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In this paper, we study the Dirichlet problem of Hessian quotient equations of the form $S_k(D^2u)/S_l(D^2u)=g(x)$ in exterior domains. For $g\\equiv \\mbox {const.}$ , we obtain the necessary and sufficient conditions on the existence of radially symmetric solutions. For g being a perturbation of a generalized symmetric function at infinity, we obtain the existence of viscosity solutions by Perron’s method. The key technique we develop is the construction of sub- and supersolutions to deal with the non-constant right-hand side g.

We investigate the existence, uniqueness and multiplicity of one-signed rotationally symmetric solutions of singular Dirichlet problems with the prescribed higher mean curvature operator in Minkowski spacetime. The main tools are the Schauder fixed point theorem along with cut-off technique and the  Leggett-Williams fixed point theorem.  In addition, we give some practical models to illustrate the effectiveness of our results.

We propose a new monotone finite difference discretization for the variational p -Laplace operator, Δ p u = div ( | ∇ u | p - 2 ∇ u ) , and present a convergent numerical scheme for related Dirichlet problems. The resulting nonlinear system is solved using two different methods: one based on Newton-Raphson and one explicit method. Finally, we exhibit some numerical simulations supporting our theoretical results. To the best of our knowledge, this is the first monotone finite difference discretization of the variational p -Laplacian and also the first time that nonhomogeneous problems for this operator can be treated numerically with a finite difference scheme.

The fourth-order properly elliptic equation with multiple root is considered in the elliptic domain. The conditions, necessary and sufficient for the unique solvability of the Dirichlet problem for this equation are found, and if these conditions fail the defect numbers of this problem are determined. The solution of the problem is found in explicit form.

The strong solvability of the Cauchy-Dirichlet problem for parabolic quasilinear equations with discontinuous data is investigated. The leading coefficients depend on the point (x, t) and on the solution u, the dependence on x is of the VMO type, while with respect to t only measurability is required from them. Under suitable structural conditions on the nonlinear terms, the existence and uniqueness of a strong solution is proved, which also turns out to be Hölder continuous.

In an unbounded domain of RN, N≥2, we prove existence and Stampacchia type regularity of solutions to some noncoercive nonlinear Dirichlet problems whose model case appears in stationary convection–diffusion phenomena. The drift term is controlled through a function in a suitable functional space, strictly containing Lebesgue one. We obtain some a priori estimates, by contradiction, via a weak maximum principle.

In this paper, we deduce a new formula that explicitly expresses the cylindrical Poisson kernel of any degree and order using cylindrical Poisson polynomials. We then demonstrate how this formula can be applied to solve the Dirichlet boundary value problems for cylindrical Laplace equations. Furthermore, we examine and discuss the behaviors of this solution approach at infinity.

In this paper, we study conformal solitons for the mean curvature flow in hyperbolic space Hn+1. Working in the upper half-space model, we focus on horo-expanders, which relate to the conformal field -∂0. We classify cylindrical and rotationally symmetric examples, finding appropriate analogues of grim-reaper cylinders, bowl and winglike solitons. Moreover, we address the Plateau and the Dirichlet problems at infinity. For the latter, we provide the sharp boundary convexity condition to guarantee its solvability and address the case of non-compact boundaries contained between two parallel hyperplanes of ∂∞Hn+1. We conclude by proving rigidity results for bowl and grim-reaper cylinders.

In this paper, by using critical point theory, we obtain some sufficient conditions on the existence of infinitely many positive solutions of the discrete Dirichlet problem involving the mean curvature operator. We show that the suitable oscillating behavior of the nonlinear term near at the origin and at infinity will lead to the existence of a sequence of pairwise distinct nontrivial positive solutions. We also give two examples to illustrate our main results.

We deal with a class of nonlinear elliptic Dirichlet problems with terms having supercritical growth from the viewpoint of the Sobolev embedding. We prove that for every planar set Γ , which is contractible in itself and consists of a finite number of curves, there exist suitable bounded domains arbitrarily close to Γ (as thin neighbourhoods of Γ ) such that in these domains these Dirichlet problems do not have any nontrivial solution. Domains of this type may have a shape very different from the starshaped bounded domains where a well known Pohozaev nonexistence result holds. Indeed, our result suggests that, in dimension n = 2 , Pohozaev nonexistence result might be extended from the bounded starshaped domains to all bounded contractible domains. Notice that this fact is not true in higher dimensions n ≥ 3 . In fact, for example, in a domain as a pierced annulus of R 2 our result guarantees nonexistence of nontrivial solutions while in a pierced annulus of R n with n ≥ 3 there exist many nontrivial solutions when the size of the perforation is small enough.