In this paper we study the Kirchhoff problem{−m(‖u‖2)Δu=f(u)in Ω,u=0on ∂Ω, in a bounded domain, allowing the function m to vanish in many different points. Under an appropriated area condition, by ...using a priori estimates, truncation techniques and variational methods, we prove a multiplicity result of positive solutions which are ordered in the H01(Ω)-norm.
In this paper, we focus on the existence of positive solutions to the following planar Schrödinger-Newton system with general subcritical growth{−Δu+u+ϕu=f(u)inR2,Δϕ=u2inR2, where f is a smooth ...reaction. We introduce a new variational approach, which enables us to study the above problem in the Sobolev space H1(R2). The analysis developed in this paper also allows to investigate the relationship between a Schrödinger-Newton system of Riesz-type and a Schrödinger-Newton system of logarithmic-type. Furthermore, this new approach can provide a new look at the planar Schrödinger-Newton system and may it have some potential applications in various related problems.
In this paper, we study an elliptic variational problem regarding the
p$$ p $$‐fractional Laplacian in
ℝN$$ {\mathrm{\mathbb{R}}}^N $$ on the basis of recent result which generalizes some ...nice published work, and then give some sufficient conditions under which some weak solutions to our studied elliptic variational problem are continuous in
ℝN$$ {\mathrm{\mathbb{R}}}^N $$. In the final appendix, we correct the proofs of two published lemmas for
1<p<2$$ 1<p<2 $$.
On a Kirchhoff type problem in RN Wu, Yuanze; Huang, Yisheng; Liu, Zeng
Journal of mathematical analysis and applications,
05/2015, Volume:
425, Issue:
1
Journal Article
Peer reviewed
Open access
In this paper, we investigate the following Kirchhoff type problem:(Pα,β){(α∫RN(|∇u|2+u2)dx+β)(−Δu+u)=|u|p−2uin RN,u∈H1(RN), where N≥1, 2<p<2⁎ (2⁎=2N/(N−2) if N≥3, 2⁎=∞ if N=1,2) and α, β are two ...positive parameters. By studying the decomposition of the Nehari manifold to (Pα,β) and using the scaling technique, we give a total description on the positive solutions to (Pα,β). We also make an observation on the sign-changing solutions to (Pα,β) in the current paper.
Dense motion estimations obtained from optical flow techniques play a significant role in many image processing and computer vision tasks. Remarkable progress has been made in both theory and its ...application in practice. In this paper, we provide a systematic review of recent optical flow techniques with a focus on the variational method and approaches based on Convolutional Neural Networks (CNNs). These two categories have led to state-of-the-art performance. We discuss recent modifications and extensions of the original model, and highlight remaining challenges. For the first time, we provide an overview of recent CNN-based optical flow methods and discuss their potential and current limitations.
•Introducing optical flow: the basic concepts, the characteristics of the variational and CNN-based techniques, and the evaluation measures.•Discussing developments of the variational method, analyzing the challenges and illustrating the corresponding treating strategies of it.•Describing the conception of the CNN-based technique, and give a detailed discussion of the issues of this technique.
In this paper, we study the existence and regularity of normalized solutions to the following doubly nonlocal equation−Δu=λu+μ1(Iα⁎|u|p)|u|p−2u+μ2(Iβ⁎|u|p)|u|p−2uinR3, having a prescribed mass ...∫R3u2=a>0, where λ∈R will arise as a Lagrange multiplier, α,β∈(0,3) with α⩽β, p∈2⁎β,2α⁎, Iα and Iβ are Riesz potentials, μ1,μ2>0 are constants. In particular, we estimate the energy level ingeniously and consider the existence of normalized solutions to the above equation with Hardy–Littlewood–Sobolev lower critical exponent p=2⁎β or upper critical exponent p=2α⁎. The results are supplements to the works of Cao et al. (2021) 5.
In this paper, we study the existence of positive solutions to a semilinear nonlocal Hénon type elliptic problem with the fractional s-Laplacian on Rn, 1∕2<s<n∕2. By showing a new inequality, we show ...that the problem has at least one positive solution in Hs(Rn). We also show that in some cases, via the use of Hardy type inequality for 0<s<n∕2, there is a nontrivial non-negative Hs(Rn) weak solution to the semilinear nonlocal elliptic problem with the fractional s-Laplacian on Rn and with Hardy type singularity term.
The aim of this paper is to analyze the multiplicity of solutions for a nonhomogeneous Kirchhoff-type problem on R4 with a critical exponent. The first two positive solutions are deduced by using the ...variational method, and the third solution is found with the help of Brezis-Lieb's lemma and Mazur's lemma. Moreover, we obtain a new compactness condition and improve on the existing results in the literature.
Let m≥1 and d≥2 be integers and consider a strip-like domain O×Rd, where O⊂Rm is a bounded Euclidean domain with smooth boundary. Furthermore, let p:O¯×Rd→R be a uniformly continuous and ...cylindrically symmetric function. We prove that the subspace of W1,p(x,y)(O×Rd) consisting of the cylindrically symmetric functions is compactly embedded into L∞(O×Rd) provided thatm+d<p−:=inf(x,y)∈O¯×Rdp(x,y)≤p+:=sup(x,y)∈O¯×Rdp(x,y)<+∞. As an application, we study a Neumann problem involving the p(x,y)-Laplacian operator and an oscillating nonlinearity, proving the existence of infinitely many cylindrically symmetric weak solutions. Our approach is based on variational and topological methods in addition to the principle of symmetric criticality.
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