Existence and concentration of solution for a class of fractional elliptic equation in $$\mathbb {R}^N$$ R N via penalization method

Alves, Claudianor O.; Miyagaki, Olı́mpio H.

doi:10.1007/s00526-016-0983-x

Cited by 153 publications

(216 citation statements)

References 25 publications

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“…One is the L ∞ ‐estimate, owing to the work of Dipierro et al; similarly, we can get the L ∞ ‐estimate. The other is the decay estimate of solutions; with the help of previous works,() we can establish the decay estimate at infinity. Besides above, the novelty of this paper is that we add a potential K ( x ) on the nonlinearity term f ( t ); this will need more careful analysis.…”

Section: Introductionmentioning

confidence: 95%

“…Proof For any ϵ n →0,

w_{n} : = w_{ϵ_{n}}

is a positive ground state solution of problem ; then

f false(x, w_{n} false) : = K false(ϵ_{n} x + ϵ_{n} y_{ϵ_{n}} false) f false(w_{n} false) + Q false(ϵ_{n} x + ϵ_{n} y_{ϵ_{n}} false) false| w_{n} |^{2_{s}^{*} - 2} w_{n} - V false(ϵ_{n} x + ϵ_{n} y_{ϵ_{n}} false) w_{ϵ_{n}} + φ_{w_{n}}^{t} w_{n} \leq C false(1 + false| w_{n} |^{2_{s}^{*} - 1} false)

, where C is independent of n and w n ; by the estimate , we have that ‖ w n ‖ ∞ ≤ C ‖ w n ‖ α ≤ C , where C > 0 is a constant independent of n . Now, we borrow the ideas in Alves and Miyagaki to complete the proof. For this purpose, we rewrite problem as follows:

{(- Δ)}^{s} w_{n} + w_{n} = g_{n} (x),

where g n ( x ): = w n + f ( x , w n ).…”

Section: Concentration Behaviormentioning

confidence: 99%

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Existence and concentration of positive ground state solutions for nonlinear fractional Schrödinger‐Poisson system with critical growth

Teng

Agarwal

2018

Math Methods in App Sciences

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In this paper, we study the following fractional Schrödinger‐Poisson system involving competing potential functions ϵ2sfalse(−normalΔfalse)su+Vfalse(xfalse)u+φu=Kfalse(xfalse)ffalse(ufalse)+Qfalse(xfalse)false|u|2s∗−2u,in0.1emR3,ϵ2tfalse(−normalΔfalse)tφ=u2,in0.1emR3, where ϵ > 0 is a small parameter, f is a function of C1 class, superlinear and subcritical nonlinearity, 2s∗=63−2s, s>34, t ∈ (0,1), V(x), K(x), and Q(x) are positive continuous functions. Under some suitable assumptions on V, K, and Q, we prove that there is a family of positive ground state solutions with polynomial growth for sufficiently small ϵ > 0, of which it is concentrating on the set of minimal points of V(x) and the sets of maximal points of K(x) and Q(x).

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Section: Introductionmentioning

confidence: 95%

“…Proof For any ϵ n →0,

w_{n} : = w_{ϵ_{n}}

is a positive ground state solution of problem ; then

f false(x, w_{n} false) : = K false(ϵ_{n} x + ϵ_{n} y_{ϵ_{n}} false) f false(w_{n} false) + Q false(ϵ_{n} x + ϵ_{n} y_{ϵ_{n}} false) false| w_{n} |^{2_{s}^{*} - 2} w_{n} - V false(ϵ_{n} x + ϵ_{n} y_{ϵ_{n}} false) w_{ϵ_{n}} + φ_{w_{n}}^{t} w_{n} \leq C false(1 + false| w_{n} |^{2_{s}^{*} - 1} false)

{(- Δ)}^{s} w_{n} + w_{n} = g_{n} (x),

where g n ( x ): = w n + f ( x , w n ).…”

Section: Concentration Behaviormentioning

confidence: 99%

Existence and concentration of positive ground state solutions for nonlinear fractional Schrödinger‐Poisson system with critical growth

Teng

Agarwal

2018

Math Methods in App Sciences

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“…Motivated by the fact that the fractional Laplacian appears in a lot of application, several authors have dedicated a special attention for problems involving this important operator. The reader can find some recent results in Alves and Miyagaki [1,2], Brändler, Colorado and Sánchez [10], Cabré and Sire [14], Caffarelli and Silvestre [15], Chang and Wang [16], Cotsiolis and Tavoularis [18], Dávila, del Pino and Wei [19], Dipierro, Palatucci and Valdinoci [21], Fall, Mahmoudi and Valdinoci [25], Felmer, Quaas and Tan [26], Palatucci and Pisante [30], Secchi [32], Silvestre [33] and their references. In the most part of the above references the variational method was used to show the existence of solution.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Bifurcation properties for a class of fractional Laplacian equations in

Alves

Lima

Nóbrega

2018

Mathematische Nachrichten

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This paper concerns with the study of some bifurcation properties for the following class of nonlocal problems trueright90.0pt{(−Δ)su=λffalse(xfalse)false(u+h(u)false),indouble-struckRN,ufalse(xfalse)>0,forallx∈double-struckRN,falseprefixlim|x|→∞ufalse(xfalse)=0,where N>2s, s∈(0,1), λ>0, f:double-struckRN→R is a positive continuous function, h:R→R is a bounded continuous function and (−Δ)su is the fractional Laplacian. The main tools used are the Leray–Shauder degree theory and the global bifurcation result due to Rabinowitz.

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“…The proof of the Theorem is obtained by applying critical point theory after transforming into an elliptic nonlinear Neumann problem in a half‐cylinder via a suitable variant of the extension method introduced by Caffarelli and Silvestre in . This approach has been brilliantly used by several authors to study existence, multiplicity, regularity and symmetry properties of solutions for different fractional problems (in

R^{N}

or in bounded domains) involving subcritical and critical nonlinearities; see for instance . Taking into account this fact, instead of , we are lead to consider the following degenerate elliptic problem with a nonlinear Neumann boundary condition

({, \begin{matrix} - prefixdiv (y^{1 - 2 s} \nabla U) = 0 & in C : = Ω \times (0, \infty), \\ U = 0 & on \partial_{L} C : = \partial Ω \times false[0, \infty false), \\ \frac{\partial U}{\partial ν^{1 - 2 s}} = κ_{s} false[f (x, u) + t φ_{1} + h false] & on \partial^{0} C : = Ω \times false{0 false}, \end{matrix})

where

κ_{s}

is a suitable positive constant (see ) and u is the trace of U (see Section 2).…”

Section: Introductionmentioning

confidence: 99%

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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Existence and concentration of solution for a class of fractional elliptic equation in $$\mathbb {R}^N$$ R N via penalization method

Abstract: In this paper, we study the existence and concentration of positive solution for the following class of fractional elliptic equationwhere ǫ is a positive parameter, f has a subcritical growth, V possesses a local minimum, N > 2s, s ∈ (0, 1), and (−∆) s u is the fractional laplacian.

Cited by 153 publications

References 25 publications

Existence and concentration of positive ground state solutions for nonlinear fractional Schrödinger‐Poisson system with critical growth

Existence and concentration of positive ground state solutions for nonlinear fractional Schrödinger‐Poisson system with critical growth

Bifurcation properties for a class of fractional Laplacian equations in

An Ambrosetti–Prodi type result for fractional spectral problems

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