scholarly journals Discontinuous perturbations of nonhomogeneous strongly-singular Kirchhoff problems

2021 ◽  
Vol 28 (6) ◽  
Author(s):  
Vicenţiu D. Rădulescu ◽  
Carlos Alberto Santos ◽  
Lais Santos ◽  
Marcos L. M. Carvalho
Keyword(s):  
Sobolev Space ◽  
Convex Analysis ◽  
Singular Term ◽  
Finite Energy ◽  
Energy Functional ◽  
Singular Problem ◽  
Discontinuous Nonlinearity ◽  
Generalized Gradient ◽  
Whole Space ◽  
Kirchhoff Problem

AbstractIn this paper, we are concerned with a Kirchhoff problem in the presence of a strongly-singular term perturbed by a discontinuous nonlinearity of the Heaviside type in the setting of Orlicz–Sobolev space. The presence of both strongly-singular and non-continuous terms brings up difficulties in associating a differentiable functional to the problem with finite energy in the whole space $$W_0^{1,\Phi }(\Omega )$$ W 0 1 , Φ ( Ω ) . To overcome this obstacle, we establish an optimal condition for the existence of $$W_0^{1,\Phi }(\Omega )$$ W 0 1 , Φ ( Ω ) -solutions to a strongly-singular problem, which allows us to constrain the energy functional to a subset of $$W_0^{1,\Phi }(\Omega )$$ W 0 1 , Φ ( Ω ) in order to apply techniques of convex analysis and generalized gradient in the sense of Clarke.

scholarly journals Periodic and asymptotically periodic fourth-order Schrödinger equations with critical and subcritical growth

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Edcarlos D. Silva ◽  
Marcos L. M. Carvalho ◽  
Claudiney Goulart
Keyword(s):  
Fourth Order ◽  
Elliptic Problem ◽  
Principal Part ◽  
Existence Of Solutions ◽  
Energy Functional ◽  
Critical Nonlinearities ◽  
Asymptotically Periodic ◽  
Whole Space ◽  
Continuous Potential ◽  
Order Elliptic Problem

<p style='text-indent:20px;'>It is established existence of solutions for subcritical and critical nonlinearities considering a fourth-order elliptic problem defined in the whole space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>. The work is devoted to study a class of potentials and nonlinearities which can be periodic or asymptotically periodic. Here we consider a general fourth-order elliptic problem where the principal part is given by <inline-formula><tex-math id="M2">\begin{document}$ \alpha \Delta^2 u + \beta \Delta u + V(x)u $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M3">\begin{document}$ \alpha &gt; 0, \beta \in \mathbb{R} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ V: \mathbb{R}^N \rightarrow \mathbb{R} $\end{document}</tex-math></inline-formula> is a continuous potential. Hence our main contribution is to consider general fourth-order elliptic problems taking into account the cases where <inline-formula><tex-math id="M5">\begin{document}$ \beta $\end{document}</tex-math></inline-formula> is negative, zero or positive. In order to do that we employ some fine estimates proving the compactness for the associated energy functional.</p>

scholarly journals A decomposition of additive functionals of finite energy

1979 ◽  
Vol 74 ◽  
pp. 137-168 ◽  
Author(s):  
Masatoshi Fukushima
Keyword(s):  
Brownian Motion ◽  
Sobolev Space ◽  
Bounded Variation ◽  
Finite Energy ◽  
Signed Measure ◽  
Ito Formula ◽  
Tempered Distribution ◽  
Additive Functionals ◽  
Right Hand ◽  
The Right

The celebrated Ito formula for the n-dimensional Brownian motion Xt and for u ∈ C2(Rn) runs as follows:(0.1) In § 6 of this paper we extend this to the case where u is any element of the Sobolev space H1R(n) and accordingly Δu is a tempered distribution which is not even a signed measure in general. As a consequence the second term of the right hand side of (0.1) may not be of bounded variation in t.

scholarly journals A fractional Kirchhoff problem involving a singular term and a critical nonlinearity

2017 ◽  
Vol 8 (1) ◽  
pp. 645-660 ◽  
Author(s):  
Alessio Fiscella
Keyword(s):  
Singular Term ◽  
Nonlocal Problem ◽  
Critical Sobolev Exponent ◽  
Critical Nonlinearity ◽  
Bounded Subset ◽  
Sobolev Exponent ◽  
Continuous Boundary ◽  
Open Bounded Subset ◽  
Fractional Laplace Operator ◽  
Kirchhoff Problem

Abstract In this paper, we consider the following critical nonlocal problem: \left\{\begin{aligned} &\displaystyle M\bigg{(}\iint_{\mathbb{R}^{2N}}\frac{% \lvert u(x)-u(y)\rvert^{2}}{\lvert x-y\rvert^{N+2s}}\,dx\,dy\biggr{)}(-\Delta)% ^{s}u=\frac{\lambda}{u^{\gamma}}+u^{2^{*}_{s}-1}&&\displaystyle\phantom{}\text% {in }\Omega,\\ \displaystyle u&\displaystyle>0&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{in }\mathbb{R}^{% N}\setminus\Omega,\end{aligned}\right. where Ω is an open bounded subset of {\mathbb{R}^{N}} with continuous boundary, dimension {N>2s} with parameter {s\in(0,1)} , {2^{*}_{s}=2N/(N-2s)} is the fractional critical Sobolev exponent, {\lambda>0} is a real parameter, {\gamma\in(0,1)} and M models a Kirchhoff-type coefficient, while {(-\Delta)^{s}} is the fractional Laplace operator. In particular, we cover the delicate degenerate case, that is, when the Kirchhoff function M is zero at zero. By combining variational methods with an appropriate truncation argument, we provide the existence of two solutions.

Some conditions ensuring the vanishing of harmonic differential forms with applications to harmonic maps and Yang-Mills theory

1982 ◽  
Vol 91 (3) ◽  
pp. 441-452 ◽  
Author(s):  
H. C. J. Sealey
Keyword(s):  
Harmonic Maps ◽  
Harmonic Map ◽  
Differential Forms ◽  
Finite Energy ◽  
Energy Functional ◽  
Conformal Transformations ◽  
Second Variation ◽  
Yang Mills

In (5) it is shown that if m ≽ 3 then there is no non-constant harmonic map φ: ℝm → Sn with finite energy. The method of proof makes use of the fact that the energy functional is not invariant under conformal transformations. This fact has also allowed Xin(9), to show that any non-constant-harmonic map φ:Sm → (N, h), m ≽ 3, is not stable in the sense of having non-negative second variation.

A CLASS OF CRITICAL KIRCHHOFF PROBLEM ON THE HYPERBOLIC SPACE

2019 ◽  
Vol 62 (1) ◽  
pp. 109-122
Author(s):  
PAULO CESAR CARRIÃO ◽  
AUGUSTO CÉSAR DOS REIS COSTA ◽  
OLIMPIO HIROSHI MIYAGAKI
Keyword(s):  
Nontrivial Solution ◽  
Hyperbolic Space ◽  
Stereographic Projection ◽  
Mountain Pass Theorem ◽  
Singular Problem ◽  
Nonlocal Term ◽  
Open Ball ◽  
Kirchhoff Type Problems ◽  
The Hardy Inequality ◽  
Kirchhoff Problem

AbstractWe investigate questions on the existence of nontrivial solution for a class of the critical Kirchhoff-type problems in Hyperbolic space. By the use of the stereographic projection the problem becomes a singular problem on the boundary of the open ball $B_1(0)\subset \mathbb{R}^n$ Combining a version of the Hardy inequality, due to Brezis–Marcus, with the mountain pass theorem due to Ambrosetti–Rabinowitz are used to obtain the nontrivial solution. One of the difficulties is to find a range where the Palais Smale converges, because our equation involves a nonlocal term coming from the Kirchhoff term.

scholarly journals SPHERICALLY SYMMETRIC SOLUTION FOR TORSION AND THE DIRAC EQUATION IN 5D SPACETIME

1998 ◽  
Vol 07 (06) ◽  
pp. 909-915 ◽  
Author(s):  
V. D. DZHUNUSHALIEV
Keyword(s):  
Spinor Field ◽  
Symmetric Solution ◽  
Finite Energy ◽  
Heisenberg Equation ◽  
Spherically Symmetric ◽  
Square Root ◽  
Algebraic Relation ◽  
Spherically Symmetric Solution ◽  
Whole Space ◽  
Spacetime Foam

Torsion in a 5D spacetime is considered. In this case, gravitation is defined by the 5D metric and the torsion. It is conjectured that torsion is connected with a spinor field. In this case, Dirac's equation becomes the nonlinear Heisenberg equation. It is shown that this equation has a discrete spectrum of solutions with each solution being regular on the whole space and having finite energy. Every solution is concentrated on the Planck region and hence we can say that torsion should play an important role in quantum gravity in the formation of bubbles of spacetime foam. On the basis of the algebraic relation between torsion and the classical spinor field in Einstein–Cartan gravity, the geometrical interpretation of the spinor field is considered as "the square root" of torsion.

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