Existence of multiple positive solutions for nonhomogeneous fractional Laplace problems with critical growth

Wang, Fang; Zhang, Yajing

doi:10.1186/s13661-019-1287-9

Cited by 3 publications

(2 citation statements)

References 17 publications

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“…where f ≥ 0, f ∈ L ∞ (Ω) has been studied in [1] and existence of two positive solutions have been established in [20] when f is a continuous function with compact support in Ω.…”

Section: Introductionmentioning

confidence: 99%

On multiplicity of positive solutions for nonlocal equations with critical nonlinearity

Bhakta

Pucci

2020

Nonlinear Analysis

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This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity:where s ∈ (0, 1), N > 2s, 2 * s := 2N N−2s , 0 < a ∈ L ∞ (R N ) and f is a nonnegative nontrivial functional in the dual space ofḢ s i.e., (Ḣ s ) ′ f, u Ḣs ≥ 0, whenever u is a nonnegative function inḢ s . We prove existence of a positive solution whose energy is negative. Further, under the additional assumption that a is a continuous function, a(x) ≥ 1 in R N , a(x) → 1 as |x| → ∞ and f Ḣs (R N ) ′ is small enough (but f ≡ 0), we establish existence of at least two positive solutions to (E). 2010 MSC: 35R11, 35A15, 35B33, 35J60 LetḢ s (R N ) := u ∈ L 2 * s (R N ) :¨R 2N |u(x) − u(y)| 2 |x − y| N +2s dx dy < ∞ , be the homogeneous fractional Sobolev space, endowed with the inner product ·, · Ḣs and corresponding Gagliardo norm u Ḣs (R N ) := ¨R 2N |u(x) − u(y)| 2 |x − y| N +2s dx dy

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“…where f ≥ 0, f ∈ L ∞ (Ω) has been studied in [1] and existence of two positive solutions have been established in [20] when f is a continuous function with compact support in Ω.…”

Section: Introductionmentioning

confidence: 99%

On multiplicity of positive solutions for nonlocal equations with critical nonlinearity

Bhakta

Pucci

2020

Nonlinear Analysis

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“…In the nonlocal case, when the domain is a bounded subset of R N , existence of positive solutions of (E γ K,t,f ) in Ω with γ = 0 = t (i.e., without Hardy and Hardy-Sobolev terms) and Dirichlet boundary condition has been proved in [23]. Existence of sign changing solutions of [4] and existence of two positive solutions have been established in [27] when f is a continuous function with compact support in Ω. In the nonlocal case, when the domain is the entire space R N , but γ = 0, we refer to [6,7], where multiplicity of positive solutions have been studied in presence of a nonhomogeneous term.…”

Section: Introductionmentioning

confidence: 99%

Fractional Hardy-Sobolev equations with nonhomogeneous terms

Bhakta¹,

Chakraborty²,

Pucci³

2020

Preprint

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This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity:whereN−2s . Here 0 < γ < γ N,s and γ N,s is the best Hardy constant in the fractional Hardy inequality. The coefficient K is a positive continuous function on R N , with K(0) = 1 = lim |x|→∞ K(x). The perturbation f is a nonnegative nontrivial functional in the dual space Ḣs (R N ) of Ḣs (R N ) i.e., ( Ḣs ) ′ f, u Ḣs ≥ 0, whenever u is a nonnegative function in Ḣs (R N ). We establish the profile decomposition of the Palais-Smale sequence associated with the functional. Further, if K ≥ 1 and f ( Ḣs ) ′ is small enough (but f ≡ 0), we establish existence of at least two positive solutions to the above equation.

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Fractional Hardy-Sobolev equations with nonhomogeneous terms

Bhakta

Chakraborty

Pucci

2021

Advances in Nonlinear Analysis

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This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity: ( − Δ ) s u − γ u | x | 2 s = K ( x ) | u | 2 s ∗ ( t ) − 2 u | x | t + f ( x ) in R N , u ∈ H ˙ s ( R N ) , $$\begin{array}{} \displaystyle \begin{cases} (-{\it\Delta})^s u -\gamma\dfrac{u}{|x|^{2s}}=K(x)\dfrac{|u|^{2^*_s(t)-2}u}{|x|^t}+f(x) \quad\mbox{in}\quad\mathbb R^N,\\ \qquad\qquad\qquad\quad u\in \dot{H}^s(\mathbb R^N), \end{cases} \end{array}$$ where N > 2s, s ∈ (0, 1), 0 ≤ t < 2s < N and 2 s ∗ ( t ) := 2 ( N − t ) N − 2 s $\begin{array}{} \displaystyle 2^*_s(t):=\frac{2(N-t)}{N-2s} \end{array}$ . Here 0 < γ < γ N,s and γ N,s is the best Hardy constant in the fractional Hardy inequality. The coefficient K is a positive continuous function on ℝ N , with K(0) = 1 = lim|x|→∞ K(x). The perturbation f is a nonnegative nontrivial functional in the dual space Ḣs (ℝ N )′ of Ḣs (ℝ N ). We establish the profile decomposition of the Palais-Smale sequence associated with the functional. Further, if K ≥ 1 and ∥f∥(Ḣs )′ is small enough (but f ≢ 0), we establish existence of at least two positive solutions to the above equation.

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Existence of multiple positive solutions for nonhomogeneous fractional Laplace problems with critical growth

Abstract: We prove the existence of multiple positive solutions of fractional Laplace problems with critical growth by using the method of monotonic iteration and variational methods.

Cited by 3 publications

References 17 publications

On multiplicity of positive solutions for nonlocal equations with critical nonlinearity

On multiplicity of positive solutions for nonlocal equations with critical nonlinearity

Fractional Hardy-Sobolev equations with nonhomogeneous terms

Fractional Hardy-Sobolev equations with nonhomogeneous terms

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