arXiv subjects

Lele Du

Publications and source records attributed to Lele Du.

6 recordsLinked to original sources

arXiv2024

On elliptic equations with N-independent stable operators

We investigate the positive solutions of the semilinear elliptic equation \begin{align*} \sum^{N}_{i=1}\left(-\partial_{ii}\right)^{s}u=u^{p} \end{align*} with one-dimensional symmetric $2s$-stable operators. Firstly, in the whole space $\R^{N}$, we establish the nonexistence of positive supersolutions for $1 \frac{N}{N-2s}$. It is crucial for these solutions to exhibit suitable decay at infinity to compensate for the absence of the Kelvin transform. Notably, while these solutions are symmetric, they are not radially symmetric due to the non-rotational invariance of the operator involved. Next, in the half space $\R_{+}^{N}$, we observe the nonexistence of positive supersolutions in the region $1<p\leq\frac{N+s}{N-s}$. Additionally, we find that positive solutions with appropriate decay for the Dirichlet boundary problem do not exist. Finally, we present the symmetry of positive solutions in the unit ball $B_{1}$.

math.AP↗

arXiv2024

Liouville results for semilinear integral equations with conical diffusion

Nonexistence results for positive supersolutions of the equation $$-Lu=u^p\quad\text{in $\mathbb R^N_+$}$$ are obtained, $-L$ being any symmetric and stable linear operator, positively homogeneous of degree $2s$, $s\in(0,1)$, whose spectral measure is absolutely continuous and positive only in a relative open set of the unit sphere of $\mathbb R^N$. The results are sharp: $u\equiv 0$ is the only nonnegative supersolution in the subcritical regime $1\leq p\leq\frac{N+s}{N-s}\,$, while nontrivial supersolutions exist, at least for some specific $-L$, as soon as $p>\frac{N+s}{N-s}$. \\ The arguments used rely on a rescaled test function's method, suitably adapted to such nonlocal setting with weak diffusion; they are quite general and also employed to obtain Liouville type results in the whole space.

math.AP↗

arXiv2023

Positive solutions to the planar logarithmic Choquard equation via asymptotic approximation

In this paper we study the following nonlinear Choquard equation $$ -Δu+u=\left(\ln\frac{1}{|x|}\ast F(u)\right)f(u),\quad\text{ in }\,\mathbb{R}^2, $$ where $f\in C^1(\mathbb{R})$ and $F$ is the primitive of the nonlinearity $f$ vanishing at zero. We use an asymptotic approximation approach to establish the existence of positive solutions to the above problem in the standard Sobolev space $H^1(\mathbb{R}^2)$. We give a new proof and at the same time extend part of the results established in [Cassani-Tarsi, Calc. Var. P.D.E. (2021)].

math.AP↗

arXiv2023

Fine bounds for best constants of fractional subcritical Sobolev embeddings and applications to nonlocal PDEs

We establish fine bounds for best constants of the fractional subcritical Sobolev embeddings \begin{align*} W_{0}^{s,p}\left(Ω\right)\hookrightarrow L^{q}\left(Ω\right), \end{align*} where $N\geq1$, $0 2s$ and the so-called Sobolev limiting case $N=1$, $s=\frac{1}{2}$ and $p=2$, where a sharp asymptotic estimate is given by means of a limiting procedure. We apply the obtained results to prove existence and non-existence of solutions for a wide class of nonlocal partial differential equations.

math.AP↗

arXiv2022

On elliptic equations with Stein-Weiss type convolution parts

The aim of this paper is to study the critical elliptic equations with Stein-Weiss type convolution parts $$ \displaystyle-Δu =\frac{1}{|x|^α}\left(\int_{\mathbb{R}^{N}}\frac{|u(y)|^{2_{α, μ}^{\ast}}}{|x-y|^μ|y|^α}dy\right) |u|^{2_{α, μ}^{\ast}-2}u,~~~x\in\mathbb{R}^{N}, $$ where the critical exponent is due to the weighted Hardy-Littlewood-Sobolev inequality and Sobolev embedding. We develop a nonlocal version of concentration-compactness principle to investigate the existence of solutions and study the regularity, symmetry of positive solutions by moving plane arguments. In the second part, the subcritical case is also considered, the existence, symmetry, regularity of the positive solutions are obtained.

math.AP↗

arXiv2018

Uniqueness and nondegeneracy of solutions for a critical nonlocal equation

The aim of this paper is to classify the positive solutions of the nonlocal critical equation: $$ -Δu=\left(I_μ\ast u^{2^{\ast}_μ}\right)u^{{2}^{\ast}_μ-1},~~x\in\mathbb{R}^{N}, $$ where $0<μ 0$ and $2^{\ast}_μ=\frac{2N-μ}{N-2}$ is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. We apply the moving plane method in integral forms to prove the symmetry and uniqueness of the positive solutions and prove the nondegeneracy of the unique solutions for the equation when $μ$ close to $N$.

math.AP↗