Normalized solutions for fractional Choquard equation with critical growth on bounded domain

Some general foundational issues of quantum mechanics are considered and are related to aspects of quantum computation. The importance of quantum entanglement and quantum information is discussed a...

In this paper, we study the following fractional Choquard equation with critical or supercritical growth where 0<s<1, denotes the fractional Laplacian of order s, N>2s, and . Under some suitable conditions, we prove that the equation has a nontrivial solution for small by the variational method.

In this paper, we are concerned with the following fractional Choquard equation with critical growth: $$(-\Delta)^s u+\lambda V(x)u=(|x|^{-\mu} \ast F(u))f(u)+|u|^{2^*_s-2}u ~\hbox{in}~\mathbb{R}^N,$$ where $s\in (0,1)$, $N>2s$, $\mu\in (0,N)$, $2^*_s=\frac{2N}{N-2s}$ is the fractional critical exponent, $V$ is a steep well potential, $F(t)=\int_0^tf(s)ds$. Under some assumptions on $f$, the existence and asymptotic behavior of the positive ground states are established. In particular, if $f(u)=|u|^{p-2}u$, we obtain the range of $p$ when the equation has the positive ground states for three cases $2s<N<4s$ or $N=4s$ or $N>4s$.

Existence and multiplicity of solutions for fractional Choquard equations

<abstract> In this paper, we investigate the existence of the least energy sign-changing solutions for nonlinear elliptic equations driven by nonlocal integro-differential operators with critical nonlinearity. By using constrained minimization method and topological degree theory, we obtain a least energy sign-changing solution for them under much weaker conditions. As a particular case, we drive an existence theorem of sign-changing solutions for the fractional Laplacian equations with critical growth. </abstract>

Infinitely many solutions for p-fractional Choquard type equations involving general nonlocal nonlinearities with critical growth via the concentration compactness method

We study the existence and regularity of the solution to the multivalued equation -ΔΦu ∈ ∂j(u) + λh in Ω, where Ω ⊂ RN is a bounded smooth domain, Φ is an N-function, ΔΦ is the corresponding Φ-Laplacian, λ &gt; 0 is a parameter, h is a measurable function, and j is a continuous function with critical growth where ∂j(u) denotes its subdifferential. We apply the Ekeland Variational Principle to an associated locally Lipschitz energy functional. A major point in our study is that in order to deal with the obtained Ekeland sequence we developed a generalized version for the framework of Orlicz–Sobolev spaces of a well-known Brézis–Lieb lemma which was employed together with a variant of the Lions concentration-compactness theory to get a solution of the equation.

Multiplicity of semiclassical solutions for fractional Choquard equations with critical growth

In this paper, we study the singularly perturbed fractional Choquard equation ε 2 s ( − Δ ) s u + V ( x ) u = ε μ − 3 ( ∫ R 3 | u ( y ) | 2 μ , s ∗ + F ( u ( y ) ) | x − y | μ d y ) ( | u | 2 μ , s ∗ − 2 u + 1 2 μ , s ∗ f ( u ) ) in R 3 , $$\begin{equation*}\varepsilon^{2s}(-{\it\Delta})^su+V(x)u=\varepsilon^{\mu-3}(\int\limits_{\mathbb{R}^3}\frac{|u(y)|^{2^*_{\mu,s}}+F(u(y))}{|x-y|^\mu}dy)(|u|^{2^*_{\mu,s}-2}u+\frac{1}{2^*_{\mu,s}}f(u)) \, \text{in}\, \mathbb{R}^3, \end{equation*}$$ where ε &gt; 0 is a small parameter, (− △ ) s denotes the fractional Laplacian of order s ∈ (0, 1), 0 &lt; μ &lt; 3, 2 μ , s ⋆ = 6 − μ 3 − 2 s $2_{\mu ,s}^{\star }=\frac{6-\mu }{3-2s}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and fractional Laplace operator. F is the primitive of f which is a continuous subcritical term. Under a local condition imposed on the potential V , we investigate the relation between the number of positive solutions and the topology of the set where the potential attains its minimum values. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory.

In this paper we consider the following class of fractional Kirchhoff equations with critical growth: [Formula: see text] where [Formula: see text] is a small parameter, [Formula: see text] are constants, [Formula: see text], [Formula: see text] is the fractional critical exponent, [Formula: see text] is the fractional Laplacian operator, V is a positive continuous potential and f is a superlinear continuous function with subcritical growth. Using penalization techniques and variational methods, we prove the existence of a family of positive solutions [Formula: see text] which concentrates around a local minimum of V as [Formula: see text].

For a class of sub-elliptic equations on Heisenberg group \(\mathbb{H}^N\) with Hardy type singularity and critical nonlinear growth, we prove the existence of least energy solutions by developing new techniques based on the Nehari constraint. This result extends previous works, e.g., by Han et al. [Hardy-Sobolev type inequalities on the H-type group, Manuscripta Math. 118 (2005), 235–252].

This research explores the existence and multiplicity of solutions to N-Laplacian equations with discontinuous exponential nonlinearities in the whole Euclidean space. Through combining symmetric rearrangement techniques and variational methods for non-differentiable functionals, it identifies sufficient conditions for the existence of weak solutions when perturbation parameters are small, and uncovers the rich solution structure caused by discontinuous growth and non-smooth operators.These studies connect critical Sobolev growth and exponential nonlinearities, which is an important link in phase transition models and nonlinear analysis.We have proven the existence and multiplicity of weak solutions for the N-Laplacian equation with discontinuous exponential growth . Notably, when the perturbation parameter is sufficiently small, there exist at least multiple weak solutions, which stem from the interaction between the discontinuous exponential nonlinearity and the N-Laplacian operator. Compared to previous findings, our results extend the existing literature on elliptic equations with critical growth and discontinuous nonlinearities. Additionally, the combination of priori estimates with non-differentiable variational methods constitutes a novel approach, distinct from traditional techniques in earlier studies.

<p style='text-indent:20px;'>We are concerned with nonlinear fractional Choquard equations involving critical growth in the sense of the Hardy-Littlewood-Sobolev inequality. Without the Ambrosetti-Rabinowitz condition or monotonicity condition on the nonlinearity, we establish the existence of radially symmetric ground state solutions.

The first aim of this paper is to classify the positive solutions of the fractional Choquard equation where is the upper critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. Moreover, based on the uniqueness and non-degeneracy of the solution of the above equation, we then study the perturbed Choquard equation By using the finite-dimensional reduction, we obtain the existence of at least one positive solution if is suitable small.

This paper presents a method to solve the problems of solutions for integer differential and partial differential equations using the convergence of Adomian’s Method. In this paper, we firstly use the convergence of Adomian’s Method to derive the solutions of high order linear fractional equations, and then the numerical solutions for nonlinear fractional equations. we also get the solutions of two fractional reaction-diffusion equations. We can see the advantage of this method to deal with fractional differential equations.