Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSymmetry and monotonicity properties of singular solutions to some cooperative semilinear elliptic systems involving critical nonlinearities
112
0
0.0
(
0
)
Authors
Francesco Esposito
Ask ChatGPT about the research
No Arabic abstract
We investigate qualitative properties of positive singular solutions of some elliptic systems in bounded and unbounded domains. We deduce symmetry and monotonicity properties via the moving plane procedure. Moreover, in the unbounded case, we study some cooperative elliptic systems involving critical nonlinearities in $mathbb{R}^n$.
rate research
Read More
We consider positive singular solutions to semilinear elliptic problems with possibly singular nonlinearity. We deduce symmetry and monotonicity properties of the solutions via the moving plane procedure.
Let $ngeq 3$, $0le m<frac{n-2}{n}$, $rho_1>0$, $beta>beta_0^{(m)}=frac{mrho_1}{n-2-nm}$, $alpha_m=frac{2beta+rho_1}{1-m}$ and $alpha=2beta+rho_1$. For any $lambda>0$, we prove the uniqueness of radially symmetric solution $v^{(m)}$ of $La(v^m/m)+alpha_m v+beta xcdot abla v=0$, $v>0$, in $R^nsetminus{0}$ which satisfies $lim_{|x|to 0}|x|^{frac{alpha_m}{beta}}v^{(m)}(x)=lambda^{-frac{rho_1}{(1-m)beta}}$ and obtain higher order estimates of $v^{(m)}$ near the blow-up point $x=0$. We prove that as $mto 0^+$, $v^{(m)}$ converges uniformly in $C^2(K)$ for any compact subset $K$ of $R^nsetminus{0}$ to the solution $v$ of $Lalog v+alpha v+beta xcdot abla v=0$, $v>0$, in $R^nbs{0}$, which satisfies $lim_{|x|to 0}|x|^{frac{alpha}{beta}}v(x)=lambda^{-frac{rho_1}{beta}}$. We also prove that if the solution $u^{(m)}$ of $u_t=Delta (u^m/m)$, $u>0$, in $(R^nsetminus{0})times (0,T)$ which blows up near ${0}times (0,T)$ at the rate $|x|^{-frac{alpha_m}{beta}}$ satisfies some mild growth condition on $(R^nsetminus{0})times (0,T)$, then as $mto 0^+$, $u^{(m)}$ converges uniformly in $C^{2+theta,1+frac{theta}{2}}(K)$ for some constant $thetain (0,1)$ and any compact subset $K$ of $(R^nsetminus{0})times (0,T)$ to the solution of $u_t=Lalog u$, $u>0$, in $(R^nsetminus{0})times (0,T)$. As a consequence of the proof we obtain existence of a unique radially symmetric solution $v^{(0)}$ of $La log v+alpha v+beta xcdot abla v=0$, $v>0$, in $R^nsetminus{0}$, which satisfies $lim_{|x|to 0}|x|^{frac{alpha}{beta}}v(x)=lambda^{-frac{rho_1}{beta}}$.
We consider positive singular solutions of PDEs arising from double phase functionals. Exploiting a rather new version of the moving plane method originally developed by Sciunzi, we prove symmetry and monotonicity properties of such solutions.
It is established existence of bound and ground state solutions for quasilinear elliptic systems driven by (phi 1, phi 2)-Laplacian operator. The main feature here is to consider quasilinear elliptic systems involving both nonsingular nonlinearities combined with indefinite potentials and singular cases perturbed by superlinear and subcritical couple terms. These prevent us to use arguments based on Ambrosetti-Rabinowitz condition and variational methods for differentiable functionals. By exploring the Nehari method and doing a fine analysis on the fibering map associated, we get estimates that allow us unify the arguments to show multiplicity of semi-trivial solutions in both cases.
This paper deals with existence and regularity of positive solutions of singular elliptic problems on a smooth bounded domain with Dirichlet boundary conditions involving the $Phi$-Laplacian operator. The proof of existence is based on a variant of the generalized Galerkin method that we developed inspired on ideas by Browder and a comparison principle. By using a kind of Moser iteration scheme we show $L^{infty}(Omega)$-regularity for positive solutions
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
