Uniqueness of higher integrable solution to the Landau equation with Coulomb interactions

We are concerned with the uniqueness of weak solution to the spatially homogeneous Landau equation with Coulomb interactions under the assumption that the solution is bounded in the space $L^\infty(0,T,L^p(\R^3))$ for some $p>3/2$. The proof uses a weighted Poincar\'e-Sobolev inequality recently introduced in \cite{GG18}

The symmetry of least-energy solutions for semilinear elliptic equations

AbstractIn this paper we will apply the method of rotating planes (MRP) to investigate the radial and axial symmetry of the least-energy solutions for semilinear elliptic equations on the Dirichlet and Neumann problems, respectively. MRP is a variant of the famous method of moving planes. One of our main results is to consider the least-energy solutions of the following equation: (∗)Δu+K(x)up=0,x∈B1,u>0inB1,u|∂B1=0,where 1<p<n+2n−2 and B1 is the unit ball of Rn with n⩾3. Here K(x)=K(|x|) is not assumed to be decreasing in |x|. In this paper, we prove that any least-energy solution of (∗) is axially symmetric with respect to some direction. Furthermore, when p is close to n+2n−2, under some reasonable condition of K, radial symmetry is shown for least-energy solutions. This is the example of the general phenomenon of the symmetry induced by point-condensation. A fine estimate for least-energy solution is required for the proof of symmetry of solutions. This estimate generalizes the result of Han (Ann. Inst. H. Poincaré Anal. Nonlinéaire 8 (1991) 159) to the case when K(x) is nonconstant. In contrast to previous works for this kinds of estimates, we only assume that K(x) is continuous

Uniqueness and structure of solutions to the Dirichlet problem for an elliptic system

AbstractIn this paper, we consider the Dirichlet problem for an elliptic system on a ball in R2. By investigating the properties for the corresponding linearized equations of solutions, and adopting the Pohozaev identity and Implicit Function Theorem, we show the uniqueness and the structure of solutions

ON THE UNIQUENESS OF SINGULAR SOLUTIONS FOR A HARDY-SOBOLEV EQUATION

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