1,720,971 research outputs found
On positive solutions of fully nonlinear degenerate Lane–Emden type equations
Full text linkWe prove existence and uniqueness results of positive viscosity solutions of fully nonlinear degenerate elliptic equations with power-like zero order perturbations in bounded domains. The principal part of such equations is either Pk−(D2u) or Pk+(D2u), some sort of “truncated Laplacians”, given respectively by the smallest and the largest partial sum of k eigenvalues of the Hessian matrix. New phenomena with respect to the semilinear case occur. Moreover, for P−k, we explicitly find the critical exponent p of the power nonlinearity that separates the existence and nonexistence range of nontrivial solutions with zero Dirichlet boundary condition
Removable singularities for degenerate elliptic Pucci operators
No full textIn this paper we introduce some fully nonlinear second order operators defined as weighted partial sums of the eigenvalues of the Hessian matrix, arising in geometrical contexts, with the aim to extend maximum principles and removable singularities results to cases of highly degenerate ellipticity
Positivity sets of supersolutions of degenerate elliptic equations and the strong maximum principle
No full textWe investigate positivity sets of nonnegative supersolutions of the
fully nonlinear elliptic equations F(x, u,Du,D2u) = 0 in Ω, where Ω is an
open subset of RN, and the validity of the strong maximum principle for
F(x, u,Du,D2u) = f in Ω, with f ∈ C(Ω) being nonpositive. We obtain
geometric characterizations of positivity sets {x ∈ Ω : u(x) > 0} of nonnegative
supersolutions u and establish the strong maximum principle under some
geometric assumption on the set {x ∈ Ω : f(x) = 0}
Viscosity Solutions of Uniformly Elliptic Equations without Boundary and Growth Conditions at Infinity
Full text linkWe deal with fully nonlinear second-order equations assuming a superlinear growth in u with the aim to generalize previous existence and uniqueness results of viscosity solutions in the whole space without conditions at infinity. We also consider the solvability of the Dirichlet problem in bounded and unbounded domains and show a blow-up result
Propagation of minima for nonlocal operators
Full text linkIn this paper we state some sharp maximum principle, i.e. we characterize the geometry of the sets of minima for supersolutions of equations involving the k-th fractional truncated Laplacian or the k-th fractional eigenvalue which are fully nonlinear integral operators whose nonlocality is somehow k-dimensional
On the uniqueness of blow-up solutions of fully nonlinear elliptic equations
Full text linkThis paper contains new uniqueness results of the boundary blow-up viscosity solutions of second order elliptic equations, generalizing a well known result of Marcus-Veron for the Laplace operator
A regularity result for a class of non-uniformly elliptic operators
Full text linkWe obtain an explicit Hölder regularity result for viscosity solutions of a class of second order fully nonlinear equations led by operators
that are neither convex/concave nor uniformly elliptic
Riesz capacity, maximum principle and removable sets of fully nonlinear second order elliptic operators
No full textIn this paper we show sufficient conditions for the extended maximum principle and the removable singularities for viscosity solutions of fully nonlinear second-order elliptic equations via Riesz and logarithmic capacity
Existence results for fully nonlinear equations in radial domains
No full textWe consider the fully nonlinear problem
-F(x,D^2u)=|u|^{p-1}u in Ω
u=0 on partial Ω
where F is uniformly elliptic, p>1 and Ω is either an annulus or a ball in Rn, n≥2.
We prove the following results:
i. existence of a positive/negative radial solution for every exponent p > 1, if Ω is an annulus;
ii. existence of infinitely many sign changing radial solutions for every p > 1, characterized by the number of nodal regions, if Ω is an annulus;
iii. existence of infinitely many sign changing radial solutions characterized by the number of nodal regions if Ω is a ball and p is subcritical
A family of degenerate elliptic operators: maximum principle and its consequences
Full text linkIn this paper we investigate the validity and the consequences of the maximum principle for degenerate elliptic operators whose higher order term is the sum of k eigenvalues of the Hessian. In particular we shed some light on some very unusual phenomena due to the degeneracy of the operator. We prove moreover Lipschitz regularity results and boundary estimates under convexity assumptions on the domain. As a consequence we obtain the existence of solutions of the Dirichlet problem and of principal eigenfunctions
- …
