) + | u | 2 s ∗ − 2 u in R 3 , u ∈ H s (R 3) , u > 0 in R 3 , where ε > 0 is a small parameter, a , b > 0 are constants, s ∈ ( 3 4 , 1) , 2 s ∗ = 6 3 − 2 s is the fractional critical exponent, (− Δ) s is the fractional Laplacian operator, V is a positive continuous potential and f is a superlinear continuous function with subcritical growth. Using penalization techniques and variational methods, we prove the existence of a family of positive solutions u ε which concentrates around a local minimum of V as ε → 0. [ABSTRACT FROM AUTHOR]

We deal with the multiplicity and concentration of positive solutions for the following fractional Schrödinger–Poisson-type system with critical growth: 𝜀 2 s (− Δ) s u + V (x) u + ϕ u = f (u) + | u | 2 s ∗ − 2 u in  ℝ 3 , 𝜀 2 t (− Δ) t ϕ = u 2 in  ℝ 3 , where 𝜀 > 0 is a small parameter, s ∈ (3 4 , 1) , t ∈ (0 , 1) , (− Δ) α , with α ∈ { s , t } , is the fractional Laplacian operator, V is a continuous positive potential and f is a superlinear continuous function with subcritical growth. Using penalization techniques and Ljusternik–Schnirelmann theory, we investigate the relation between the number of positive solutions with the topology of the set where the potential attains its minimum value. [ABSTRACT FROM AUTHOR]

In this note we extend the well-known limiting formulas due to Bourgain-Brezis- -Mironescu and Maz'ya-Shaposhnikova, to the setting of fractional Sobolev spaces on the torus. We also give a Γ-convergence result in the spirit of Ponce. The main theorems are obtained by using the nice structure of Fourier series. [ABSTRACT FROM AUTHOR]

We prove that sequences of functions (u n) ⊂ W s , p (R N) , with s ∈ (0 , 1) and p ∈ (1 , N s) , bounded in W s , p (R N) , strongly convergent in L N p N − s p (R N) and solving nonlinear fractional p -Laplacian Schrödinger equations in R N , must vanish at infinity uniformly with respect to n ∈ N. [ABSTRACT FROM AUTHOR]

We deal with the following nonlinear Schrödinger equation with magnetic field and critical growth: { (ε ı ∇ − A (x)) 2 u + V (x) u = f (| u | 2) u + | u | 2 ⁎ − 2 u in R N , u ∈ H 1 (R N , C) , where ε > 0 is a small parameter, N ≥ 3 , 2 ⁎ = 2 N N − 2 is the critical Sobolev exponent, A ∈ C 1 (R N , R N) is a magnetic vector potential, V : R N → R is a continuous positive potential having a local minimum and f : R → R is a superlinear continuous function with subcritical growth. Using penalization techniques and variational methods, we investigate the existence and concentration of nontrivial solutions for ε > 0 small enough. [ABSTRACT FROM AUTHOR]

In this paper we deal with the following fractional Choquard equation ε s p (− Δ) p s u + V (x) | u | p − 2 u = ε μ − N 1 | x | μ ∗ F (u) f (u) in R N , u ∈ W s , p (R N) , u > 0 in R N , where ε > 0 is a small parameter, s ∈ (0 , 1) , p ∈ (1 , ∞) , N > s p , (− Δ) p s is the fractional p -Laplacian, V is a positive continuous potential, 0 < μ < s p , and f is a continuous superlinear function with subcritical growth. Using minimax arguments and the Ljusternik–Schnirelmann category theory, we obtain the existence, multiplicity and concentration of positive solutions for ε > 0 small enough. [ABSTRACT FROM AUTHOR]

In this paper we deal with the following fractional Kirchhoff problem M ∬ R n × R n | u (x) − u (y) | p | x − y | n + s p d x d y p − 1 (− Δ) p s u = f (x , u) + λ | u | r − 2 u in Ω , u = 0 in R n ∖ Ω. $$\begin{array}{} \displaystyle \left\{ {\begin{array}{l} \left[M\left(\displaystyle \iint_{\mathbb R^n\times \mathbb R^n} \frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}} dx dy\right)\right]^{p-1}(-\Delta)^{s}_{p}u = f(x, u)+\lambda |u|^{r-2}u \\\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \mbox{ in } \, \Omega, \\ \\\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad u=0 \, \, ~\mbox{ in } \, \mathbb R^n\setminus \Omega. \end{array}} \right. \end{array}$$ Here Ω ⊂ ℝn is a smooth bounded open set with continuous boundary ∂Ω, p ∈ (1, +∞), s ∈ (0, 1), n > sp, (− Δ) p s $\begin{array}{} (-\Delta)^{s}_{p} \end{array}$ is the fractional p-Laplacian, M is a Kirchhoff function, f is a continuous function with subcritical growth, λ is a nonnegative parameter and r > p s ∗ $\begin{array}{} p^*_s \end{array}$ , where p s ∗ = n p n − s p $\begin{array}{} p^*_s=\frac{np}{n-sp} \end{array}$ is the fractional critical Sobolev exponent. By combining variational techniques and a truncation argument, we prove two existence results for this problem, provided that the parameter λ is sufficiently small. [ABSTRACT FROM AUTHOR]

In this note we improve the recent results established in Ambrosio and Isernia (2018) concerning the multiplicity and concentration of positive solutions for the following class of p -fractional Schrödinger equations: ε s p (− Δ) p s u + V (x) | u | p − 2 u = f (u) + γ | u | p s ∗ − 2 u in R N where ε > 0 , s ∈ (0 , 1) , p ∈ (1 , ∞) , N > s p , γ ∈ { 0 , 1 } , p s ∗ = N p N − s p is the fractional critical exponent, (− Δ) p s is the fractional p -Laplacian operator, V is a continuous positive potential satisfying the Rabinowitz condition (Rabinowitz, 1992), and f is a superlinear continuous function with subcritical growth at infinity. Here we do not assume the well-known Ambrosetti–Rabinowitz condition (Ambrosetti and Rabinowitz, 1973) on f and we complete the study in the critical case γ = 1. [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]

Abstract We deal with a class of fractional magnetic Schrödinger equations in the whole line with exponential critical growth. Under a local condition on the potential, we use penalization methods and Ljusternik–Schnirelmann category theory to investigate the existence, multiplicity and concentration of nontrivial weak solutions. [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]

We investigate a class of critical stationary Kirchhoff fractional p-Laplacian problems in presence of a Hardy potential. By using a suitable version of the symmetric mountain-pass lemma due to Kajikiya, we obtain the existence of a sequence of infinitely many arbitrarily small solutions converging to zero. [ABSTRACT FROM AUTHOR]

In this paper, we study the regularity of weak solutions for a class of nonlinear Choquard equations driven by the p -Laplacian operator. We also establish a Pohozaev type identity. [ABSTRACT FROM AUTHOR]

In this paper we study the existence and the multiplicity of positive solutions for the following class of fractional Schrödinger equations ϵ 2 s ( − Δ ) s u + V ( x ) u = f ( u ) in R N , where ϵ > 0 is a parameter, s ∈ ( 0 , 1 ) , N > 2 s , V : R N → R is a continuous positive potential, and f : R → R is a C 1 superlinear nonlinearity which does not satisfy the Ambrosetti–Rabinowitz condition. The main result is established by using minimax methods and Ljusternik–Schnirelmann theory of critical points. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence and multiplicity of periodic solutions for a class of parametric nonlocal equations with critical and supercritical growth. It is well known that these equations can be realized as local degenerate elliptic problems in a half-cylinder of R+N+1 together with a nonlinear Neumann boundary condition, through the extension technique in periodic setting. Exploiting this fact, and by combining the Moser iteration scheme in the nonlocal framework with an abstract multiplicity result valid for differentiable functionals due to Ricceri, we show that the problem under consideration admits at least three periodic solutions with the property that their Sobolev norms are bounded by a suitable constant. Finally, we provide a concrete estimate of the range of these parameters by using some properties of the fractional calculus on a specific family of test functions. This estimate turns out to be deeply related to the geometry of the domain. [ABSTRACT FROM AUTHOR]

We present an elementary proof of the existence of a nontrivial weak periodic solution for a nonlinear fractional problem driven by a relativistic Schrödinger operator with periodic boundary conditions and involving a periodic continuous subcritical nonlinearity satisfying a more general Ambrosetti-Rabinowitz condition. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following fractional Schrödinger equation (-Δ)su+u=k(x)f(u)+h(x)inRNu∈Hs(RN),u>0inRN,where s∈(0,1),N>2s,(-Δ)s is the fractional Laplacian, k is a bounded positive function, h∈L2(RN),h≢0 is nonnegative and f is either asymptotically linear or superlinear at infinity. By using the s-harmonic extension technique and suitable variational methods, we prove the existence of at least two positive solutions for the problem under consideration, provided that |h|2 is sufficiently small. [ABSTRACT FROM AUTHOR]

In this paper we deal with a one-dimensional free-boundary problem arising in the modeling of concrete carbonation process. More precisely, we investigate global existence, uniqueness and large-time behavior of weak solutions for the problem under consideration. We also obtain the existence of a weak solution when the measure of the initial domain vanishes. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the following fractional Kirchhoff equation p + q ( 1 − s ) ∬ ℝ 2 N | u ( x ) − u ( y ) | 2 | x − y | N + 2 s d x d y ( − Δ ) s u = g ( u ) in  ℝ N , where s ∈ ( 0 , 1 ) , N ≥ 2 , p > 0 , q is a small positive parameter and g : ℝ → ℝ is an odd function satisfying Berestycki–Lions type assumptions. By using minimax arguments, we establish a multiplicity result for the above equation, provided that q is sufficiently small. [ABSTRACT FROM AUTHOR]

In this work we study the following fractional scalar field equation: { ( - Δ ) s ⁢ u = g ′ ⁢ ( u ) in ⁢ ℝ N , u > 0 , \left\{\begin{aligned} \displaystyle(-\Delta)^{s}u&\displaystyle=g^{\prime}(u)% \quad\mbox{in }\mathbb{R}^{N},\\ \displaystyle u&\displaystyle>0,\end{aligned}\right. where N ≥ 2 {N\geq 2} , s ∈ ( 0 , 1 ) {s\in(0,1)} , ( - Δ ) s {(-\Delta)^{s}} is the fractional Laplacian and the nonlinearity g ∈ C 2 ⁢ ( ℝ ) {g\in C^{2}(\mathbb{R})} is such that g ′′ ⁢ ( 0 ) = 0 {g^{\prime\prime}(0)=0}. By using variational methods, we prove the existence of a positive solution which is spherically symmetric and decreasing in r = | x | {r=|x|}. [ABSTRACT FROM AUTHOR]

By using minimax arguments we prove the existence of a nontrivial solution for a fractional Kirchhoff-Schrödinger- Poisson system in R3 involving a Berestycki-Lions type nonlinearity. [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]

In this paper we investigate the existence of nontrivial ground state solutions for the following fractional scalar field equation (-Δ )su + V (x)u = f (u) in RN, where s ∊ (0, 1), N > 2s, (-Δ )s is the fractional Laplacian, V : RN → R is a bounded potential satisfying suitable assumptions and f ∊ C1,β (R,R) has critical growth. We first analyze the case V constant and then we develop a Jeanjean- Tanaka argument [Indiana Univ. Math. J. 54 (2005), 443-464] to deal with the non autonomous case. As far as we know, all results presented here are new. [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]

We use a variant of the fountain Theorem to prove the existence of infinitely many weak solutions for the fractional p-Laplace equation (-Δ)spu + V (x)|u|p-2u = ƒ(x, u) in RN, where s ∈ (0, 1), p ≼ 2, N ≼ 2, (-Δ)sp is the fractional p-Laplace operator, the nonlinearity ƒ is p-superlinear at infinity and the potential V (x) is allowed to be sign-changing. [ABSTRACT FROM AUTHOR]

In this paper we study ground states of the following fractional Schrödinger equation {(-Δ)su + V(x)u = f(x, u) in RN, u ∈ HS(RN), where s ∈ (0, 1), N > 2s and f is a continuous function satisfying a suitable growth assumption weaker than the Ambrosetti-Rabinowitz condition. We consider the cases when the potential V(x) is 1-periodic or has a bounded potential well. [ABSTRACT FROM AUTHOR]

In this paper we prove the existence, regularity, and symmetry of a ground state for a nonlinear equation in the whole space, involving a pseudo-relativistic Schrödinger operator. [ABSTRACT FROM AUTHOR]

We study the existence and the regularity of non trivial T -periodic solutions to the following nonlinear pseudo-relativistic Schrödinger equation (0.1) ( − Δ x + m 2 − m ) u ( x ) = f ( x , u ( x ) ) in ( 0 , T ) N where T > 0 , m is a non negative real number, f is a regular function satisfying the Ambrosetti–Rabinowitz condition and a polynomial growth at rate p for some 1 < p < 2 ♯ − 1 . We investigate such problem using critical point theory after transforming it to an elliptic equation in the infinite half-cylinder ( 0 , T ) N × ( 0 , ∞ ) with a nonlinear Neumann boundary condition. By passing to the limit as m → 0 in (0.1) we also prove the existence of a non trivial T -periodic weak solution to (0.1) with m = 0 . [ABSTRACT FROM AUTHOR]