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In this paper, we consider the uniformly elliptic nonlocal Bellman problem F s u (x) = f (u (x) , v (x)) , F s v (x) = g (u (x) , v (x)). Firstly, we study narrow region principles for the uniformly elliptic nonlocal Bellman operators in bounded and unbounded domains, which play key roles in obtaining the main results by the process of sliding method. Then we deal with monotonicity properties of solutions to the uniformly elliptic nonlocal Bellman system. [ABSTRACT FROM AUTHOR]

The first part of this paper deals with existence of solutions to the quasilinear elliptic problem (P) − div a (x , ∇ u) = f (x , u , ∇ u) in Ω , a (x , ∇ u) ⋅ ν = g (x , u) − ζ | u | p − 2 u on ∂ Ω , involving a general nonhomogeneous differential operator, namely div a , and Carathéodory functions f : Ω × R × R N → R and g : ∂ Ω × R → R. Under appropriate conditions on the perturbations, we show that (P) possesses a bounded solution. In the second part, we consider the special case when div a is the (p , q) -Laplacian with a parameter μ > 0 , and study the asymptotic behavior of solutions as μ goes to zero or to infinity. A uniqueness result is also provided. [ABSTRACT FROM AUTHOR]

We investigate the Kirchhoff type elliptic problem with critical nonlinearity; − 1 + α ∫ Ω | ∇ u | 2 d x Δ u = λ u q + u 2 ∗ − 1 , u > 0 in Ω , u = 0 on ∂ Ω , where N ≥ 5 , Ω ⊂ R N is a bounded domain with smooth boundary ∂ Ω , α > 0 , λ ∈ R , 2 ∗ = 2 N ∕ (N − 2) , and q ∈ [ 1 , 2 ∗ − 1). We prove the existence of two solutions of it via the variational method. Since N ≥ 5 and α > 0 , the uniqueness assertion for the associated limiting problem may fail. This causes serious difficulties in controlling concentrating Palais–Smale sequences. We overcome these by introducing new techniques. For a mountain pass type solution, we utilize the limit function of the fibering maps of the concentrating Palais–Smale sequence. This tool is based on our careful setting of Nehari type sets. On the other hand, a suitable modification to a concentrating minimizing sequence enables us to obtain a global minimum solution. This is the first work which proves the multiplicity of positive solutions of the Kirchhoff type critical problem in high dimension. [ABSTRACT FROM AUTHOR]

We study the relationship between the Regularity and Dirichlet boundary value problems for parabolic equations of the form L u = div (A ∇ u) − u t = 0 in Lip (1 , 1 ∕ 2) time-varying cylinders, where the coefficient matrix A = a i j (X , t) is uniformly elliptic and bounded. We show that if the Regularity problem (R) p for the equation L u = 0 is solvable for some 1 < p < ∞ then the Dirichlet problem (D ∗) p ′ for the adjoint equation L ∗ v = 0 is also solvable, where p ′ = p ∕ (p − 1). This result is an analogue of the result established in the elliptic case by Kenig and Pipher (1993). In the parabolic settings in the special case of the heat equation in slightly smoother domains this has been established by Hofmann and Lewis (1996) and Nyström (2006) for scalar parabolic systems. In comparison, our result is abstract with no assumption on the coefficients beyond the ellipticity condition and is valid in more general class of domains. [ABSTRACT FROM AUTHOR]

We prove the validity of maximum principles for a class of fully nonlinear operators on unbounded subdomains Ω ⊂ R n of cylindrical type. The main structural assumption is the uniform ellipticity of the operator along the bounded directions of Ω , with possible degeneracy along the unbounded directions. [ABSTRACT FROM AUTHOR]

Abstract We prove various representation formulae of solutions of second order differential inequalities on R N. As an outcome we obtain positivity results of the solutions of − L u ≥ 0 , where L is a general second order elliptic operator in divergence form. Results of this type are very useful for studying related Liouville theorems of second order semilinear inequalities. [ABSTRACT FROM AUTHOR]

Abstract An irregular obstacle problem with non-uniformly elliptic operator in divergence form of (p , q) -growth is studied. We find an optimal regularity for such a double phase obstacle problem by essentially proving that the gradient of a solution is as integrable as both the gradient of the assigned obstacle function and the associated nonhomogeneous term in the divergence. Calderón–Zygmund type estimates are also obtained under minimal regularity requirements of the prescribed data. [ABSTRACT FROM AUTHOR]

We present some recent advances in the productive and symbiotic interplay between general potential theories (subharmonic functions associated to closed subsets F ⊂ J 2 (X) of the 2-jets on X ⊂ R n open) and subsolutions of degenerate elliptic and parabolic PDEs of the form F (x , u , D u , D 2 u) = 0. We will implement the monotonicity-duality method begun by Harvey and Lawson (2009) (in the pure second order constant coefficient case) for proving comparison principles for potential theories where F has sufficient monotonicity and fiberegularity (in variable coefficient settings) and which carry over to all differential operators F which are compatible with F in a precise sense for which the correspondence principle holds. We will consider both elliptic and parabolic versions of the comparison principle in which the effect of boundary data is seen on the entire boundary or merely on a proper subset of the boundary. Particular attention will be given to gradient dependent examples with the requisite sufficient monotonicity of proper ellipticity and directionality in the gradient. Examples operators we will discuss include the degenerate elliptic operators of optimal transport in which the target density is strictly increasing in some directions as well as operators which are weakly parabolic in the sense of Krylov. Further examples, modeled on hyperbolic polynomials in the sense of Gårding give a rich class of examples with directionality in the gradient. Moreover we present a model example in which the comparison principle holds, but standard viscosity structural conditions fail to hold. [ABSTRACT FROM AUTHOR]

We obtain a global extension of the classical weak Harnack inequality which extends and quantifies the Hopf–Oleinik boundary-point lemma, for uniformly elliptic equations in divergence form. Among the consequences is a boundary gradient estimate, due to Krylov and well-studied for non-divergence form equations, but completely novel in the divergence framework. Another consequence is a new more general version of the Hopf–Oleinik lemma. [ABSTRACT FROM AUTHOR]

Abstract We compute estimates for eigenvalues of a class of linear second-order elliptic differential operators in divergence form (with Dirichlet boundary condition) on a bounded domain in a complete Riemannian manifold. Our estimates are based upon the Weyl's asymptotic formula. As an application, we find a lower bound for the mean of the first k eigenvalues of the drifting Laplacian. In particular, we have extended for this operator a partial solution given by Cheng and Yang for the generalized conjecture of Pólya. We also derive a second-Yang type inequality due to Chen and Cheng, and other two inequalities which generalize results by Cheng and Yang obtained for a domain in the unit sphere and for a domain in the projective space. [ABSTRACT FROM AUTHOR]

We consider a class of nonlinear non-diagonal elliptic systems with p -growth and establish the L q -integrability for all q ∈ [ p , p + 2 ] of any weak solution provided the corresponding right hand side belongs to the corresponding Lebesgue space and the involved elliptic operator asymptotically satisfies the p -uniform ellipticity, the so-called splitting condition and it is continuous with respect to the spatial variable. For operators satisfying the uniform p -ellipticity condition the higher integrability is known for q ∈ [ p , d p ∕ ( d − 2 ) ] and for operators having the so-called Uhlenbeck structure, the theory is valid for all q ∈ [ p , ∞ ) . The key novelty of the paper is twofold. First, the statement uses only the information coming from the asymptotic operator and second, and more importantly, by using the splitting condition, we are able to extend the range of possible q ’s significantly whenever p < d − 2 . [ABSTRACT FROM AUTHOR]

For ν ∈ ( 0 , 1 ) , we consider the semilinear integro-differential equation on the one-dimensional domain Ω = ( a , b ) in the unknown u = u ( x , t ) D t ν u − L 1 u − ∫ 0 t K ( t − s ) L 2 u ( ⋅ , s ) d s = f ( x , t , u ) + g ( x , t ) where D t ν is the Caputo fractional derivative and L 1 and L 2 are uniform elliptic operators with time-dependent smooth coefficients. Under certain structural conditions on the nonlinearity f , the global existence and uniqueness of classical solutions to the related initial–boundary value problems are established, via the so-called continuation argument approach. The key point is looking for suitable a priori estimates of the solution in the fractional Hölder spaces. [ABSTRACT FROM AUTHOR]

Let Ω ⊆ R N be a bounded domain with the extension property, whose boundary Γ ≔ ∂ Ω is an upper d -set (with respect to a measure μ ), for N ≥ 2 and d ∈ ( N − p , N ) . We investigate the solvability of the Ambrosetti–Prodi problem for the p -Laplace operator Δ p , with Neumann boundary conditions. Using a priori estimates, regularity theory, a sub-supersolution method, and the Leray–Schauder degree theory, we obtain a necessary condition for the non-existence of solutions (in the weak sense), the existence of at least one minimal solution, and the existence of at least two distinct solutions. Moreover, we establish global Hölder continuity for weak solutions of the Neumann problem of Ambrosetti–Prodi type on a large class of “bad” domain, extending the solvability of this type of elliptic problem for the first time, to a wide class of non-smooth domains. [ABSTRACT FROM AUTHOR]

We study the qualitative properties of functions belonging to the corresponding De Giorgi classes ∫ B r (1 − σ) (x 0) Φ (x , | ∇ (u − k) ± |) d x ⩽ γ ∫ B r (x 0) Φ (x , (u − k) ± σ r ) d x , where σ , r ∈ (0 , 1) , k ∈ R and the function Φ satisfies the non-logarithmic condition ( r − n ∫ B r (x 0) [ Φ (x , v r ) ] s d x) 1 s ( r − n ∫ B r (x 0) [ Φ (x , v r ) ] − t d x) 1 t ⩽ c (K) Λ (x 0 , r) , r ⩽ v ⩽ K λ (r) , under some assumptions on the functions λ (r) and Λ (x 0 , r) and the numbers s , t > 1. These conditions generalize the known logarithmic, non-logarithmic and non uniformly elliptic conditions. In particular, our results cover new cases of non uniformly elliptic double-phase, degenerate double-phase functionals and functionals with variable exponents. [ABSTRACT FROM AUTHOR]

This note concerns the existence of positive solutions for the boundary problems { − L A u = u 2 ∗ − 1 + λ u in Ω u = 0 on ∂ Ω , { − L a , p u = u p ∗ − 1 + λ u p − 1 in Ω u = 0 on ∂ Ω , where L A u = div ( A ( x ) ∇ u ) and L a , p u = div ( a ( x ) | ∇ u | p − 2 ∇ u ) are, respectively, linear and quasilinear uniformly elliptic operators in divergence form in a non-smooth bounded open subset Ω of R n , 1 < p < n , p ∗ = n p / ( n − p ) is the critical Sobolev exponent and λ is a real parameter. Both problems have been quite studied when the ellipticity of L A and L a , p concentrate in the interior of Ω . We here focus on the borderline case, namely we assume that the determinant of A ( x ) has a global minimum point x 0 on the boundary of Ω such that A ( x ) − A ( x 0 ) is locally comparable to | x − x 0 | γ I n in the bilinear forms sense, where I n denotes the identity matrix of order n . Similarly, we assume that a ( x ) has a global minimum point x 0 on the boundary of Ω such that a ( x ) − a ( x 0 ) is locally comparable to | x − x 0 | σ . We provide a linking between the exponents γ and σ and the order of singularity of the boundary of Ω at x 0 so that these problems admit at least one positive solution for any λ ∈ ( 0 , λ 1 ( − L A ) ) and λ ∈ ( 0 , λ 1 ( − L a , p ) ) , respectively, where λ 1 denotes the first Dirichlet eigenvalue of the corresponding operator. [ABSTRACT FROM AUTHOR]

We examine Hamilton–Jacobi equations driven by fully nonlinear degenerate elliptic operators in the presence of superlinear Hamiltonians. By exploring the Ishii–Jensen inequality, we prove that viscosity solutions are locally Lipschitz-continuous, with estimates depending on the structural conditions of the problem. We close the paper with an application of our findings to a two-phase free boundary problem. [ABSTRACT FROM AUTHOR]

In this note, we prove the interior C 2 , α regularity for viscosity solutions of fully nonlinear uniformly elliptic equations with the ellipticity constants close to each other enough. In particular, the regularity for Isaacs equations can be derived. The main idea is the compactness technique. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the uniformly elliptic nonlocal Bellman system. Firstly, we study different maximum principles for the uniformly elliptic nonlocal Bellman operators and obtain key ingredients for carrying on the method of moving planes. Then we derive qualitative properties of solutions to the uniformly elliptic nonlocal Bellman in bounded domains, unbounded domains, and a coercive epigraph domain Ω in R n , respectively, such as monotonicity, symmetry, and Liouville-type results. [ABSTRACT FROM AUTHOR]

We show an abstract method to construct different realizations of abstract operators generating a strongly continuous semigroup and apply it to several second order elliptic operators with singular coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we establish the optimal interior regularity and the C 1 , γ smoothness of the regular part of the free boundary in the thin obstacle problem for a class of degenerate elliptic equations with variable coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we deal with an overdetermined problem of Serrin-type with respect to a two-phase elliptic operator in divergence form with piecewise constant coefficients. In particular, we consider the case where the two-phase overdetermined problem is close to the one-phase setting. First, we show quantitative stability estimates for the two-phase problem via a one-phase stability result. Furthermore, we prove non-existence for the corresponding inner problem by the aforementioned two-phase stability result. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem for the semilinear nonautonomous parabolic equation u t = A ( t ) u + ψ ( t , u ) in [ s , τ ] × R d , τ > s , in the spaces C b ( [ s , τ ] × R d ) and in L p ( ( s , τ ) × R d , ν ) . Here ν is a Borel measure defined via a tight evolution system of measures for the evolution operator G ( t , s ) associated to the family of time depending second order uniformly elliptic operators A ( t ) . Sufficient conditions for existence in the large and stability of the null solution are also given in both C b and L p contexts. The novelty with respect to the literature is that the coefficients of the operators A ( t ) are allowed to be unbounded. [ABSTRACT FROM AUTHOR]

This paper deals with the existence and the asymptotic behavior of non-negative solutions for a class of stationary Kirchhoff problems driven by a fractional integro-differential operator L K and involving a critical nonlinearity. In particular, we consider the problem − M ( | | u | | 2 ) L K u = λ f ( x , u ) + | u | 2 s ∗ − 2 u in Ω , u = 0 in R n ∖ Ω , where Ω ⊂ R n is a bounded domain, 2 s ∗ is the critical exponent of the fractional Sobolev space H s ( R n ) , the function f is a subcritical term and λ is a positive parameter. The main feature, as well as the main difficulty, of the analysis is the fact that the Kirchhoff function M could be zero at zero, that is the problem is degenerate . The adopted techniques are variational and the main theorems extend in several directions previous results recently appeared in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a uniform W 2 , ε -estimate of solutions to the fully nonlinear uniformly elliptic equations on Riemannian manifolds with a lower bound of sectional curvature using the ABP method. [ABSTRACT FROM AUTHOR]

Optimal regularity estimates are established for the gradient of solutions to non-uniformly elliptic equations of Orlicz double phase with variable exponents type in divergence form under sharp conditions on such highly nonlinear operators for the Calderón–Zygmund theory. [ABSTRACT FROM AUTHOR]

We study the behavior near the boundary corner point of bounded weak solutions to the oblique derivative problem for singular p (x) -Laplacian equation with strong nonlinearities singular absorption term in a bounded plane sector. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the existence and some qualitative properties of solutions of variational inequalities of the form: and of quasi-variational inequalities of the form: where is a second-order elliptic operator of Leray–Lions type, is a multivalued lower order term, and are convex functionals, and also depends on . We concentrate here in noncoercive cases and use sub-supersolution methods to study the existence and enclosure of solutions, and also the existence of extremal solutions between sub and supersolutions. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the existence of solutions for equations driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary conditions. We make use of homological linking and Morse theory. [Copyright &y& Elsevier]

Abstract: This study deals with the analysis of the Cauchy problem of a general class of nonlocal nonlinear equations modeling the bi-directional propagation of dispersive waves in various contexts. The nonlocal nature of the problem is reflected by two different elliptic pseudodifferential operators acting on linear and nonlinear functions of the dependent variable, respectively. The well-known doubly dispersive nonlinear wave equation that incorporates two types of dispersive effects originated from two different dispersion operators falls into the category studied here. The class of nonlocal nonlinear wave equations also covers a variety of well-known wave equations such as various forms of the Boussinesq equation. Local existence of solutions of the Cauchy problem with initial data in suitable Sobolev spaces is proven and the conditions for global existence and finite-time blow-up of solutions are established. [Copyright &y& Elsevier]

Abstract: We prove some existence, nonexistence and regularity results for the boundary value problem where is a bounded subset of , , and is a -Laplacian, i.e. a “degenerate” elliptic operator of the kind Together with some assumptions made in Franchi and Lanconelli (1984) , the family is supposed to verify a condition making homogeneous of degree two with respect to a group of dilations in . [Copyright &y& Elsevier]

Abstract: In this paper we establish the existence of two nontrivial weak solutions of some eigenvalue problems, involving a general elliptic operator in divergence form. First we study a one parameter problem under homogeneous Dirichlet boundary conditions in bounded domains, and then a two parameters problem under inhomogeneous nonlinear Robin boundary conditions in regular bounded domains. These results complete in several directions some recent papers, where a Ricceri type critical point theorem was used. The main argument of this paper is based on two recent theorems due to Arcoya and Carmona (2007) in which we use in a slightly modified version, see Section . [Copyright &y& Elsevier]

Abstract: Let be the -dimensional Heisenberg group, be its sub-elliptic gradient operator, and for be the distance function in . Denote and . It is proved in this paper that there exists a positive constant such that for any pair and satisfying and , where is the Sobolev space on . When , the above integral is still finite for any . Furthermore the supremum is infinite if , where , . Actually if we replace and by unbounded domain and respectively, the above inequality still holds. As an application of this inequality, a sub-elliptic equation with exponential growth is considered. [Copyright &y& Elsevier]

Abstract: Picone identities are established for a class of half-linear elliptic operators with -Laplacians, and Sturmian comparison theorems are obtained on the basis of the Picone identities. Specializations to half-linear partial or ordinary differential inequalities with -Laplacians are shown. [Copyright &y& Elsevier]

Abstract: Given a non-negative, continuous function on such that for all , in , and non-decreasing in for each , we study the boundary value problem where is a bounded domain and is the -Laplacian, a degenerate elliptic operator. We provide sufficient conditions on under which the above problem admits a solution, or fails to admit a solution. A necessary and sufficient condition on is given for a solution to exist in the special case when . In the latter case an asymptotic boundary behavior of solutions will be studied. As an application a sufficient condition on will be given to ensure the uniqueness of solutions in case is a constant. [Copyright &y& Elsevier]

Abstract: Spectral properties of some integro-differential operators on are studied. Characterisation of the principal eigenvalue is obtained in terms of the positive eigenfunction. These results are used to prove local and global stability of travelling waves and to find their speed. [Copyright &y& Elsevier]

Abstract: The results by Palagachev (2009)  regarding global Hölder continuity for the weak solutions to quasilinear divergence form elliptic equations are generalized to the case of nonlinear terms with optimal growths with respect to the unknown function and its gradient. Moreover, the principal coefficients are discontinuous with discontinuity measured in terms of small norms and the underlying domain is supposed to have fractal boundary satisfying a condition of Reifenberg flatness. The results are extended to the case of parabolic operators as well. [ABSTRACT FROM AUTHOR]

Abstract: This paper is devoted to providing a sufficient condition for the maximality of the sum of subdifferential operators defined on reflexive Banach spaces and proving the maximal monotonicity in of the nonlinear elliptic operator with a maximal monotone graph . [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper is to extend some results of the potential theory of an elliptic operator to the fractional Laplacian , , in a bounded domain in . In particular, we introduce a new Kato class and we exploit the properties of this class to study the existence of positive solutions of some Dirichlet problems for the fractional Laplacian. [ABSTRACT FROM AUTHOR]

Abstract: We consider a singular parametric elliptic equation driven by the -Laplacian and with a -superlinear potential which does not satisfy the Ambrosetti–Rabinowitz condition. Using variational arguments, combined with truncation techniques and the method of upper and lower solutions, we show that there exists a such that if the parameter is in , then the problem has two ordered smooth positive strong solutions. [Copyright &y& Elsevier]

Abstract: In this paper, we deal with the existence and nonexistence of nonnegative nontrivial weak solutions for a class of degenerate quasilinear elliptic problems with weights and nonlinearity involving the critical Hardy–Sobolev exponent and a sign-changing function. Some existence results are obtained by splitting the Nehari manifold and by exploring some properties of the best Hardy–Sobolev constant together with an approach developed by Brezis and Nirenberg. [Copyright &y& Elsevier]

Abstract: In , endowed with coordinates , we consider the PDE for which we prove a symmetry result. [Copyright &y& Elsevier]

Abstract: A Picone identity is established for a class of half-linear elliptic operators, and Sturmian comparison and oscillation results are derived on the basis of the Picone identity. Wirtinger inequalities and Riccati inequalities are also obtained as corollaries of the Picone identity, and a new class of half-linear elliptic equations is introduced. [Copyright &y& Elsevier]

Abstract: The goal of this paper is to establish the existence of a continuous family of eigenvalues lying in a neighborhood at the right of the origin for some nonlinear, nonhomogeneous, degenerate and singular elliptic operators on the whole space . Our proofs rely essentially on the Caffarelli–Kohn–Nirenberg inequality combined with the Banach fixed point theorem. [Copyright &y& Elsevier]

Abstract: We establish a comparison principle for quasilinear elliptic operators in unbounded cylindrical domains by combining the traditional test function method with nonlinear capacity techniques. Some applications to monotonicity and symmetry of solutions to nonlinear elliptic problems are given. [Copyright &y& Elsevier]

Abstract: The goal of this paper is to study the existence of non-trivial weak solutions for the nonuniformly nonlinear elliptic equation in an unbounded domain , where . The solutions will be obtained in a subspace of the Sobolev space and the proofs rely essentially on a variation of the Mountain pass theorem in [D.M. Duc, Nonlinear singular elliptic equations, J. London. Math. Soc. 40 (2) (1989) 420–440]. [Copyright &y& Elsevier]

Abstract: In this paper, by using fixed point theory, under quite general conditions on the nonlinear term, we obtain an existence result on unbounded positive solutions of certain quasilinear elliptic equations in two-dimensional exterior domains. [Copyright &y& Elsevier]

Abstract: This paper deals with a class of semilinear elliptic Dirichlet boundary value problems at resonance. We introduce a sufficient Landesman–Lazer condition for the existence of pair positive–negative solutions. Furthermore, developing the fibering method in the framework of the Leray–Schauder degree theory we can prove the existence of branches for positive and negative solutions. [Copyright &y& Elsevier]

Abstract: We prove a Liouville-type theorem for a class of degenerate elliptic operators of the form is supposed to be -elliptic, with respect to a family of locally Lipschitz continuous vector fields, in the sense introduced in [E. Lanconelli, A.E. Kogoj, -elliptic operators and -control distances, Contributions in Honor of the Memory of Ennio De Giorgi, Ricerche di Matematica 49 (Suppl.) (2000) 223–243]. [Copyright &y& Elsevier]

Abstract: We establish the complete bifurcation diagram for a class of nonlinear problems on the whole space. Our model corresponds to a class of semilinear elliptic equations with logistic type nonlinearity and absorption. Since this problem arises in population dynamics or in fishery or hunting management, we are interested only in situations allowing the existence of positive solutions. The proofs combine elliptic estimates with the method of sub- and super-solutions. [Copyright &y& Elsevier]

Abstract: We extend the refined maximum principle in [H. Berestycki, L. Nirenberg, S.R.S. Varadhan, The principal eigenvalue and the maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47 (1994) 47–92] to degenerate elliptic and parabolic equations with unbounded coefficients. Then we discuss the well-posedness of the corresponding Dirichlet boundary value problems. [Copyright &y& Elsevier]