Sobolev Inequality Research Articles

Multiple solutions for Schrödinger equations on Riemannian manifolds via $$\nabla $$-theorems

We consider a smooth, complete and non-compact Riemannian manifold \((\mathcal {M},g)\) of dimension \(d \ge 3\), and we look for solutions to the semilinear elliptic equation $$\begin{aligned} -\varDelta _g w + V(\sigma ) w = \alpha (\sigma ) f(w) + \lambda w \quad \hbox {in }\mathcal {M}. \end{aligned}$$The potential \(V :\mathcal {M} \rightarrow \mathbb {R}\) is a continuous function which is coercive in a suitable sense, while the nonlinearity f has a subcritical growth in the sense of Sobolev embeddings. By means of \(\nabla \)-theorems introduced by Marino and Saccon, we prove that at least three non-trivial solutions exist as soon as the parameter \(\lambda \) is sufficiently close to an eigenvalue of the operator \(-\varDelta _g\).

A remark on the regularity criterion for the 3D Navier–Stokes equations in terms of two vorticity components

In this paper, we establish a new regularity criterion for the 3D Navier–Stokes equations in terms of only two vorticity components. By making use of a logarithmic Sobolev inequality in the Besov spaces with negative indices and well known commutator estimate, we prove that a unique local strong solution does not blow-up at time T if two components of vorticity belong to L2(0,T;Ḃ∞,∞−1). This result is the further extension of the previous works by Guo et al. (2018) and by O (2021), and becomes the improvement of the work by Gala (2011).

On Cauchy problem for pseudo-parabolic equation with Caputo-Fabrizio operator

Abstract In this article, we considered the pseudo-parabolic equation with Caputo-Fabrizio fractional derivative. This equation has many applications in different fields, such as science, technology, and so on. In this article, we gave the formula of mild solution, which is represented in the form of Fourier series by some operators . In the linear case, we investigated the continuity of the mild solution with respect to the fractional order. For the nonlinear case, we investigated the existence and uniqueness of a global solution. The main proof technique is based on the Banach fixed point theorem combined with some Sobolev embeddings. For more detailed, we obtained two other interesting results: the continuity of mild solution with respect to the derivative order and the convergence of solution as the coefficient k approaches to zero.

Multiple solutions for a fractional Choquard problem with slightly subcritical exponents on bounded domains

This paper is devoted to study a fractional Choquard problem with slightly subcritical exponents on bounded domains. When the exponent of the convolution type nonlinearity tends to the fractional critical one in the sense of Hardy–Littlewood–Sobolev inequality, we obtain the existence of multiple positive solutions via Lusternik–Schnirelmann category and nonlocal global compactness. Moreover, we prove that the topology of the domain furnishes a lower bound for the number of positive solutions.

On the maximizing problem associated with critical Sobolev inequality under inhomogeneous constraints

In this paper, we study a maximizing problem associated with the critical Sobolev inequality under inhomogeneous constraints. The problem of this type was previously studied by Ishiwata and Wadade [On the effect of equivalent constrain on a maximizing problem associated with the Sobolev type embedding in . Math Ann. 2016;364:1043–1068; On the maximizing problem associated with Sobolev type embeddings under inhomogeneous constraints. Appl Anal. 2019;98(10):1916–1934], Ishiwata [On variational problems associated with Sobolev type inequalities and related topics. Available from: http://www.rism.it/doc/Ishiwata.pdf] and Nguyen [Maximizers for the variational problems associated with Sobolev type inequalities under constraints. Math Ann. 2018;372(1–2):229–255]. Our results give a complete picture of the effect of the constraints on the attainability and non-attainability of the problem. The sharp Sobolev inequality plays a crucial role in our argument. Our method also provides a new and simple proof for the recent results of Ishiwata and Wadade [On the maximizing problem associated with Sobolev type embeddings under inhomogeneous constraints. Appl Anal. 2019;98(10):1916–1934] concerning the sub-critical Sobolev-type inequalities (or Gagliardo–Nirenberg inequalities).

Shannon, Sobolev and uncertainty inequalities for the Weinstein transform

ABSTRACT The objective of this paper is to extend some weighted inequalities for the Weinstein transform. Especially, we are interesting in the study of the Beckner logarithmic inequality, the Shannon inequality. We give a sharp version of this inequality. Next, we establish different Sobolev type embedding theorems in our context. Finally, we deal with some uncertainty inequalities and a logarithmic uncertainty inequality for the Weistein transform.

Existence and continuity results for Kirchhoff parabolic equation with Caputo–Fabrizio operator

This paper deals with the solution of the Kirchhoff parabolic equation involving the Caputo–Fabrizio fractional derivative with non-singular kernel. We represent the mild solution by equation operators which are defined via Fourier series. The main results of this work are about the existence, uniqueness and the continuity property with respect to the derivative order of the mild solution. The result of the existence of a solution consists of two parts: The existence of a local solution with a nonlinear source function. The existence of a global solution in the case of a linear source function. To achieve global results, we had to introduce a new space with norm contains weighted function. The key analysis is based on the Banach fixed point theorem and some Sobolev embeddings. The order-continuity problem has two main results: The first result proves the continuity related to order α∈(0,1). The remaining results examine the continuity when the fractional order is approaching 1−. Finally, we illustrate the achieved theoretical results by some numerical examples.

A Note on Sharp Affine Poincaré-Sobolev Inequalities and Exact in Minimization of Zhang’s Energy on Bounded Variation and Exactness

As for the affine energy, Edir Junior and Ferreira Leite establish the existence of minimizers for particular restricted subcritical and critical variational issues on BV(Ω). Similar functionals exhibit deeper weak* topological traits including lower semicontinuity and affine compactness, and their geometry is non-coercive. Our work also proves the result that extremal functions exist for certain affine Poincaré-Sobolev inequalities.

Laguerre BV spaces, Laguerre perimeter and their applications

<abstract><p>In this paper, we introduce the Laguerre bounded variation space and the Laguerre perimeter, thereby investigating their properties. Moreover, we prove the isoperimetric inequality and the Sobolev inequality in the Laguerre setting. As applications, we derive the mean curvature for the Laguerre perimeter.</p></abstract>

The ε-ε property and the boundedness of isoperimetric sets with different monomial weights

We consider a class of monomial weights 𝑥A = |𝑥1|𝑎1…|𝑥N|𝑎N in ℝN, where ai is a nonnegative real number for each i ∈ {1,…,N}, and we establish the ε — ε property and the boundedness of isoperimetric sets with different monomial weights for the perimeter and volume. Moreover, we present cases of nonexistence of the isoperimetric inequality when it is not possible to associate the corresponding Sobolev inequality. Finally, for N = 2, we developed an original type of symmetrization, which we call star-shaped Steiner symmetrization, and we apply it to a class of isoperimetric problems with different monomial weights.

NEW RESULTS ON CONTINUITY BY ORDER OF DERIVATIVE FOR CONFORMABLE PARABOLIC EQUATIONS

In this paper, we study the continuity problem by an order of derivative for conformable parabolic equations. The problem is examined in both the linear and nonlinear cases. For the input data in suitable Hilbert scale spaces, we consider the continuity problem for the linear problem. In the nonlinear case, we prove the existence of mild solutions for a class of conformable parabolic equations once the source function is a global Lipschitz type in the [Formula: see text] space sense. The main results are based on semigroup theory combined with the Banach fixed point theorem and Sobolev embeddings. We also inspect the continuity problem for the nonlinear model, and prove the convergence of the mild solution to the nonlinear problem as [Formula: see text] tends to [Formula: see text].

WELL-POSEDNESS AND REGULARIZATION FOR NONLOCAL DIFFUSION EQUATION WITH RIEMANN–LIOUVILLE DERIVATIVE

In this paper, we are interested in studying the fractional diffusion equation with Riemann–Liouville as follows: [Formula: see text] with nonlocal in time condition. We are going to study the well-posedness of the above problem with some assumptions of the input data. On the other hand, in Hilbert scale and [Formula: see text] spaces, we provide several estimates of regularity results of the mild solution. We also establish the evaluation for gradient term of the mild solution. We also show that the nonlocal problem is ill-posed in the sense of Hadamard. We also derive the regularity result by applying Fourier truncation method. The main tool of the paper is to use some estimates of Wright functions and Sobolev embeddings. In addition, we also obtain a lower bound of the solution according to the input data.

Uniform large deviation principles of fractional stochastic reaction-diffusion equations on unbounded domains

This paper is concerned with uniform large deviation principles of fractional stochastic reaction-diffusion equations driven by additive noise defined on unbounded domains where the solution operator is non-compact and hence the result of [32] does not apply. The nonlinear drift is assumed to be locally Lipschitz continnous instead of being globally Lipschitz continuous. We first prove a large deviation principle for a fractional linear stochastic equation by the weak convergence method, and then show a uniform large deviation principle for the fractional nonlinear equation by a uniform contraction principle, despite the Sobolev embeddings are non-compact in unbounded domains. The result of the paper regrading the uniform large deviations can be applied to investigate the exit time and exit place of the solutions of the stochastic reaction-diffusion equations from a given domain in the phase space.

Verification of the Landau Equation and Hardy’s Inequality

We prove the L estimate for the isotropic version of the homogeneous landau problem, which was explored by M. Gualdani and N. Guillen. As shown in a region of the smooth potentials range under values of the interaction exponent (2), a weighted Poincaré inequality is a natural consequence of the traditional weighted Hardy inequality, which in turn implies that the norms of solutions propagate in the L1 space. Now, the L estimate is based on the work of De Giorgi, Nash, and Moser, as well as a few weighted Sobolev inequalities.

Sharp reversed Hardy–Littlewood–Sobolev inequality with extension kernel

In this paper, we prove the following reversed Hardy–Littlewood–Sobolev inequality with extension kernel: $$\int _{\mathbb R_+^n}\int _{\partial\mathbb R^n_+}\frac{x_n^\beta}{|x-y|^{n-\alpha}}f(y)g(x)\,dy\,dx\geq C_{n,\alpha ,\beta ,p}\|f\|_{L^{p}(\partia

Homoclinic solutions for a class of asymptotically autonomous Hamiltonian systems with indefinite sign nonlinearities

In this paper, we obtain the multiplicity of homoclinic solutions for a class of asymptotically autonomous Hamiltonian systems with indefinite sign potentials. The concentration-compactness principle is applied to show the compactness. As a byproduct, we obtain the uniqueness of the positive ground state solution for a class of autonomous Hamiltonian systems and the best constant for Sobolev inequality which are of independent interests.

The coupling system of Kirchhoff and Euler-Bernoulli plates with logarithmic source terms: Strong damping versus weak damping of variable-exponent type

<abstract><p>In this paper, we study the asymptotic behavior of solutions of the dissipative coupled system where we have interactions between a Kirchhoff plate and a Euler-Bernoulli plate. We investigate the interaction between the internal strong damping acting in the Kirchhoff equation and internal weak damping of variable-exponent type acting in the Euler-Bernoulli equation. By using the potential well, the energy method (multiplier method) combined with the logarithmic Sobolev inequality, we prove the global existence and derive the stability results. We show that the solutions of this system decay to zero sometimes exponentially and other times polynomially. We find explicit decay rates that depend on the weak damping of the variable-exponent type. This outcome extends earlier results in the literature.</p></abstract>

The extremal problem for Sobolev inequalities with upper order remainder terms

The extremal problem for Sobolev inequalities with upper order remainder terms

Ground state solutions and decay estimation of Choquard equation with critical exponent and Dipole potential

<p style='text-indent:20px;'>In this paper, we study a class of Choquard equations with critical exponent and Dipole potential. We prove the existence of radial ground state solutions for Choquard equations by using the refined Sobolev inequality with the Morrey norm, and show that any nonnegative weak solutions of Choquard equations have additional decay properties in terms of ground state representation.</p>

The separation property for 2D Cahn-Hilliard equations: Local, nonlocal and fractional energy cases

We study the separation property for Cahn-Hilliard type equations with constant mobility and (physically relevant) singular potentials in two dimensions. That is, any solution with initial finite energy stays uniformly away from the pure phases $ \pm 1 $ from a certain time on. Beyond its physical interest, this property plays a crucial role to achieve high order Sobolev and analytic regularity of the solutions and to analyze their longtime behavior. In the local case, we streamline known arguments by exploiting the Sobolev inequality to obtain direct entropy estimates. In the nonlocal case, we provide a new proof based on De Giorgi estimates rather than the Alikakos-Moser type argument. Finally, in the spectral-fractional case, we prove nonlinear estimates and the separation property for any fractional index $ s\in (0,1) $ filling the gap between first-order (local) and zero-order (nonlocal) energy cases. In all of the aforementioned cases, our new proofs neither make use of the Trudinger-Moser inequality nor of any assumptions involving the third derivative of the entropy, as in the previous contributions. In particular, they apply for a more general class of singular potentials than the Flory-Huggins (Boltzmann-Gibbs) logarithmic density. Besides, the new methods present a series of technical advantages, which can be useful to the analysis of important physical systems that couple Cahn-Hilliard equations with other equations (e.g., reaction-diffusion equations and/or Navier-Stokes type systems) as well as their stochastic counterparts.