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
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Book•
07 Jan 2013
TL;DR: In this article, Leray-Schauder and Harnack this article considered the Dirichlet Problem for Poisson's Equation and showed that it is a special case of Divergence Form Operators.
Abstract: Chapter 1. Introduction Part I: Linear Equations Chapter 2. Laplace's Equation 2.1 The Mean Value Inequalities 2.2 Maximum and Minimum Principle 2.3 The Harnack Inequality 2.4 Green's Representation 2.5 The Poisson Integral 2.6 Convergence Theorems 2.7 Interior Estimates of Derivatives 2.8 The Dirichlet Problem the Method of Subharmonic Functions 2.9 Capacity Problems Chapter 3. The Classical Maximum Principle 3.1 The Weak Maximum Principle 3.2 The Strong Maximum Principle 3.3 Apriori Bounds 3.4 Gradient Estimates for Poisson's Equation 3.5 A Harnack Inequality 3.6 Operators in Divergence Form Notes Problems Chapter 4. Poisson's Equation and Newtonian Potential 4.1 Holder Continuity 4.2 The Dirichlet Problem for Poisson's Equation 4.3 Holder Estimates for the Second Derivatives 4.4 Estimates at the Boundary 4.5 Holder Estimates for the First Derivatives Notes Problems Chapter 5. Banach and Hilbert Spaces 5.1 The Contraction Mapping 5.2 The Method of Cintinuity 5.3 The Fredholm Alternative 5.4 Dual Spaces and Adjoints 5.5 Hilbert Spaces 5.6 The Projection Theorem 5.7 The Riesz Representation Theorem 5.8 The Lax-Milgram Theorem 5.9 The Fredholm Alternative in Hilbert Spaces 5.10 Weak Compactness Notes Problems Chapter 6. Classical Solutions the Schauder Approach 6.1 The Schauder Interior Estimates 6.2 Boundary and Global Estimates 6.3 The Dirichlet Problem 6.4 Interior and Boundary Regularity 6.5 An Alternative Approach 6.6 Non-Uniformly Elliptic Equations 6.7 Other Boundary Conditions the Obliue Derivative Problem 6.8 Appendix 1: Interpolation Inequalities 6.9 Appendix 2: Extension Lemmas Notes Problems Chapter 7. Sobolev Spaces 7.1 L^p spaces 7.2 Regularization and Approximation by Smooth Functions 7.3 Weak Derivatives 7.4 The Chain Rule 7.5 The W^(k,p) Spaces 7.6 DensityTheorems 7.7 Imbedding Theorems 7.8 Potential Estimates and Imbedding Theorems 7.9 The Morrey and John-Nirenberg Estimes 7.10 Compactness Results 7.11 Difference Quotients 7.12 Extension and Interpolation Notes Problems Chapter 8 Generalized Solutions and Regularity 8.1 The Weak Maximum Principle 8.2 Solvability of the Dirichlet Problem 8.3 Diferentiability of Weak Solutions 8.4 Global Regularity 8.5 Global Boundedness of Weak Solutions 8.6 Local Properties of Weak Solutions 8.7 The Strong Maximum Principle 8.8 The Harnack Inequality 8.9 Holder Continuity 8.10 Local Estimates at the Boundary 8.11 Holder Estimates for the First Derivatives 8.12 The Eigenvalue Problem Notes Problems Chapter 9. Strong Solutions 9.1 Maximum Princiles for Strong Solutions 9.2 L^p Estimates: Preliminary Analysis 9.3 The Marcinkiewicz Interpolation Theorem 9.4 The Calderon-Zygmund Inequality 9.5 L^p Estimates 9.6 The Dirichlet Problem 9.7 A Local Maximum Principle 9.8 Holder and Harnack Estimates 9.9 Local Estimates at the Boundary Notes Problems Part II: Quasilinear Equations Chapter 10. Maximum and Comparison Principles 10.1 The Comparison Principle 10.2 Maximum Principles 10.3 A Counterexample 10.4 Comparison Principles for Divergence Form Operators 10.5 Maximum Principles for Divergence Form Operators Notes Problems Chapter 11. Topological Fixed Point Theorems and Their Application 11.1 The Schauder Fixes Point Theorem 11.2 The Leray-Schauder Theorem: a Special Case 11.3 An Application 11.4 The Leray-Schauder Fixed Point Theorem 11.5 Variational Problems Notes Chapter 12. Equations in Two Variables 12.1 Quasiconformal Mappings 12.2 holder Gradient Estimates for Linear Equations 12.3 The Dirichlet Problem for Uniformly Elliptic Equations 12.4 Non-Uniformly Elliptic Equations Notes Problems Chapter 13. Holder Estimates for
18,443 citations
21 Mar 2001
1,723 citations
TL;DR: In this article, the Hartree-Fock theory of a plasma was used to prove existence and uniqueness of a minimization of the functional function of an electron trapped in its own hole.
Abstract: The equation dealt with in this paper is in three dimensions. It comes from minimizing the functional which, in turn, comes from an approximation to the Hartree-Fock theory of a plasma. It describes an electron trapped in its own hole. The interesting mathematical aspect of the problem is that & is not convex, and usual methods to show existence and uniqueness of the minimum do not apply. By using symmetric decreasing rearrangement inequalities we are able to prove existence and uniqueness (modulo translations) of a minimizing Φ. To prove uniqueness a strict form of the inequality, which we believe is new, is employed.
1,039 citations
TL;DR: In this paper, the authors considered a semilinear elliptic problem and proved the existence of a positive groundstate solution of the Choquard or nonlinear Schr\"odinger--Newton equation for an optimal range of parameters.
658 citations
Book•
01 Mar 2016TL;DR: A thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators can be found in this paper, where the authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of equations, plus their application to various processes arising in the applied sciences.
Abstract: This book provides researchers and graduate students with a thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations, plus their application to various processes arising in the applied sciences. The equations are examined from several viewpoints, with the calculus of variations as the unifying theme. Part I begins the book with some basic facts about fractional Sobolev spaces. Part II is dedicated to the analysis of fractional elliptic problems involving subcritical nonlinearities, via classical variational methods and other novel approaches. Finally, Part III contains a selection of recent results on critical fractional equations. A careful balance is struck between rigorous mathematics and physical applications, allowing readers to see how these diverse topics relate to other important areas, including topology, functional analysis, mathematical physics, and potential theory.
613 citations