Ground state solutions for nonlinear fractional Schrödinger equations in RN
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In this paper, a variational approach is proposed to solve a class of Schrodinger equations involving the fractional Laplacian, which is variational in nature and based on minimization on the Nehari manifold.Abstract:
We construct solutions to a class of Schrodinger equations involving the fractional Laplacian. Our approach is variational in nature, and based on minimization on the Nehari manifold.read more
Citations
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Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional p -Laplacian in $${\mathbb {R}}^N$$ R N
TL;DR: In this article, the existence of multiple solutions for the nonhomogeneous fractional p-Laplacian equations of Schrodinger-Kirchhoff type was investigated, and multiplicity results were obtained by using the Ekeland variational principle and the Mountain Pass theorem.
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Concentrating standing waves for the fractional nonlinear Schrödinger equation
TL;DR: In this article, the authors considered the semilinear equation e 2 s ( − Δ ) s u + V ( x ) u − u p = 0, u > 0, u ∈ H 2 s n (R N ) where 0 s 1, 1 p N + 2 s N − 2 s, V (x ) is a sufficiently smooth potential with inf R V(x ) > 0, and e > 0 is a small number.
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Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity
TL;DR: In this article, a scalar field equation involving a fractional Laplacian was studied and a positive ground state was obtained under the general Berestycki-Lions type assumptions.
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Elliptic problems involving the fractional Laplacian in RN
TL;DR: In this paper, the existence and multiplicity of solutions for elliptic equations in R N, driven by a non-local integro-differential operator, which main prototype is the fractional Laplacian, was studied.
Journal ArticleDOI
Ground state solutions of scalar field fractional Schrödinger equations
TL;DR: In this article, the existence of multiple ground state solutions for a class of parametric fractional Schrodinger equations whose simplest prototype is (−�) s u + V (x)u = λ f (x, u) in R n, where n > 2, s stands for the fractional Laplace operator of order s ∈ (0, 1),a ndλ is a positive real parameter.
References
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Hitchhiker's guide to the fractional Sobolev spaces
TL;DR: In this article, the authors deal with the fractional Sobolev spaces W s;p and analyze the relations among some of their possible denitions and their role in the trace theory.
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An Extension Problem Related to the Fractional Laplacian
TL;DR: In this article, the square root of the Laplacian (−△) 1/2 operator was obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition.
Book
Critical Point Theory and Hamiltonian Systems
Jean Mawhin,Michel Willem +1 more
TL;DR: The direct method of the Calculus of Variations, Fenchel Transform and duality, Minimax Theorems for Indefinite Functional, Borsuk-Ulam Theorem and Index Theories, Lusternik-Schnirelman Theory and Multiple Periodic Solution with Fixed Energy, Morse-Ekeland Index, Morse Theory, and Morse Theory for Second Order Systems as discussed by the authors.
Journal ArticleDOI
On a class of nonlinear Schro¨dinger equations
TL;DR: In this paper, the existence of standing wave solutions of nonlinear Schrodinger equations was studied and sufficient conditions for nontrivial solutionsu ∈W¯¯¯¯1,2(ℝ�姫 n ) were established.
Journal ArticleDOI
Fractional Schrödinger equation.
TL;DR: The Hermiticity of the fractional Hamilton operator and the parity conservation law for fractional quantum mechanics are established and the energy spectra of a hydrogenlike atom and of a fractional oscillator in the semiclassical approximation are found.