Critical nonlocal systems with concave-convex powers

By using the fibering method jointly with Nehari manifold techniques, we obtain the existence of multiple solutions to a fractional $p$-Laplacian system involving critical concave-convex nonlinearities provided that a suitable smallness condition on the parameters involved is assumed. The result is obtained despite there is no general classification for the optimizers of the critical fractional Sobolev embedding.Comment: 22 page

Nonlocal problems with critical Hardy nonlinearity

By means of variational methods we establish existence and multiplicity of solutions for a class of nonlinear nonlocal problems involving the fractional p-Laplacian and a combined Sobolev and Hardy nonlinearity at subcritical and critical growth.Comment: 36 pages, revised versio

Existence of solution for a Kirchhoff type problem involving the fractional p-Laplace operator

This paper is concerned with the existence of solutions to a Kirchhoff type problem involving the fractional $p$-Laplacian operator. We obtain the existence of solutions by Ekeland's variational principle

Regional Government Competition

This monograph provides a coherent and systematic explanation of China’s regional economic development from the perspective of regional government competition. It gives an almost unknown exposition of the mechanisms of China's regional economic development, with numerous supporting cases drawn from both China and elsewhere. This book is an invaluable resource for anyone interested to learn more particularly the development and transformation of China’s regional economy from both the Chinese and global perspectives

Normalized solutions for a fractional N/sN/s-Laplacian Choquard equation with exponential critical nonlinearities

In this paper, we are concerned with the following fractional $N/s$-Laplacian Choquard equation \begin{align*} \begin{cases} (-\Delta)^s_{N/s}u=\lambda |u|^{\frac{N}{s}-2}u +(I_\mu*F(u))f(u),\ \ \mbox{in}\ \mathbb{R}^N, \displaystyle\int_{\mathbb{R}^N}|u|^{N/s} \mathrm{d}x=a^{N/s}, \end{cases} \end{align*} where $s\in(0,1)$, $10$ is a prescribed constant, $\lambda\in \mathbb{R}$, $I_\mu(x)=\frac{1}{|x|^{\mu}}$ with $\mu\in(0,N)$, $F$ is the primitive function of $f$, and $f$ is a continuous function with exponential critical growth of Trudinger-Moser type. Under some suitable assumptions on $f$, we prove that the above problem admits a ground state solution for any given $a>0$, by using the constraint variational method and minimax technique

Normalized ground states for a biharmonic Choquard system in R4\mathbb{R}^4

In this paper, we study the existence of normalized ground state solutions for the following biharmonic Choquard system \begin{align*} \begin{split} \left\{ \begin{array}{ll} \Delta^2u=\lambda_1 u+(I_\mu*F(u,v))F_u (u,v), \quad\mbox{in}\ \ \mathbb{R}^4, \Delta^2v=\lambda_2 v+(I_\mu*F(u,v)) F_v(u,v), \quad\mbox{in}\ \ \mathbb{R}^4, \displaystyle\int_{\mathbb{R}^4}|u|^2dx=a^2,\quad \displaystyle\int_{\mathbb{R}^4}|v|^2dx=b^2,\quad u,v\in H^2(\mathbb{R}^4), \end{array} \right. \end{split} \end{align*} where $a,b>0$ are prescribed, $\lambda_1,\lambda_2\in \mathbb{R}$, $I_\mu=\frac{1}{|x|^\mu}$ with $\mu\in (0,4)$, $F_u,F_v$ are partial derivatives of $F$ and $F_u,F_v$ have exponential subcritical or critical growth in the sense of the Adams inequality. By using a minimax principle and analyzing the behavior of the ground state energy with respect to the prescribed mass, we obtain the existence of ground state solutions for the above problem.Comment: arXiv admin note: text overlap with arXiv:2211.1370

Normalized ground states for a biharmonic Choquard equation with exponential critical growth

In this paper, we consider the normalized ground state solution for the following biharmonic Choquard type problem \begin{align*} \begin{split} \left\{ \begin{array}{ll} \Delta^2u-\beta\Delta u=\lambda u+(I_\mu*F(u))f(u), \quad\mbox{in}\ \ \mathbb{R}^4, \displaystyle\int_{\mathbb{R}^4}|u|^2dx=c^2,\quad u\in H^2(\mathbb{R}^4), \end{array} \right. \end{split} \end{align*} where $\beta\geq0$, $c>0$, $\lambda\in \mathbb{R}$, $I_\mu=\frac{1}{|x|^\mu}$ with $\mu\in (0,4)$, $F(u)$ is the primitive function of $f(u)$, and $f$ is a continuous function with exponential critical growth in the sense of the Adams inequality. By using a minimax principle based on the homotopy stable family, we obtain that the above problem admits at least one ground state normalized solution.Comment: arXiv admin note: text overlap with arXiv:2210.0088