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In this paper, we study the semilinear subelliptic equation where is the self-adjoint Hörmander operator associated with the vector fields X = (Xl, X2, ..., Xm) satisfying the Hörmander's condition, is a Carathéodory function on Ω × ℝ, and Ω is an open bounded domain in ℝn with smooth boundary. Combining the perturbation from the symmetry method with the approaches involving the eigenvalue estimate and the Morse index in estimating the minimax values, we obtain two kinds of existence results for multiple weak solutions to the problem above. Furthermore, we discuss the difference between the eigenvalue estimate approach and the Morse index approach in degenerate situations. Compared with the classical elliptic cases, both approaches here have their own strengths in the degenerate cases. This new phenomenon implies that the results in general degenerate cases would be quite different from the situations in classical elliptic cases. [ABSTRACT FROM AUTHOR]

We consider the problem of supersonic flow of a Chaplygin gas past a delta wing with a shock or a rarefaction wave attached to the leading edges. The flow under study is described by the three-dimensional steady Euler system. In conical coordinates, this problem can be reformulated as a boundary value problem for a nonlinear equation of mixed type. The type of this equation depends fully on the solutions of the problem itself, and thus it cannot be determined in advance. We overcome the difficulty by establishing a crucial Lipschitz estimate, and finally prove the unique existence of the solution via the method of continuity. [ABSTRACT FROM AUTHOR]

The present paper is concerned with a class of quasi-linear elliptic degenerate equations. The degenerate operator arises from analysis of manifolds with singularities. The variational methods are applied here to verify the existence of infinitely many solutions for the problem. [ABSTRACT FROM AUTHOR]

In this paper, we give interior gradient and Hessian estimates for systems of semi-linear degenerate elliptic partial differential equations on bounded domains, using both tools of backward stochastic differential equations and quasi-derivatives. [ABSTRACT FROM AUTHOR]

We study a degenerate elliptic system with variable exponents. Using the variational approach and some recent theory on weighted Lebesgue and Sobolev spaces with variable exponents, we prove the existence of at least two distinct nontrivial weak solutions of the system. Several consequences of the main theorem are derived; in particular, the existence of at lease two distinct nontrivial non-negative solutions is established for a scalar degenerate problem. One example is provided to show the applicability of our results. [ABSTRACT FROM AUTHOR]

In this paper, we study the effect of lower order perturbations in the existence of positive solutions to the fractional Kirchhoff equation with critical growth (a + b ∫ R 3 | (− △) s 2 u | 2 d x) (− △) s u + V (x) u = μ | μ p − 1 u + | u | 2 s ∗ − 2 u , x ϵ R 3 , where a; b > 0 are constants, μ > 0 is a parameter, s ∈ (3 4 , 1) , 1 < p < 2 s ∗ − 1 = 3 + 2 s 3 − 2 s , and V : ℝ3 → ℝ is a continuous potential function. For suitable assumptions on V, we show the existence of a positive ground state solution, by using the methods of the Pohozaev-Nehari manifold, Jeanjean's monotonicity trick and the concentration-compactness principle due to Lions (1984). [ABSTRACT FROM AUTHOR]

We study the boundary behaviors of solutions f to the Dirichlet problem for minimal graphs in the hyperbolic space with singular asymptotic boundaries and characterize the boundary behaviors of f at the points strictly located in the tangent cones at the singular points on the boundary. For n = 2, we also obtain a refined estimate of f. [ABSTRACT FROM AUTHOR]

We give an alternative way to construct an entire function with quasiconformal surgery so that all its Fatou components are quasi-disks but the Julia set is non-locally connected. [ABSTRACT FROM AUTHOR]

With the rapid-growth-in-size scientific data in various disciplines, feature screening plays an important role to reduce the high-dimensionality to a moderate scale in many scientific fields. In this paper, we introduce a unified and robust model-free feature screening approach for high-dimensional survival data with censoring, which has several advantages: it is a model-free approach under a general model framework, and hence avoids the complication to specify an actual model form with huge number of candidate variables; under mild conditions without requiring the existence of any moment of the response, it enjoys the ranking consistency and sure screening properties in ultra-high dimension. In particular, we impose a conditional independence assumption of the response and the censoring variable given each covariate, instead of assuming the censoring variable is independent of the response and the covariates. Moreover, we also propose a more robust variant to the new procedure, which possesses desirable theoretical properties without any finite moment condition of the predictors and the response. The computation of the newly proposed methods does not require any complicated numerical optimization and it is fast and easy to implement. Extensive numerical studies demonstrate that the proposed methods perform competitively for various configurations. Application is illustrated with an analysis of a genetic data set. [ABSTRACT FROM AUTHOR]

We study corner-degenerate pseudo-differential operators of any singularity order and develop ellipticity based on the principal symbolic hierarchy, associated with the stratification of the underlying space. We construct parametrices within the calculus and discuss the aspect of additional trace and potential conditions along lower-dimensional strata. [ABSTRACT FROM AUTHOR]

We give a classification of second-order polynomial solutions for the homogeneous k-Hessian equation σ[ u] = 0. There are only two classes of polynomial solutions: One is convex polynomial; another one must not be ( k + 1)-convex, and in the second case, the k-Hessian equations are uniformly elliptic with respect to that solution. Base on this classification, we obtain the existence of C local solution for nonhomogeneous term f without sign assumptions. [ABSTRACT FROM AUTHOR]

In this paper, by using the idea of category, we investigate how the shape of the graph of h( x) affects the number of positive solutions to the following weighted nonlinear elliptic system: where 0 ∈ Ω is a smooth bounded domain in ℝ ( N ⩾ 3), λ, σ > 0 are parameters, $0 \leqslant \mu < \bar \mu _a \triangleq (\frac{{N - 2 - 2a}} {2})2 $; h( x), K( x) and K( x) are positive continuous functions in $\bar \Omega $, 1 ⩽ q < 2, α, β > 1 and α + β = 2*( a, b) $(2*(a,b) \triangleq \frac{{2N}} {{N - 2(1 + a - b)}}) $, is critical Sobolev-Hardy exponent). We prove that the system has at least k nontrivial nonnegative solutions when the pair of the parameters ( λ, σ) belongs to a certain subset of ℝ. [ABSTRACT FROM AUTHOR]

In this paper, we mainly discuss a priori bounds of the following degenerate elliptic equation, where a ∂ ϕ∂ ϕ = 0 on ∂Ω, and ϕ is the defining function of ∂Ω. Imposing suitable conditions on the coefficients and f( x, u), one can get the L-estimates of (*) via blow up method. [ABSTRACT FROM AUTHOR]

In this paper, we study the very weak solutions to some nonlinear elliptic systems with right-hand side integrable data with respect to the distance to the boundary. Firstly, we study the existence of the approximate solutions. Secondly, a priori estimates are given in the framework of weighted spaces. Finally, we prove the existence, uniqueness and regularity of the very weak solutions. [ABSTRACT FROM AUTHOR]

We characterize a class of physical boundary conditions that guarantee the existence and uniqueness of the subsonic Euler flow in a general finitely long nozzle. More precisely, by prescribing the incoming flow angle and the Bernoulli's function at the inlet and the end pressure at the exit of the nozzle, we establish an existence and uniqueness theorem for subsonic Euler flows in a 2-D nozzle, which is also required to be adjacent to some special background solutions. Such a result can also be extended to the 3-D asymmetric case. [ABSTRACT FROM AUTHOR]

For Ω a bounded subset of ℝ, n ⩾ 2, ψ any function in Ω with values in ℝ ∪ {±∞} and $$\theta \in W^{1,\left( {q_i } \right)} \left( \Omega \right)$$, let This paper deals with solutions to $$\mathcal{K}_{\psi ,\theta }^{\left( {q_i } \right)}$$-obstacle problems for the A-harmonic equation as well as the integral functional Local regularity and local boundedness results are obtained under some coercive and controllable growth conditions on the operator A and some growth conditions on the integrand f. [ABSTRACT FROM AUTHOR]