Nontrivial Solutions of Dirac-Laplace Equation on Compact Spin Manifolds

Multiple solutions for the $p-$laplace operator with critical growth

In this note we show the existence of at least three nontrivial solutions to the following quasilinear elliptic equation $-Δ_p u = |u|^{p^*-2}u + λf(x,u)$ in a smooth bounded domain $Ω$ of $\R^N$ with homogeneous Dirichlet boundary conditions on $\partialΩ$, where $p^*=Np/(N-p)$ is the critical Sobolev exponent and $Δ_p u =div(|\nabla u|^{p-2}\nabla u)$ is the $p-$laplacian. The proof is based on variational arguments and the classical concentrated compactness method.

Spectral boundary conditions for generalizations of Laplace and Dirac operators

Spectral boundary conditions for Laplace-type operators, of interest in string and brane theory, are partly Dirichlet, partly Neumann-type conditions, partitioned by a pseudodifferential projection. We give sufficient conditions for existence of associated heat trace expansions with power and power-log terms. The first log coefficient is a noncommutative residue, vanishing when the smearing function is 1. For Dirac operators with general well-posed spectral boundary conditions, it follows that the zeta function is regular at 0. In the selfadjoint case, the eta function has a simple pole at zer…

Laplace's equation on n-dimensional singular manifolds

We show that the Sobolev embedding is compact on punctured manifolds with conical singularities. On the other hand, we find the Sobolev inequality does not hold on punctured manifolds with Poincaré like metric, on which one has Poincaré inequality. Applying the results to the Laplace's equation on the singular manifolds, we obtain the existences of the solution in both cases. In conical singularity case, we prove further that the solution can be extended to the singular points and it is Hölder continuous. However, the solution can not be continuously extended to the singular points in Poincaré…

Stationary solutions and connecting orbits for $p$-Laplace equation

We deal with one dimensional $p$-Laplace equation of the form $$ u_t = (|u_x|^{p-2} u_x )_x + f(x,u), \ x\in (0,l), \ t>0, $$ under Dirichlet boundary condition, where $p>2$ and $f\colon [0,l]\times \mathbb{R}\to \mathbb{R}$ is a continuous function with $f(x,0)=0$. We will prove that if there is at least one eigenvalue of the $p$-Laplace operator between $\lim_{u\to 0} f(x,u)/|u|^{p-2}u$ and $\lim_{|u|\to +\infty} f(x,u)/|u|^{p-2}u$, then there exists a nontrivial stationary solution. Moreover we show the existence of a connecting orbit between stationary solutions. The results are obtained b…

Profile expansion for the first nontrivial Steklov eigenvalue in Riemannian manifolds

We study the problem of maximizing the first nontrivial Steklov eigenvalue of the Laplace-Beltrami Operator among subdomains of fixed volume of a Riemannian manifold. More precisely, we study the expansion of the corresponding profile of this isoperimetric (or isochoric) problem as the volume tends to zero. The main difficulty encountered in our study is the lack of existence results for maximizing domains and the possible degeneracy of the first nontrivial Steklov eigenvalue, which makes it difficult to tackle the problem with domain variation techniques. As a corollary of our results, we ded…

On sufficient "local" conditions for existence results to generalized $p(\cdot)$-Laplace equations involving critical growth

In this paper, we study the existence of multiple solutions to a generalized $p(\cdot)$-Laplace equation with two parameters involving critical growth. More precisely, we give sufficient "local" conditions, which mean that growths between the main operator and nonlinear term are locally assumed for the cases $p(\cdot)$-sublinear, $p(\cdot)$-superlinear, and sandwich-type. Compared to constant exponent problems (for examples, $p$-Laplacian and $(p,q)$-Laplacian), this characterizes the study of variable exponent problems. We show this by applying variants of the Mountain Pass Theorem for $p(\cd…

关于在 $\mathbb{H}^N$ 中具有临界 Trudinger-Moser 非线性的 $(p, Q)$-Laplace Choquard 方程的研究

On the study of $(p, Q)$-Laplace Choquard equations with critical Trudinger-Moser nonlinearity in $\mathbb{H}^N$

本文研究Heisenberg群上具有临界Trudinger-Moser非线性与Stein-Weiss反应的(p,Q)-Laplace Choquard方程解的存在性与多重性，结合变分原理与全新Brézis-Lieb引理证明。

Generalized Picone inequalities and their applications to $(p,q)$-Laplace equations

We obtain a generalization of the Picone inequality which, in combination with the classical Picone inequality, appears to be useful for problems with the $(p,q)$-Laplace type operators. With its help, as well as with the help of several other known generalized Picone inequalities, we provide some nontrivial facts on the existence and nonexistence of positive solutions to the zero Dirichlet problem for the equation $-Δ_p u -Δ_q u = f_μ(x,u,\nabla u)$ in a bounded domain $Ω\subset \mathbb{R}^N$ under certain assumptions on the nonlinearity and with a special attention to the resonance case $f_μ…

Dirac concentrations in Lotka-Volterra parabolic PDEs

We consider parabolic partial differential equations of Lotka-Volterra type, with a non-local nonlinear term. This models, at the population level, the darwinian evolution of a population; the Laplace term represents mutations and the nonlinear birth/death term represents competition leading to selection. Once rescaled with a small diffusion, we prove that the solutions converge to a moving Dirac mass. The velocity and weights cannot be obtained by a simple expression, e.g., an ordinary differential equation. We show that they are given by a constrained Hamilton-Jacobi equation. This extends s…

Optimal lower bounds for eigenvalues of linear and nonlinear Neumann problems

In this paper we prove a sharp lower bound for the first nontrivial Neumann eigenvalue $μ_1(Ω)$ for the $p$-Laplace operator in a Lipschitz, bounded domain $Ω$ in $\R^n$. Our estimate does not require any convexity assumption on $Ω$ and it involves the best isoperimetric constant relative to $Ω$.

具有Soler型非线性项的非线性Dirac方程的周期解

Periodic Solutions for a Nonlinear Dirac Equation with Soler-Type Nonlinearity

针对三维环面上带Soler型非线性的非线性Dirac方程，通过引入强制扰动恢复Palais–Smale条件，结合谱估计与联结论证，证明了质量以下区间内任意谱参数对应的非平凡连续可微周期解存在性。

Nonlinear Dirichlet problems with unilateral growth on the reaction

We consider a nonlinear Dirichlet problem driven by the $p$-Laplace differential operator with a reaction which has a subcritical growth restriction only from above. We prove two multiplicity theorems producing three nontrivial solutions, two of constant sign and the third nodal. The two multiplicity theorems differ on the geometry near the origin. In the semilinear case (that is, $p=2$), using Morse theory (critical groups), we produce a second nodal solution for a total of four nontrivial solutions. As an illustration, we show that our results incorporate and significantly extend the multipl…

On the Spectrum of geometric operators on Kähler manifolds

On a compact Kähler manifold there is a canonical action of a Lie-superalgebra on the space of differential forms. It is generated by the differentials, the Lefschetz operator and the adjoints of these operators. We determine the asymptotic distribution of irreducible representations of this Lie-superalgebra on the eigenspaces of the Laplace-Beltrami operator. Because of the high degree of symmetry the Laplace-Beltrami operator on forms can not be quantum ergodic. We show that after taking these symmetries into account quantum ergodicity holds for the Laplace-Beltrami operator and for the Spin…

A sharp criterion for zero modes of the Dirac equation

It is shown that $\Vert A \Vert_{L^d}^2 \ge \frac{d}{d-2}\, S_d$ is a necessary condition for the existence of a nontrivial solution of the Dirac equation $γ\cdot (-i\nabla -A)ψ= 0$ in $d$ dimensions. Here, $S_d$ is the sharp Sobolev constant. If $d$ is odd and $\Vert A \Vert_{L^d}^2= \frac{d}{d-2}\, S_d$, then there exist vector potentials that allow for zero modes. A complete classification of these vector potentials and their corresponding zero modes is given.

Dispersion for the Schrödinger equation on the line with multiple Dirac delta potentials and on delta trees

In this paper we consider the time dependent one-dimensional Schrödinger equation with multiple Dirac delta potentials {of different strengths}. We prove that the classical dispersion property holds under some restrictions on the strengths and on the lengths of the finite intervals. The result is obtained in a more general setting of a Laplace operator on a tree with $δ$-coupling conditions at the vertices. The proof relies on a careful analysis of the properties of the resolvent of the associated Hamiltonian. With respect to the analysis done in \cite{MR2858075} for Kirchhoff conditions, here…

The Aharonov-Bohm effect for massless Dirac fermions and the spectral flow of Dirac type operators with classical boundary conditions

We compute, in topological terms, the spectral flow of an arbitrary family of self-adjoint Dirac type operators with classical (local) boundary conditions on a compact Riemannian manifold with boundary under the assumption that the initial and terminal operators of the family are conjugate by a bundle automorphism. This result is used to study conditions for the existence of nonzero spectral flow of a family of self-adjoint Dirac type operators with local boundary conditions in a two-dimensional domain with nontrivial topology. Possible physical realizations of nonzero spectral flow are discus…