Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userPointwise gradient bounds for entire solutions of elliptic equations with non-standard growth conditions and general nonlinearities
151
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We give pointwise gradient bounds for solutions of (possibly non-uniformly) elliptic partial differential equations in the entire Euclidean space. The operator taken into account is very general and comprises also the singular and degenerate nonlinear case with non-standard growth conditions. The sourcing term is also allowed to have a very general form, depending on the space variables, on the solution itself, on its gradient, and possibly on higher order derivatives if additional structural conditions are satisfied.
rate research
Read More
We investigate partial symmetry of solutions to semi-linear and quasi-linear elliptic problems with convex nonlinearities, in domains that are either axially symmetric or radially symmetric.
We classify positive solutions to a class of quasilinear equations with Neumann or Robin boundary conditions in convex domains. Our main tool is an integral formula involving the trace of some relevant quantities for the problem. Under a suitable condition on the nonlinearity, a relevant consequence of our results is that we can extend to weak solutions a celebrated result obtained for stable solutions by Casten and Holland and by Matano.
We study inverse problems for semilinear elliptic equations with fractional power type nonlinearities. Our arguments are based on the higher order linearization method, which helps us to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not known. By using a fractional order adaptation of this method, we show that the results of [LLLS20a, LLLS20b] remain valid for general power type nonlinearities.
We prove, by a shooting method, the existence of infinitely many solutions of the form $psi(x^0,x) = e^{-iOmega x^0}chi(x)$ of the nonlinear Dirac equation {equation*} iunderset{mu=0}{overset{3}{sum}} gamma^mu partial_mu psi- mpsi - F(bar{psi}psi)psi = 0 {equation*} where $Omega>m>0,$ $chi$ is compactly supported and [F(x) = {{array}{ll} p|x|^{p-1} & text{if} |x|>0 0 & text{if} x=0 {array}.] with $pin(0,1),$ under some restrictions on the parameters $p$ and $Omega.$ We study also the behavior of the solutions as $p$ tends to zero to establish the link between these equations and the M.I.T. bag model ones.
We obtain asymptotic mean value formulas for solutions of second-order elliptic equations. Our approach is very flexible and allows us to consider several families of operators obtained as an infimum, a supremum, or a combination of both infimum and supremum, of linear operators. We study both when the set of coefficients is bounded and unbounded (each case requires different techniques). The families of equations that we consider include well-known operators such as Pucci, Issacs, and $k-$Hessian operators.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
