Nonexistence Of Minimizers Research Articles

Minimizers of L^2-critical inhomogeneous variational problems with a spatially decaying nonlinearity in bounded domains

We consider the minimizers of $L^{2}$-critical inhomogeneous variational problems with a spatially decaying nonlinear term in an open bounded domain $\Omega$ of $\mathbb{R}^{N}$ which contains $0$. We prove that there is a threshold $a^{*}> 0$ such that minimizers exist for $0< a< a^{*}$ and the minimizer does not exist for any $a> a^{*}$. In contrast to the homogeneous case, we show that both the existence and nonexistence of minimizers may occur at the threshold $a^*$ depending on the value of $V(0)$, where $V(x)$ denotes the trapping potential. Moreover, under some suitable assumptions on $V(x)$, based on a detailed analysis on the concentration behavior of minimizers as $a\nearrow a^*$, we prove local uniqueness of minimizers when $a$ is close enough to $a^*$.

The existence and asymptotic behavior of normalized solutions for Kirchhoff equation with singular potential

In this paper, the existence, non-existence and asymptotic behavior of minimizers for a constrained Kirchhoff type minimization problem with singular potential were considered. Using concentration-compactness lemma, the existence of minimizers of this minimization problem was obtained. By truncation function technique, the non-existence of minimizers was got. Compared with the known result for this problem with trapping potential (Guo, Zhang, Zhou, 2018, [4]) that there exists minimizer for the case of critical exponent p=p⁎ and critical parameter β=βp⁎, our problem has no minimizers in the same condition. Therefore, we discuss the blow-up behavior of minimizer for p=p⁎ as β approaches βp⁎ or β=β˜p and p approaches p⁎, and they appear to be new.

Concentration and local uniqueness of minimizers for mass critical degenerate Kirchhoff energy functional

In this paper, we consider the L2-norm prescribed minimizer of the mass critical Kirchhoff type energy functional with a weight function a(x),E(u)=∫RNa(x)|∇u|2dx+b2(∫RN|∇u|2dx)2−NN+4∫RN|u|2N+8Ndx,N=1,2,3. Making use of the Gagliardo–Nirenberg inequality, we firstly give the classification of existence and non-existence of minimizers. Then the mass concentration of minimizers as c↗c⁎:=(b‖Q‖28/N2)N8−2N is investigated, where Q>0 is the unique radially symmetric positive solution of 2ΔQ−(4N−1)Q+Q8N+1=0 in RN. It is surprise that the concentrating point of a minimizer is possibly determined by the weight function a(x). Finally, we analyze the local uniqueness of minimizers induced by concentration.

On the existence and limit behavior of ground states for two coupled Hartree equations

We study the ground states for a system of two coupled Hartree equations with both attractive intraspecies and attractive interspecies interactions in RN (N≥3), which can also be described by L2-constraint minimizers of mass critical Hartree energy functionals with trapping potentials. The existence and nonexistence of minimizers are classified completely by establishing the refined Gagliardo-Nirenberg type inequality. Based on some delicate estimates of the Hartree energy functional, the limit behavior of minimizers is also analyzed as the interspecies interaction strength goes to a critical value, where each component of minimizers blows up and concentrates at a flattest common minimum point of trapping potentials. An optimal blow-up rate of minimizers for the Hartree energy functional is concomitantly obtained.

A Remark on Elastic Graphs with the Symmetric Cone Obstacle

This paper is concerned with the variational problem for the elastic energy defined on symmetric graphs under the unilateral constraint. Assuming that the obstacle function satisfies the symmetric cone condition, we prove (i) uniqueness of minimizers and (ii) loss of regularity of minimizers and give (iii) complete classification of existence and nonexistence of minimizers in terms of the size of obstacle. As an application, we characterize the solution of obstacle problem as equilibrium of the corresponding dynamical problem.

Convergence of the TFDW Energy to the Liquid Drop Model

We consider two nonlocal variational models arising in physical contexts. The first is the Thomas--Fermi--Dirac--von Weizsäcker (TFDW) model, introduced in the study of ionization of atoms and molecules, and the second is the liquid drop model with external potential, proposed by Gamow in the context of nuclear structure. It has been observed that the two models exhibit many of the same properties, especially in regard to the existence and nonexistence of minimizers. We show that, under a “sharp interface” scaling of the coefficients, the TFDW energy with constrained mass $\Gamma$-converges to the liquid drop model for a general class of external potentials. Finally, we present some consequences for global minimization of each model.

A Degenerate Isoperimetric Problem in the Plane

We establish sufficient conditions for existence of curves minimizing length as measured with respect to a degenerate metric on the plane while enclosing a specified amount of Euclidean area. Non-existence of minimizers can occur and examples are provided. This continues the investigation begun in Alama et al. (Commun Pure Appl Math 70:340–377, 2017) where the metric $$ds^2$$ near the singularities equals a quadratic polynomial times the standard metric. Here, we allow the conformal factor to be any smooth non-negative potential vanishing at isolated points provided the Hessian at these points is positive definite. These isoperimetric curves, appropriately parametrized, arise as traveling wave solutions to a bi-stable Hamiltonian system.

Existence and Nonexistence of Ground State Solutions to Singular Elliptic Systems

In this paper, a system of semi-linear elliptic equations is investigated, which involves multiple critical nonlinearities and multiple singular points. By variational methods, the existence and nonexistence of minimizers to the Rayleigh quotient and ground state solutions to the system are established.

Nonexistence in Thomas-Fermi-Dirac-von Weizsäcker Theory with Small Nuclear Charges

We study the ionization problem in the Thomas-Fermi-Dirac-von Weizsacker theory for atoms and molecules. We prove the nonexistence of minimizers for the energy functional when the number of electrons is large and the total nuclear charge is small. This nonexistence result also applies to external potentials decaying faster than the Coulomb potential. In the case of arbitrary nuclear charges, we obtain the nonexistence of stable minimizers and radial minimizers.

On a variational problem associated with a Hardy type inequality involving a mean oscillation

We consider a minimizing problem associated with a critical Hardy type inequality involving a mean oscillation. We first introduce a dilation-invariant critical Hardy type inequality which contains a nonlocal term and then analyze the associated Euler–Lagrange equation with the aid of the scaling argument. We prove the nonexistence of minimizers for $$1<p<\infty $$ and the existence for $$p=1$$ .

A class of minimization problems related to quasilinear equations

For a class of minimization problems related to quasilinear equations we establish the existence and nonexistence of minimizers by a minimization argument and the Pohozaev type identity.

Sharp nonexistence results of prescribed L 2-norm solutions for some class of Schrödinger–Poisson and quasi-linear equations

In this paper, we study the existence of minimizers for $$F(u) = \frac{1}{2} \int_{\mathbb{R}^3} |\nabla u|^{2} {\rm d}x + \frac{1}{4} \int_{\mathbb{R}^3} \int_{\mathbb{R}^3} \frac{| u(x)|^2 | u(y)|^2}{| x-y|} {\rm d}x{\rm d}y-\frac{1}{p} \int_{\mathbb{R}^3}|u|^p {\rm d}x$$ on the constraint $$S(c) = \{u \in H^1(\mathbb{R}^3) : \int_{\mathbb{R}^3}|u|^2 {\rm d}x = c\}$$ , where c > 0 is a given parameter. In the range $${p \in [3,\frac{10}{3}]}$$ , we explicit a threshold value of c > 0 separating existence and nonexistence of minimizers. We also derive a nonexistence result of critical points of F(u) restricted to S(c) when c > 0 is sufficiently small. Finally, as a by-product of our approaches, we extend some results of Colin et al. (Nonlinearity 23(6):1353–1385, 2010) where a constrained minimization problem, associated with a quasi-linear equation, is considered.

On a notion of subdifferentiability for non-convex functions†

In this paper, a notion of subdifferentiability for non-convex functions is introduced (based in a previous work by the author: The lack of lower semicontinuity and non-existence of minimizers, Preprint SISSA, to appear in Nonlinear Analysis TMA), such a notion arises in a very natural way for l.s.c. functions. We exploit the special structure of the metric space and our particular coupling function to derive many topological properties regarding subdifferentiability. In particular, we present an analogous result to the well known extremal relation in convex optimization

The lack of lower semicontinuity and nonexistence of minimizers

The lack of lower semicontinuity and nonexistence of minimizers