Symmetry and Nonexistence of Positive Solutions for a Fractional Laplacion System with Coupled Terms

In this paper, we study the problem for a nonlinear elliptic system involving fractional Laplacion: (equation 1.1) where 0 < α, β < 2, p, q > 0 and max{p, q} ≥ 1, α + γ > 0, β + τ > 0, n ≥ 2. First of all, while in the subcritical case, i.e. n + α + γ − p(n − α) − (q + 1)(n − β) > 0, n + β + τ − (p + 1)(n − α) − q(n − β) > 0, we prove the nonexistence of positive solution for the above system in R n . Moreover, though Doubling Lemma to obtain the singularity estimates of the positive solution on bounded domain Ω. In addition, while in the critical case, i.e. n+α+γ −p(n−α)−(q + 1)(n−β) = 0, n+β +τ −(p+ 1)(n−α)−q(n−β) = 0, we show that the positive solution of above system are radical symmetric and decreasing about some point by using the method of Moving planes in Rn Mathematics Subject Classification (2020): 35R11, 35A10, 35B06.

Singular quasilinear convective elliptic systems in ℝ N

The existence of a positive entire weak solution to a singular quasi-linear elliptic system with convection terms is established, chiefly through perturbation techniques, fixed point arguments, and a priori estimates. Some regularity results are then employed to show that the obtained solution is actually strong.

Second Order Regularity for a Linear Elliptic System Having BMO Coefficients

We consider linear elliptic systems whose prototype is $$\begin{aligned} div \, \Lambda \left[ \,\exp (-|x|) - \log |x|\,\right] I \, Du = div \, F + g \text { in}\, B. \end{aligned}$$ d i v Λ exp ( - | x | ) - log | x | I D u = d i v F + g in B . Here B denotes the unit ball of $$\mathbb {R}^n$$ R n , for $$n > 2$$ n > 2 , centered in the origin, I is the identity matrix, F is a matrix in $$W^{1, 2}(B, \mathbb {R}^{n \times n})$$ W 1 , 2 ( B , R n × n ) , g is a vector in $$L^2(B, \mathbb {R}^n)$$ L 2 ( B , R n ) and $$\Lambda $$ Λ is a positive constant. Our result reads that the gradient of the solution $$u \in W_0^{1, 2}(B, \mathbb {R}^n)$$ u ∈ W 0 1 , 2 ( B , R n ) to Dirichlet problem for system (0.1) is weakly differentiable provided the constant $$\Lambda $$ Λ is not large enough.

Homogenization of a nonlinear elliptic system with a transport term in L 2

We consider the homogenization of a non-linear elliptic system of two equations related to some models in chemotaxis and flows in porous media. One of the equations contains a convection term where the transport vector is only in L 2 and a right-hand side which is only in L 1 . This makes it necessary to deal with entropy or renormalized solutions. The existence of solutions for this system has been proved in reference (Comm. Partial Differential Equations 45(7) (2020) 690–713). Here, we prove its stability by homogenization and that the correctors corresponding to the linear diffusion terms still provide a corrector for the solutions of the non-linear system.

Solutions to indefinite weakly coupled cooperative elliptic systems

We study the elliptic system \begin{equation*} \begin{cases} -\Delta u_1 - \kappa_1u_1 = \mu_1|u_1|^{p-2}u_1 + \lambda\alpha|u_1|^{\alpha-2}|u_2|^\beta u_1, \\ -\Delta u_2 - \kappa_2u_2 = \mu_2|u_2|^{p-2}u_2 + \lambda\beta|u_1|^\alpha|u_2|^{\beta-2}u_2, \\ u_1,u_2\in D^{1,2}_0(\Omega), \end{cases} \end{equation*} where $\Omega$ is a bounded domain in $\mathbb R^N$, $N\geq 3$, $\kappa_1,\kappa_2\in\mathbb R$, $\mu_1,\mu_2,\lambda> 0$, $\alpha,\beta> 1$, and $\alpha + \beta = p\le 2^*:={2N}/({N-2})$. For $p\in (2,2^*)$ we establish the existence of a ground state and of a prescribed number of fully nontrivial solutions to this system for $\lambda$ sufficiently large. If $p=2^*$ and $\kappa_1,\kappa_2> 0$ we establish the existence of a ground state for $\lambda$ sufficiently large if, either $N\ge5$, or $N=4$ and neither $\kappa_1$ nor $\kappa_2$ are Dirichlet eigenvalues of $-\Delta$ in $\Omega$.