In this article, we consider the following boundary value problem: {F(x,u,Du,D2u)+c(x)u+p(x)u−α=0inΩ,u=0on∂Ω, where Ω 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.

This article investigates the exceptional set of the boundary for the following problem: (Formula presented.) 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.

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 Ω is a smooth and bounded domain in RN and fi:[0,∞)×[0,∞)⋯×[0,∞)→[0,∞) are Cα 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.

In this paper, we establish C1,α regularity up to the boundary for a class of degenerate fully nonlinear elliptic equations with Neumann boundary conditions. Our main result Theorem 2.1 constitutes the boundary analogue of the interior C1,α regularity result established in Imbert and Silvestre (Adv. Math. 233: 196–206, 2013) for equations with similar structural assumptions. The proof of our main result is achieved via compactness arguments combined with new boundary Hölder estimates for equations which are uniformly elliptic when the gradient is either small or large.

In this article, 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.

In this article we prove a three solution type theorem for the following boundary value problem: (formula presented) where is a bounded smooth domain in RN and f: [0,∞] → [0,∞] is a Cα 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.

In this article, we establish a Lyapunov-type inequality for the following extremal Pucci's equation: (M+λ, Λ (D2u) + b(x)|Du| + a(x)u = 0 in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝN, N≥2. This work generalizes the well-known works on the Lyapunov inequality for extremal Pucci's equations with gradient nonlinearity.

In this paper, we prove H2+α regularity for viscosity solutions to non-convex fully nonlinear parabolic equations near the boundary. This constitutes the parabolic counterpart of a similar C2,α regularity result due to Silvestre and Sirakov proved in [17] for solutions to non-convex fully nonlinear elliptic equations.

In this paper, we establish the existence of a positive solution to − M + λ Λ (D 2 u) + H(x, Du) = k (x)f(u ) in Ω, u α u > 0 in Ω, u = 0 on ∂Ω, under certain conditions on k, f and H, using viscosity sub-and supersolution method. The main feature of this problem is that it has singularity as well as a superlinear growth in the gradient term. We use Hopf-Cole transformation to handle the superlinear gradient term and an approximation method combined with suitable stability result for viscosity solution to outfit the singular nonlinearity. This work extends and complements the recent works on elliptic equations involving singular as well as superlinear gradient nonlinearities.

In this paper, we show the existence of a classical solution to a class of fractional logistic equations in an open bounded subset with smooth boundary. We use the method of sub-and super-solutions with variational arguments to establish the existence of a unique positive solution. We also establish the stability and non-degeneracy of the positive solution.

In this paper, we consider the bifurcation problem for the fractional Laplace equation (Formula presented.) where Ω ⊂ Rn , n > 2s (0 < s < 1) is an open bounded subset with smooth boundary, (−∆)s stands for the fractional Laplacian. We show that a continuum of solutions bifurcates out from the principal eigenvalue λ1 of the problem (Formula presented.) and, conversely.

In this article, we study the existence and multiplicity of nontrivial solutions to the problem (formula presented) where Ω is a smooth bounded domain in ℝn, n > 2. We show that the problem possesses nontrivial solutions for small value of ε provided f and ψ are continuous and f has a positive zero. We employ degree theory arguments and Liouville type theorem for the multiplicity of the solutionss.

In this paper, we establish the existence of a positive solution to(Formula presented.)where Ω is a smooth bounded domain in Rn,n≥1. Under certain conditions on k,fandh, using viscosity sub- and super solution method with the aid of comparison principle, we establish the existence of a unique positive viscosity solution. This work extends and complements the earlier works on semilinear and singular elliptic equations with sublinear nonlinearity.