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.
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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
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
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
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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])....
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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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TL;DR: In this article, a dirichlet problem with singular nonlinearity is considered, and the authors present a communication in Partial Differential Equations (PDE) approach to solve it.
Abstract: (1977). On a dirichlet problem with a singular nonlinearity. Communications in Partial Differential Equations: Vol. 2, No. 2, pp. 193-222.
596 citations
TL;DR: In this paper, the relation between the Nehari manifold and fibrering maps was explored and it was shown how existence and non-existence results for positive solutions of the equation are linked to properties of the manifold.
390 citations
01 Jan 1993
TL;DR: In this paper, a fonctionelles de la forme Φ (u)=(1/2)∫ Ω |⊇u| 2 -∫ F(x, u).
Abstract: On considere de fonctionelles de la forme Φ (u)=(1/2)∫ Ω |⊇u| 2 -∫ Ω F(x, u). Sous des hypotheses convenables on prouve qu'un minimum local de Φ au sens de la norme C 1 est necessairement un minimum local au sens de la norme H 1 . Ce resultat est particulierement utile dons le cas ou l'equation correspondante admet une sous-solution et une sur-solution
268 citations
Additional excerpts
...k → ∞ as k → ∞. Therefore, u ≤ (δ), since {(uk − (δ))+} converges to (u− (δ))+ almost everywhere, as k→ ∞. Using (δ) ≤ 2α¯hδ , we obtain the conclusion. We need the following result (Theorem 3 in [6]) to prove our next result. Lemma 6.7 Let Ω be a bounded domain in Rn with smooth boundary ∂Ω. Let u∈ L1 loc(Ω) and assume that for some k≥ 0, usatisfies, in the sense of distributions −∆u+ku≥ 0 in Ω, ...
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TL;DR: In this paper, the authors prove existence, regularity and nonexistence results for problems whose model is γ > 0 and f is a nonnegative function on the boundary of an open, bounded subset of the Euclidean space.
Abstract: We prove existence, regularity and nonexistence results for problems whose model is
$$-\Delta u = \frac{f(x)}{u^{\gamma}}\quad {{\rm in}\,\Omega},$$
with zero Dirichlet conditions on the boundary of an open, bounded subset Ω of \({\mathbb{R}^{N}}\). Here γ > 0 and f is a nonnegative function on Ω. Our results will depend on the summability of f in some Lebesgue spaces, and on the values of γ (which can be equal, larger or smaller than 1).
264 citations
Book•
01 Jan 2008
TL;DR: In this paper, the authors consider singular Gierer-meinhardt systems and singular singular elliptic problems with singular nonlinear nonlinear convection terms and show that the influence of a nonlinear term in singular elliptical problems can influence the stability of the solution of a singular problem.
Abstract: II BLOW-UP SOLUTIONS 2 Blow-up solutions for semilinear elliptic equations 3 Entire solutions blowing-up at infinity for elliptic systems III ELLIPTIC PROBLEMS WITH SINGULAR NONLINEARITIES 4 Sublinear perturbations of singular elliptic problems 5 Bifurcation and asymptotic analysis The monotone case 6 Bifurcation and asymptotic analysis The nonmonotone case 7 Superlinear perturbations of singular elliptic problems 8 Stability of the solution of a singular problem 9 The influence of a nonlinear convection term in singular elliptic problems 10 Singular Gierer-Meinhardt systems A Spectral theory for differential operators B Implicit function theorem C Ekeland's variational principle D Mountain pass theorem References Index
245 citations