Periodic solutions for the non-local operator $(-\Delta+m^{2})^{s}-m^{2s}$ with $m\geq 0$

Ambrosio, Vincenzo

doi:10.12775/tmna.2016.063

Cited by 13 publications

(28 citation statements)

References 30 publications

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“…Then, using the fact that the extension technique works again in the periodic setting [6,7,36], and following the ideas discussed above, we are able to prove the next result.…”

Section: Introductionmentioning

confidence: 72%

“…Let us denote by

X_{2 π}^{s}

the completion of

C_{2 π}^{\infty} true(\bar{{double-struckR}_{+}^{N + 1}} true) : = \{U \in C^{\infty} (\bar{R_{+}^{N + 1}}) : U (x + 2 π e_{i}, y) = U (x, y) for every (x, y) \in \bar{{double-struckR}_{+}^{N + 1}}, i = 1, \dots, N\}

under the

H^{1} ({scriptS}_{2 π}, y^{1 - 2 s})

‐norm, where

{scriptS}_{2 π} = {(- π, π)}^{N} \times false(0, \infty false)

{∥ U ∥}_{X_{2 π}^{s}} = {(() {\int \int}_{S_{2 π}} y^{1 - 2 s} ({| \nabla U |}^{2} + m^{2} U^{2}) dx dy)}^{1 / 2} .

It is worth recalling the following fundamental results concerning the spaces

X_{2 π}^{s}

and

H_{2 π}^{s}

. Theorem There exists a s...…”

Section: Fractional Periodic Ambrosetti–prodi Type Problemmentioning

confidence: 99%

“…Theorem 5.1 [6,7]. There exists a surjective linear operator Tr ∶ 2 → ℍ 2 such that: (ii) Tr is bounded and…”

Section: Fractional Periodic Ambrosetti-prodi T Ype Problemmentioning

confidence: 99%

“…This operator can be extended by density on the Hilbert space

{double-struckH}_{2 π}^{s} : = \{u = \sum_{k \in Z^{N}} c_{k} \frac{e^{ı k \cdot x}}{{(2 π)}^{\frac{N}{2}}} \in L^{2} {false(- π, π false)}^{N} : \sum_{k \in Z^{N}} {{true(false| k false|}^{2} + m^{2} true)}^{s} {false| c_{k} false|}^{2} < + \infty\}

endowed with the norm

{| u |}_{H_{2 π}^{s}} : = {((), \sum_{k \in {double-struckZ}^{N}}, {{(| k |}^{2} + m^{2})}^{s}, {| c_{k} |}^{2})}^{1 / 2} .

Then, using the fact that the extension technique works again in the periodic setting , and following the ideas discussed above, we are able to prove the next result. Theorem Assume that f satisfies either – or ––.…”

Section: Introductionmentioning

confidence: 99%

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An Ambrosetti–Prodi type result for fractional spectral problems

Ambrosio

2019

Mathematische Nachrichten

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We consider the following class of fractional parametric problems lefttrue(−ΔDirtrue)su=f(x,u)+tφ1+hleft4.ptin4.ptnormalΩ,leftu=0left4.pton4.pt∂normalΩ,where Ω⊂double-struckRN is a smooth bounded domain, s∈(0,1), N>2s, true(−ΔDirtrue)s is the fractional Dirichlet Laplacian, f:normalΩ¯×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.

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“…Then, using the fact that the extension technique works again in the periodic setting [6,7,36], and following the ideas discussed above, we are able to prove the next result.…”

Section: Introductionmentioning

confidence: 72%

“…Let us denote by

X_{2 π}^{s}

the completion of

C_{2 π}^{\infty} true(\bar{{double-struckR}_{+}^{N + 1}} true) : = \{U \in C^{\infty} (\bar{R_{+}^{N + 1}}) : U (x + 2 π e_{i}, y) = U (x, y) for every (x, y) \in \bar{{double-struckR}_{+}^{N + 1}}, i = 1, \dots, N\}

under the

H^{1} ({scriptS}_{2 π}, y^{1 - 2 s})

‐norm, where

{scriptS}_{2 π} = {(- π, π)}^{N} \times false(0, \infty false)

{∥ U ∥}_{X_{2 π}^{s}} = {(() {\int \int}_{S_{2 π}} y^{1 - 2 s} ({| \nabla U |}^{2} + m^{2} U^{2}) dx dy)}^{1 / 2} .

It is worth recalling the following fundamental results concerning the spaces

X_{2 π}^{s}

and

H_{2 π}^{s}

. Theorem There exists a s...…”

Section: Fractional Periodic Ambrosetti–prodi Type Problemmentioning

confidence: 99%

“…Theorem 5.1 [6,7]. There exists a surjective linear operator Tr ∶ 2 → ℍ 2 such that: (ii) Tr is bounded and…”

Section: Fractional Periodic Ambrosetti-prodi T Ype Problemmentioning

confidence: 99%

“…This operator can be extended by density on the Hilbert space

{double-struckH}_{2 π}^{s} : = \{u = \sum_{k \in Z^{N}} c_{k} \frac{e^{ı k \cdot x}}{{(2 π)}^{\frac{N}{2}}} \in L^{2} {false(- π, π false)}^{N} : \sum_{k \in Z^{N}} {{true(false| k false|}^{2} + m^{2} true)}^{s} {false| c_{k} false|}^{2} < + \infty\}

endowed with the norm

{| u |}_{H_{2 π}^{s}} : = {((), \sum_{k \in {double-struckZ}^{N}}, {{(| k |}^{2} + m^{2})}^{s}, {| c_{k} |}^{2})}^{1 / 2} .

Section: Introductionmentioning

confidence: 99%

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An Ambrosetti–Prodi type result for fractional spectral problems

Ambrosio

2019

Mathematische Nachrichten

Self Cite

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“…On the other hand, because of the lack of scaling properties (there is no standard group action under, which ( I − Δ) s behaves as a local differential operator) for the Bessel operator, the study of does not seem to be trivial. The Bessel operator ( I − Δ) s with 0 < s < 1 is related to the pseudo‐relativistic Schördinger operator

{((), m^{2} - normalΔ)}^{1 false/ 2} - m

( m > 0) and, recently, a lot of attention is paid to equations involving it (see the works of Ambrosio for example). Secchi proved the existence and multiple existence results for the following Schördinger equation:

{(I - Δ)}^{α} u + λ V (x) u = f (x, u) + μ ξ (x) | u {false|}^{q - 2} u in {double-struckR}^{N},

where λ , μ > 0 are parameters, α ∈ (0,1), 1 < q < 2, and

ξ false(x false) \in L^{2 false/ false(2 - q false)} ((), R^{N})

.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Multiplicity and concentration results for fractional Schrödinger‐Poisson systems involving a Bessel operator

Shen

2018

Math Methods in App Sciences

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This paper is concerned with the fractional Schrödinger‐Poisson systems involving a Bessel operator. By using Mountain‐pass theorem and Ekeland's variational principle, we obtain the multiplicity and concentration of nontrivial solutions for the given problem. In particular, although there exist concave‐convex nonlinearities in our problem, it is not necessary to assume that the corresponding Lebesgue norm of the weight function of the convex term needs to be small enough.

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