Positive solutions for nonlinear Choquard equation with singular nonlinearity

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.

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.

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,...

Abstract: In this paper, we consider a nonautonomous Choquard equation with singularity − Δ u + V ( x ) u + λ ( I α ∗ | u | p ) | u | p − 2 u = f ( x ) u − γ , x ∈ R 3 , u > 0 , x ∈ R 3 , where I α is the Riesz potential of order α ∈ ( 0 , 3 ) and 1 + α 3 ≤ p 3 + α , 0 γ 1 . Under certain assumptions on V and f , we show the existence and uniqueness of positive solution for λ > 0 by using variational method. We also study the asymptotic behavior of solutions as λ → 0 .

Abstract: In the present paper, we consider the following magnetic nonlinear Choquard equation $$ \left\{ \begin{array}{ll} & (-i abla+A(x))^2u + \mu g(x)u = \lambda u + (|x|^{-\alpha} * |u|^{2^*_\alpha})|u|^{2^*_\alpha-2}u ,\; u>0 \;\text{in} \; \mathbb{ R}^n , & u \in H^1(\mathbb{R}^n, \mathbb{ C}) \end{array} \right\}.$$ where $n \geq 4$, $2^*_\alpha= \frac{2n-\alpha}{n-2}$, $\lambda>0$, $\mu \in \mathbb{ R}$ is a parameter, $\alpha \in (0,n)$, $A(x): \mathbb{R}^n \rightarrow \mathbb{ R}^n$ is a magnetic vector potential and $g(x)$ is a real valued potential function on $\mathbb{R}^n$. Using variational methods, we establish the existence of least energy solution under some suitable conditions. Moreover, the concentration behavior of solutions is also studied as $\mu \rightarrow +\infty$.

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

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.

Abstract: (1977). On a dirichlet problem with a singular nonlinearity. Communications in Partial Differential Equations: Vol. 2, No. 2, pp. 193-222.

Abstract: We consider a semilinear elliptic problem \[ - \Delta u + u = (I_\alpha \ast \abs{u}^p) \abs{u}^{p - 2} u\quad\text{in \(\R^N\),} \] where \(I_\alpha\) is a Riesz potential and \(p>1\). This family of equations includes the Choquard or nonlinear Schr\"odinger--Newton equation. For an optimal range of parameters we prove the existence of a positive groundstate solution of the equation. We also establish regularity and positivity of the groundstates and prove that all positive groundstates are radially symmetric and monotone decaying about some point. Finally, we derive the decay asymptotics at infinity of the groundstates.