In the present paper, in view of the variational approach, we consider the existence and multiplicity of weak solutions for a class of the double phase problem ...−div(|∇u|p−2∇u+a(x)|∇u|q−2∇u)=λf(x,u),inΩ,u=0,on∂Ω,where N≥2 and 1<p<q<N. Firstly, by the Fountain and Dual Theorem with Cerami condition, we obtain some existence of infinitely many solutions for the above problem under some weaker assumptions on f. Secondly, we prove that this problem has at least one nontrivial solution for any parameter λ>0 small enough, and also that the solution blows up, in the Sobolev norm, as λ→0+. Finally, by imposing additional assumptions on f, we establish the existence of infinitely many solutions by using Krasnoselskii’s genus theory for the above equation.
The paraboloid structure was first predicted based on the variational method, as an effective platform to circumvent the limitations of tip-charge induced self-discharge. Consequently, the ...metal–organic frameworks (MOFs)-derived nanowire arrays with the ideal parabola-sharp deliver an excellent specific capacitance (1483 C g−1 at 1 A g−1), high cyclic stability and an ultralow leakage current of 10.28 μA.
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•Zn-Ni-CoP@CoP NWRs with paraboloid structure are firstly synthesized.•The variational method is used.•The ideal paraboloid sharp Zn-Ni-CoP@CoP NWRs circumvents the self-discharge.•The Zn-Ni-CoP@CoP NWRs exhibit an ultralow leakage current of 10.28 μA.
Nanosized electrode materials have received wide attention in electrochemistry since their large specific surface areas are the precondition for the excellent performance of electrochemistry, such as supercapacitors, Li-ion batteries, Na-ion batteries, electrocatalysis, photoelectrocatalysis, electrolysis and electrodeposition. However it also leads to the high value of curvature of nanomaterials, causing the unavoidable and non-ignorable leakage currents, and therefore the activity degradation. Here, guided by the variational method, we first theoretically predict that, with giving a certain value of curvature, the paraboloid structure possesses the maximal surface area to greatly reduce the tip-charge, thus guaranteeing the electrochemical performance. Consequently, with using supercapacitors as the model application, this unusual morphological structure was successfully constructed in the metal-organic frameworks (MOFs)-derived Zn-Ni-Co-P@CoP nanowire arrays (ZNCP@CoP NWRs), which largely ensures the ZNCP@CoP NWRs || AC device exhibits an ultralow leakage current of 10.28 μA. The low curvature structure of ZNCP@CoP NWRs also significantly improves the electrochemical performance of supercapacitors with a large specific capacitance of 1483 C g−1 at 1 A g−1 and a high cyclic stability of 80% capacity maintained over 5000 cycles at 5 A g−1. The finding in this work highlights the great potential of the reasonable material design approach for advanced energy applications.
In this article, we study the existence of multiple solutions for nonlinear impulsive problems with small non-autonomous perturbations. We show the existence of at least three distinct classical ...solutions by using variational methods and a three critical points theorem.
In this paper, we study the critical Kirchhoff type fractional Schrödinger equation:(0.1)(1+α∫R3∫R3|u(x)−u(y)|2|x−y|3+2sdxdy)(−Δ)su+u=βf(u)+u2s⁎−1inR3, where s∈(0,1) and 2s⁎=63−2s. We establish the ...Pohozǎev type identity of (0.1). When s∈34,1), under some conditions on α, β and f(u), we obtain some results on the existence of ground state solutions. When s∈(0,34, we also prove the non-existence result. In particular, when α=0, we obtain an existence result.
In this paper, we study the desingularization of vortices for the 3D incompressible Euler equations in an infinite pipe. We construct a family of traveling-rotating helical vortices for the Euler ...equations with a general vorticity function, which tends asymptotically to singular helical vortex filament evolved by the binormal curvature flow. The results are obtained by using an improved vorticity method. Some asymptotic properties of this family of solutions have also been studied.
We are concerned with the existence of multi-peak solutions to Kirchhoff equation(0.1)−(ϵ2a+ϵb∫R3|∇u|2dx)Δu+V(x)u=f(u),x∈R3, where ϵ>0 is a small parameter, a, b>0 are constants. Under general ...conditions of f, we construct a family positive solutions uϵ∈H1(R3) which concentrates around the isolated components of positive local minima of V as ϵ→0+. Our result generalize the previous results on single peak solutions to multi-peak solutions.
We consider the existence of solutions for the following Schrödinger-Poisson system with indefinite nonlinearity{−Δu+u+μϕu=a(x)|u|p−2u+λk(x)u,x∈R3,−Δϕ=u2,ϕ∈D1,2(R3), where p∈(2,4) and the parameters ...μ,λ>0, the functions 0<k(x)∈L32(R3) and a(x)∈C(R3,R) satisfying a∞:=lim|x|→∞a(x)<0<maxx∈R3a(x)=:amax and other suitable conditions. We prove the existence and multiplicity results depending on μ, λ and p. We also study the asymptotic behavior of solutions when the parameter μ→0+.
We are interested in the asymptotic behavior of ground states for a class of quasilinear elliptic equations in
when the nonlinear term has
-critical growth. In the previous result Adachi et al. ...Asymptotic property of ground states for a class of quasilinear Schrödinger equation with
-critical growth. Calc Var Partial Differential Equations. 2019;58(3). Art. 88, 29 pp., it was shown that, after a suitable scaling, the ground state converges to the Talenti function. However, the uniqueness of the limit of the full sequence was not obtained, which was essentially owning to the fact that the Talenti function does not belong to
. In this paper, by constructing a refined test function and performing a detailed asymptotic analysis, we are able to obtain the uniqueness of asymptotic limit of ground states.
This paper establishes a dual variational framework for studying the existence and multiplicity of solutions to a class of Helmholtz systems. We first focus on a compact nonlinear potential and prove ...the existence and multiplicity of solutions via the symmetric Mountain Pass theorem. Next, we investigate the periodic nonlinear potential case and prove a nonvanishing theorem to recover the loss of compactness. We then demonstrate the existence of solutions using the Nehari manifold method and the multiplicity of solutions using the pseudoindex theory. Additionally, the regularity of solutions is given.
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