In this paper, we study the Schrödinger–Poisson system
(
SP)
{
−
Δ
u
+
u
+
K
(
x
)
ϕ
(
x
)
u
=
a
(
x
)
f
(
u
)
,
in
R
3
,
−
Δ
ϕ
=
K
(
x
)
u
2
,
in
R
3
,
and prove the existence of ground state ...solutions for system
(
SP)
under certain assumptions on the linear and nonlinear terms. Some recent results from different authors are extended.
In this paper, we prove the existence of a sign-changing solution for a p-Laplacian elliptic problem with constraint in RN. By constructing a pseudo gradient vector field and a descending flow, we ...obtain three solutions and among these solutions, one is positive, one is negative and one is sign-changing.
In this paper, we study the existence of infinitely many nontrivial solutions for a class of nonlinear Schrödinger–Kirchhoff type equation-a+b∫RN|∇u|2dxΔu+V(x)u=f(x,u)inRN,u(x)→0,as|x|→∞,where ...constants a>0,b>0 and the potential V(x) is allowed to be sign-changing. Under general superlinear assumption on nonlinearity f(x,u), we establish the existence of infinitely many solutions via variational methods, which unifies and improves the recent results of Wu (2011) 9.
This work aims to develop the variational framework for some Kirchhoff problems involving both the
p$$ p $$‐Laplace operator and the
ψ$$ \psi $$‐Hilfer derivative. Precisely, we use the mountain pass ...theorem to prove the existence of nontrivial solutions. Moreover, the multiplicity of solutions is proved by the use of the
Z2$$ {Z}_2 $$‐symmetry mountain pass theorem. Our main results generalize the paper of Torres (J Fract Calculus Appli. 2014;5(1):1‐10) and the work of Sousa et al. (Comp Appl Math. 2019;38:4).
Abstract
We consider a general class of nonlocal problems involving the fractional Laplacian and singular nonlinearities and we deal with the variational characterization of the solutions. Even ...though solutions generally have not fractional Sobolev regularity, we provide a variational characterization of the solutions via a suitable action functional.
This article concerns with a class of nonlocal fractional Laplacian problems depending on two real parameters. Our approach is based on variational methods. We establish the existence of three weak ...solutions via a recent abstract result by Ricceri about nonlocal equations.
Chiral spin textures are widely researched in condensed matter systems and have shown potential for use in spintronics and storage applications. Photonic counterparts of these textures are observed ...in various optical systems with broken inversion symmetry. Unfortunately, the resemblances are only phenomenological. This work proposes a theoretical framework based on the variational theorem to show that the photonic chiral spin textures in an optical interface originate from the system's symmetry and relativity. Analysis of the rotational symmetry in optical systems indicates that the conservation of the total angular momentum is preserved from the local variations of spin vectors. Specifically, although the integral spin momentum does not carry net energy, the local spin momentum distribution, which determines the local subluminal energy transport and minimization variation of the square of the total angular momentum, results in the chiral twisting of the local spin vectors. The findings of this study deepen the understanding of the symmetries, conservative laws, and energy transport in optical systems, reveal the similarities in the formation mechanisms and the geometries of photonic and condensed‐matter chiral spin textures, and have potential applications in chiral and spin photonics.
A theoretical framework based on the variational theorem shows that the photonic chiral spin textures are originated from the system's symmetry and relativity. The findings deepen the understanding of the symmetries, conservative laws, and energy transport in optical systems, reveal the similarities in formation mechanisms and geometries of photonic and condensed‐matter chiral spin textures, and provide application in spin photonics.
In this paper, we study the fractional Choquard equation (−Δ)su+u=(|x|−μ∗F(u))f(u),inRN,where N≥3, 0<s<1, 0<μ<min{N,4s}, and f∈C(R,R) satisfies the general Berestycki–Lions conditions. Combining ...constrained variational method with deformation lemma, we obtain a ground state solution of Pohoz̆aev type for the above equation. The result improves some ones in Shen et al. (2016).
We consider an open connected set Ω and a smooth potential U which is positive in Ω and vanishes on ∂Ω. We study the existence of orbits of the mechanical system u¨=Ux(u), that connect different ...components of ∂Ω and lie on the zero level of the energy. We allow that ∂Ω contains a finite number of critical points of U. The case of symmetric potential is also considered.
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