Multiple Critical Exponents Research Articles

Existence and nonexistence of solutions for elliptic problems with multiple critical exponents

Abstract In this article, the existence and nonexistence of solutions for the quasilinear elliptic equations involving multiple critical terms under Dirichlet boundary conditions on bounded smooth domains Ω ⊂ R N ( N ≥ 3 ) \Omega \subset {R}^{N}(N\ge 3) are proved by using the variational method and Pohozaev identity, respectively.

Existence and multiplicity of positive solutions to sub-elliptic systems with multiple critical exponents on Carnot groups

Existence and multiplicity of positive solutions to sub-elliptic systems with multiple critical exponents on Carnot groups

Excitations and ergodicity of critical quantum spin chains from non-equilibrium classical dynamics

We study a quantum spin-1/2 chain that is dual to the canonical problem of non-equilibrium Kawasaki dynamics of a classical Ising chain coupled to a thermal bath. The Hamiltonian is obtained for the general disordered case with non-uniform Ising couplings. The quantum spin chain (dubbed Ising-Kawasaki) is stoquastic, and depends on the Ising couplings normalized by the bath's temperature. We give its exact ground states. Proceeding with uniform couplings, we study the one- and two-magnon excitations. Solutions for the latter are derived via a Bethe Ansatz scheme. In the antiferromagnetic regime, the two-magnon branch states show intricate behavior, especially regarding their hybridization with the continuum. We find that that the gapless chain hosts multiple dynamics at low energy as seen through the presence of multiple dynamical critical exponents. Finally, we analyze the full energy level spacing distribution as a function of the Ising coupling. We conclude that the system is non-integrable for generic parameters, or equivalently, that the corresponding non-equilibrium classical dynamics are ergodic.

Positive radial solutions of critical Hénon equations on the unit ball in ℝN$$ {\mathbb{R}}^N $$

In this paper, we study the positive radial solutions for the Hénon equations with weighted critical exponents on the unit ball in with . We first confirm that with is the critical exponent for the embedding from into and name as the Hénon‐Sobolev critical exponent. Then, following the great ideas of Brezis and Nirenberg (Comm Pure Appl Math. 1983;36:437‐477), we establish the existence and nonexistence of positive radial solutions of the problems with single Hénon‐Sobolev critical exponent and linear or nonlinear but subcritical perturbations. We further study the problems with multiple critical exponents, which may be Hénon‐Sobolev critical exponents, Hardy‐Sobolev critical exponents, or Sobolev critical exponents. The methods and arguments involved with are the mountain pass theorem and the strong maximum principle and the Pohozaev identity.

Sub-elliptic problems with multiple critical Sobolev-Hardy exponents on Carnot groups

Sub-elliptic problems with multiple critical Sobolev-Hardy exponents on Carnot groups

Density fluctuation analysis very near above and below critical point using morphological and spatiotemporal information

We investigate the fractal nature of critical fluctuations in sulfur hexafluoride (SF $$_6$$ ) under microgravity conditions. For this purpose, we use the Bidimensional Empiric Mode Decomposition (BEMD) approach to separate the spatial scales of fluctuations in orthogonal Independent Mode Functions (IMFs). Statistical analysis of three morphology measures (area, eccentricity, and orientation of convex objects in recorded images) across different IMFs shows that critical fluctuations obey power-laws across multiple spatial scales. We also perform a spatiotemporal analysis of fluctuations by extracting one line of pixels from each image and creating a temporal stack from successive images, or “waterfalls.” The spatiotemporal section analysis along the spatial direction reveals multiple spatial scales present in the original fluctuating image. The analysis of the “waterfalls” along the temporal direction identifies a common power-law temporal behavior across all spatial scales. Our results show that critical fluctuations very near critical temperature ( $$T_c$$ ) have a fractal structure captured by power-laws with multiple critical exponents. The morphology analysis shows that very near $$T_c$$ , the fluctuating domains are mostly spherical with some anisotropy.

Existence of solution for a quasilinear elliptic Neumann problem involving multiple critical exponents

In this paper, we study the Neumann boundary value problem to a quasilinear elliptic equation with the critical Sobolev exponent and critical Hardy–Sobolev exponent, and prove the existence of nontrivial nonnegative solution by means of variational method.

Multiplicity results for biharmonic equations involving multiple Rellich-type potentials and critical exponents

In this paper, a biharmonic equation is investigated, which involves multiple Rellich-type potentials and a critical Sobolev exponent. By using variational methods and analytical techniques, the existence and multiplicity of nontrivial solutions to the equation are established.

On G-invariant solutions of a singular biharmonic elliptic system involving multiple critical exponents in R^{N}

In this work, a biharmonic elliptic system is investigated in mathbb{R}^{N}, which involves singular potentials and multiple critical exponents. By the Rellich inequality and the symmetric criticality principle, the existence and multiplicity of G-invariant solutions to the system are established. To our best knowledge, our results are new even in the scalar cases.

Positive symmetric results for a weighted quasilinear elliptic system with multiple critical exponents in mathbb{R}^{N}

This work focuses on the symmetric solutions for a weighted quasilinear elliptic system involving multiple critical exponents in mathbb{R}^{N}. Based upon the Caffarelli-Kohn-Nirenberg inequality and the symmetric criticality principle due to Palais, we prove a variety of symmetric results under certain appropriate hypotheses on the singular potentials and the parameters.

On symmetric solutions of a critical semilinear elliptic system involving the Caffarelli-Kohn-Nirenberg inequality in R N $\mathbb{R}^{N}$

This paper deals with the existence and multiplicity of symmetric solutions for a weighted semilinear elliptic system with multiple critical Hardy-Sobolev exponents and singular potentials in $\mathbb{R}^{N}$ . Applying the symmetric criticality principle and Caffarelli-Kohn-Nirenberg inequality, we establish several existence and multiplicity results of G-symmetric solutions under certain appropriate hypotheses on the parameters and the weighted functions.

Symmetric solutions for singular quasilinear elliptic systems involving multiple critical Hardy-Sobolev exponents

This paper deals with the existence and multiplicity of symmetric solutions for a class of singular quasilinear elliptic systems involving multiple critical Hardy-Sobolev exponents in a bounded symmetric domain. Based upon the symmetric criticality principle of Palais and variational methods, we establish several existence and multiplicity results of G-symmetric solutions under certain appropriate hypotheses on the weighted functions and the parameters.

Existence of positive solutions for a quasilinear elliptic system involving critical Sobolev-Hardy exponents and concave-convex nonlinearities

This paper is concerned with a quasilinear elliptic system, which involves the Caffarelli-Kohn-Nirenberg inequality and multiple critical exponents. The existence and multiplicity results of positive solutions are obtained by variational methods.

P-Laplace Equations with Multiple Critical Exponents and Singular Cylindrical Potential

In this paper, we deal with the following problem: {−Δp u − λ |y|−p|u|p−2u = |y|−s|u|P*(s)−2u+|u|p*−2u in ℝN, y ≠ 0, u≥ 0, where u(x) = u(y,z) : ℝm × ℝN−m → ℝ, N ≥3,2 <m <N,1 <p<m,λ < (m−pp)p and 0 < s < p, p*(s) = p(N−s)N−p, p* = pNN−p. By variational method, we prove the existence of a nontrivial weak solution when 0 < λ < (m−pp)p and the existence of a cylindrical weak solution when λ ≤ 0.

Elliptic systems involving multiple critical nonlinearities and symmetric multi-polar potentials

In this paper, a system of elliptic equations is investigated, which involves multiple critical Sobolev exponents and symmetric multi-polar potentials. By variational methods and analytic techniques, the relevant best constants are studied and the existence of \(\left( {\mathbb{Z}_k \times \mathbb{S}\mathbb{O}\left( {N - 2} \right)} \right)^2\)-invariant solutions to the system is established.

Systems of elliptic equations involving multiple inverse-square potentials and critical exponents

In this paper, a system of elliptic equations is investigated, which involves multiple critical Sobolev exponents and singular points. The best Sobelev constant related to the system is studied, which is verified to be independent of the location of singular points. By a variant of the concentration compactness principle and the mountain-pass argument, the existence of positive solutions to the system is proved. At last, the existence of sign-changing solutions to the system is also established on the basis of the mountain-pass-type positive solutions.

Elliptic equations with cylindrical potential and multiple critical exponents

In this paper, we deal with the following problem:\begin{eqnarray*} -\Delta u-\lambda|y|^{-2}u=|y|^{-s}u^{2^{*}(s)-1}+u^{2^{*}-1}\\ \ in \ \ R^N , y\neq 0\\u\geq 0\end{eqnarray*} where $u(x)=u(y,z): R^m\times R^{N-m}\longrightarrow R$, $N\geq 4$, $2 < m < N$, $\lambda < (\frac{m-2}{2})^2$ and$0 < s < 2$, $2^*(s)=\frac{2(N-s)}{N-2}$, $2^*=\frac{2N}{N-2}$. Using the Variational method, we proved the existence of a ground statesolution for the case $0 < \lambda < (\frac{m-2}{2})^2$ and theexistence of a cylindrical weak solution under the case $\lambda<0$.

A variational approach to a singular elliptic system involving critical Sobolev-Hardy exponents and concave-convex nonlinearities

This paper is concerned with a quasilinear elliptic system, which involves the Caffarelli-Kohn-Nirenberg inequality and multiple critical exponents. The existence and multiplicity results of positive solutions are obtained by variational methods.

Asymptotic Properties of Solutions to Semilinear Equations Involving Multiple Critical Exponents

AbstractIn this paper, we investigate a semilinear elliptic equation that involves multiple Hardy-type terms and critical Hardy–Sobolev exponents. By the Moser iteration method and analytic techniques, the asymptotic properties of its nontrivial solutions at the singular points are investigated.

Quasilinear systems involving multiple critical exponents and potentials

In this paper, a quasilinear system of elliptic equations isinvestigated, which involves multiple critical Hardy--Sobolev exponents and Hardy--type terms. By variational methods and analytictechnics, the existence of nontrivial solutions to the system isestablished.