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Journal ArticleDOI

Positive solutions for nonlinear Choquard equation with singular nonlinearity

02 Feb 2017-Complex Variables and Elliptic Equations (Taylor & Francis)-Vol. 62, Iss: 8, pp 1044-1071
TL;DR: In this article, the existence and multiplicity of positive weak solutions of the Choquard equation with singular non-linearity was studied and the regularity of these weak solutions was studied.
Abstract: In this article, we study the following non-linear Choquard equation with singular non-linearitywhere is a bounded domain in with smooth boundary , and . Using variational approach and structure of associated Nehari manifold, we show the existence and multiplicity of positive weak solutions of the above problem, if is less than some positive constant. We also study the regularity of these weak solutions.
Citations
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Journal ArticleDOI
TL;DR: In this article, the existence/multiplicity results for the variable order nonlocal Choquard problem with variable exponents were studied and the existence and multiplicity results were derived.
Abstract: In this article, we study the existence/multiplicity results for the variable order nonlocal Choquard problem with variable exponents (−Δ)p(⋅)s(⋅)u(x)=λ|u(x)|α(x)−2u(x)+∫ΩF(y,u(y))|x−y|μ(x,y)dyf(x,...

29 citations

Book ChapterDOI
TL;DR: A survey of recent developments and results on Choquard equations is given in this article, where the authors focus on the existence and multiplicity of solutions of the partial differential equations which involves the nonlinearity of the convolution type.
Abstract: This article deals with a survey of recent developments and results on Choquard equations where we focus on the existence and multiplicity of solutions of the partial differential equations which involves the nonlinearity of the convolution type. Because of its nature, these equations are categorized under the nonlocal problems. We give a brief survey on the work already done in this regard following which we illustrate the problems we have addressed. Seeking the help of variational methods and asymptotic estimates, we prove our main results.

18 citations

Journal ArticleDOI
TL;DR: In this paper, the existence of least energy sign-changing solutions by considering the Nehari nodal set is investigated by using a minimization method on the associated Nehari manifold, where the groundstate solutions are obtained by using the minimum energy sign changing solution.
Abstract: Abstract We study the equation ( - Δ ) s ⁢ u + V ⁢ ( x ) ⁢ u = ( I α * | u | p ) ⁢ | u | p - 2 ⁢ u + λ ⁢ ( I β * | u | q ) ⁢ | u | q - 2 ⁢ u in ⁢ ℝ N , (-\\Delta)^{s}u+V(x)u=(I_{\\alpha}*\\lvert u\\rvert^{p})\\lvert u\\rvert^{p-2}u+% \\lambda(I_{\\beta}*\\lvert u\\rvert^{q})\\lvert u\\rvert^{q-2}u\\quad\\text{in }{% \\mathbb{R}}^{N}, where I γ ⁢ ( x ) = | x | - γ {I_{\\gamma}(x)=\\lvert x\\rvert^{-\\gamma}} for any γ ∈ ( 0 , N ) {\\gamma\\in(0,N)} , p , q > 0 {p,q>0} , α , β ∈ ( 0 , N ) {\\alpha,\\beta\\in(0,N)} , N ≥ 3 {N\\geq 3} , and λ ∈ ℝ {\\lambda\\in{\\mathbb{R}}} . First, the existence of groundstate solutions by using a minimization method on the associated Nehari manifold is obtained. Next, the existence of least energy sign-changing solutions is investigated by considering the Nehari nodal set.

18 citations

Posted Content
TL;DR: In this article, the existence/multiplicity results for the variable order nonlocal Choquard problem with variable exponents were studied under the Hardy-Sobolev-Littlewood-type result for the fractional Sobolev space.
Abstract: In this article, we study the existence/multiplicity results for the following variable order nonlocal Choquard problem with variable exponents (-\Delta)_{p(\cdot)}^{s(\cdot)}u(x)&=\lambda|u(x)|^{\alpha(x)-2}u(x)+ \left(\DD\int_\Omega\frac{F(y,u(y))}{|x-y|^{\mu(x,y)}}dy\right)f(x,u(x)), x\in \Omega, u(x)&=0, x\in \mathbb R^N\setminus\Omega, where $\Omega\subset\mathbb R^N$ is a smooth and bounded domain, $N\geq 2$, $p,s,\mu$ and $\alpha$ are continuous functions on $\mathbb R^N\times\mathbb R^N$ and $f(x,t)$ is Carathedory function. Under suitable assumption on $s,p,\mu,\alpha$ and $f(x,t)$, first we study the analogous Hardy-Sobolev-Littlewood-type result for variable exponents suitable for the fractional Sobolev space with variable order and variable exponents. Then we give the existence/multiplicity results for the above equation.

13 citations


Cites background from "Positive solutions for nonlinear Ch..."

  • ...For more results on Choquard problem involving concaveconvex nonlinearities we refer ([30]-[31],[35]-[36])....

    [...]

Journal ArticleDOI
TL;DR: In this article, the authors considered the magnetic nonlinear Choquard equation and established the existence of least energy solution under some suitable conditions, where the concentration behavior of solutions is studied as μ → + ∞.

9 citations

References
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Posted Content
TL;DR: In this article, the authors considered the Dirichlet boundary condition of the Choquard equation and proved existence and multiplicity results for the equation by variational methods under suitable assumptions on different types of nonlinearities.
Abstract: We consider the following nonlinear Choquard equation with Dirichlet boundary condition $$-\Delta u =\left(\int_{\Omega}\frac{|u|^{2_{\mu}^{\ast}}}{|x-y|^{\mu}}dy\right)|u|^{2_{\mu}^{\ast}-2}u+\lambda f(u)\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \Omega, $$ where $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$, $\lambda>0$, $N\geq3$, $0<\mu

8 citations

Journal ArticleDOI
TL;DR: Using shooting methods and asymptotic analysis of ODEs, it is shown that if λ > 0 is small enough, (Pλ ) admits at least one weak solution (in the sense of dis- tributions), and the borderline between uniqueness and multiplicity is given by the growth condition.
Abstract: Let B1 be the unit open ball with center at the origin in IRN , N ≥ 2, . We consider the following quasilinear problem depending on a real parameter λ > 0:   −∆N u = λf (u) in Ω, u > 0 (Pλ )  u = 0 on ∂Ω, N/N −1 where f (t) is a nonlinearity that grows like et as t → ∞ and behaves like tα , + . More precisely, we require f to satisfy assump- for some α ∈ (−∞, 0), as t → 0 tions (A1)-(A2) listed in Section 1. For such a general nonlinearity we show that if λ > 0 is small enough, (Pλ ) admits at least one weak solution (in the sense of dis- tributions). We further study the question of uniqueness and multiplicity of solutions to (Pλ ) when Ω = B1 under additional structural conditions on f (see assumptions (A3)-(A8) in Section 2). Using shooting methods and asymptotic analysis of ODEs, under the additional assumptions (A3)-(A5), we prove uniqueness of solution to (Pλ ) for all λ > 0 small whereas under (A6)-(A7) or (A8), we show multiplicity of so- lutions to (Pλ ) for all λ > 0 in a maximal interval. These results clearly show that the borderline between uniqueness and multiplicity is given by the growth condition 1 lim inf h(t)te t N −1 t→∞ = ∞ ∀ > 0.

5 citations