This article investigates the exceptional set of the boundary for the following problem: We provide a sufficient condition for the exceptional set in terms of the Hausdorff measure bound of this boundary portion. This condition ensures that even if the boundary values are not nonnegative on this portion, the supersolution remains nonnegative

We consider the following boundary value problem: {F(x,u,Du,D2u)+c(x)u+p(x)u??=0in?,u=0on??, where ? is a bounded and C2 smooth domain in RN. The operator F is proper and has superlinear growth in gradient. This study examines the boundary behavior of the solutions to the above equation and establishes a global regularity result similar to that established in [11,15], which involves linear growth in the gradient

We prove a theorem concerning the existence of three solutions for a fully nonlinear boundary value problem involving the Pucci maximal operator and a natural gradient growth term. Specifically, we consider the Dirichlet problem (Formula presented.)

This article deals with the existence of multiple positive solutions to the following system of nonlinear equations involving Pucci’s extremal operators: (Formula presented.) where ? is a smooth and bounded domain in R and f:[0,?)×[0,?)?×[0,?)?[0,?) are C functions for i=1,2,?,n. The multiplicity result in this work is motivated by the work Amann (SIAM Rev 18(4):620–709, 1976), and Shivaji (Nonlinear analysis and applications (Arlington, Tex., 1986), Dekker, New York, 1987), where the three solutions theorem (multiplicity) has been proved for linear equations. Later on, it was extended for a system of equations involving the Laplace operator by Shivaji and Ali (Differ Integr Equ 19(6):669–680, 2006). Thus, the results here can be considered as a nonlinear analog of the results mentioned above. We also have applied the above results to show the existence of three positive solutions to a system of nonlinear elliptic equations having combined sublinear growth by explicitly constructing two ordered pairs of sub and supersolutions.

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We prove a three solution type theorem for the following boundary value problem: (formula presented) where is a bounded smooth domain in R and f: [0,?] ? [0,?] is a C function. This is motivated by the work of Amann [3] and Shivaji [27], where a three solutions theorem has been established for the Laplace operator. Furthermore, using this result we show the existence of three positive solutions to above boundary value by explicitly constructing two ordered pairs of sub and supersolutions when f has a sublinear growth and f(0) = 0.

We prove the continuity of the horizontal gradient near a C1,Dini non-characteristic portion of the boundary for solutions to perturbations of horizontal Laplaceans as in (1.1) below, where the scalar term is in scaling critical Lorentz space L(Q, 1) with Q being the homogeneous dimension of the group. This result can be thought of both as a sharpening of the boundary regularity result in [4] as well as a subelliptic analogue of the main result in [1] restricted to linear equations.