Search

In this paper, we study degenerate or singular elliptic equations in divergence form - div (x n α A ∇ u) = div (x n α g) in B 1 ∩ { x n > 0 }. When α > - 1 , we establish boundary Schauder type estimates under the conormal boundary condition on the flat boundary, provided that the coefficients satisfy Dini mean oscillation (DMO) type conditions. Additionally, as an application, we derive higher-order boundary Harnack principles for uniformly elliptic equations in divergence form with DMO coefficients. [ABSTRACT FROM AUTHOR]

We investigate Bernoulli free boundary problems prescribing infinite jump conditions. The mathematical set-up leads to the analysis of non-differentiable minimization problems of the form ∫ ∇ u · (A (x) ∇ u) + φ (x) 1 { u > 0 } d x → min , where A(x) is an elliptic matrix with bounded, measurable coefficients and φ is not necessarily locally bounded. We prove universal Hölder continuity of minimizers for the one- and two-phase problems. Sharp regularity estimates along the free boundary are also obtained. Furthermore, we perform a thorough analysis of the geometry of the free boundary around a point ξ of infinite jump, ξ ∈ φ - 1 (∞) . We show that it is determined by the blow-up rate of φ near ξ and we obtain an analytical description of such cusp geometries. [ABSTRACT FROM AUTHOR]

Incompressible Euler flows in narrow domains, in which the horizontal length scale is much larger than other scales, play an important role in many different applications, and their leading-order behavior can be described by the hydrostatic Euler equations. In this paper, we show that steady solutions of the hydrostatic Euler equations in an infinite strip strictly away from stagnation must be shear flows. Furthermore, we prove the existence, uniqueness, and asymptotic behavior of global steady solutions to the hydrostatic Euler equations in general nozzles. In terms of stream function formulation, the hydrostatic Euler equations can be written as a degenerate elliptic equation, for which the Liouville type theorem in a strip is a consequence of the analysis for the second order ordinary differential equation (ODE). The analysis on the associated ODE also helps determine the far field behavior of solutions in general nozzles, which plays an important role in guaranteeing the equivalence of stream function formulation. One of the key ingredients for the analysis on flows in a general nozzle is a new transformation, which combines a change of variable and an Euler–Lagrange transformation. With the aid of this new transformation, the solutions in the new coordinates enjoy explicit representations so that the regularity with respect to the horizontal variable can be gained in a clear way. [ABSTRACT FROM AUTHOR]

We study Lipschitz critical points of the energy ∫ Ω g (det D u) d x in two dimensions, where g is a strictly convex function. We prove that the Jacobian of any Lipschitz critical point is constant, and that the Jacobians of sequences of approximately critical points converge strongly. The latter result answers, in particular, an open problem posed by Kirchheim, Müller and Šverák in 2003. [ABSTRACT FROM AUTHOR]

For the problem ∂ t k u - ∑ | α | = m ∂ α a α (x , t , u) ≥ f (| u |) in R + n + 1 = R n × (0 , ∞) , u (x , 0) = u 0 (x) , ∂ t u (x , 0) = u 1 (x) , … , ∂ t k - 1 u (x , 0) = u k - 1 (x) ≥ 0 , where u i ∈ L 1 , l o c (R n) and a α are Caratheodory functions such that | a α (x , t , ζ) | ≤ A | ζ | p , A , p = c o n s t > 0 , for almost all (x , t) ∈ R + n + 1 and for all ζ ∈ R , we obtain exact conditions on the function f guaranteeing that any global weak solution is identically zero. [ABSTRACT FROM AUTHOR]

In this note, we classify solutions to a class of Monge–Ampère equations whose right hand side may be degenerate or singular in the half space. Solutions to these equations are special solutions to a class of fourth order equations, including the affine maximal hypersurface equation, in the half space. Both the Dirichlet boundary value and Neumann boundary value cases are considered. [ABSTRACT FROM AUTHOR]

The present paper is concerned by the study of a nonlinear elliptic equation which contains a Hardy potential, lower order term, singular term and L m (·) data. Our approach is based on approximating the initial problem with a non-singular problem that is well-posed. We then establish the necessary estimates to pass to the limit. [ABSTRACT FROM AUTHOR]

Let (X, G) be a G-system, where G is an infinite countable discrete amenable group and X is a compact metric space with a metric d. In this paper, we study the topological pressure of intricacy and average sample complexity for amenable group actions. We show that the topological pressure of strong sub-additive potential is equal to the pressure of intricacy and average sample complexity as taking supremum over open covers of X. We establish the definition of the pressure of intricacy and average sample complexity by using spanning sets and separated sets. In the last part, we show that the variational principle for pressure of intricacy and average sample complexity. [ABSTRACT FROM AUTHOR]

We consider the numerical approximation of variational problems with orthotropic growth, that is those where the integrand depends strongly on the coordinate directions with possibly different growth in each direction. Under realistic regularity assumptions we derive optimal error estimates. These estimates depend on the existence of an orthotropically stable interpolation operator. Over certain meshes we construct an orthotropically stable interpolant that is also a projection. Numerical experiments illustrate and explore the limits of our theory. [ABSTRACT FROM AUTHOR]

The aim of the paper is to study a quasilinear Dirichlet problem driven by a p-Laplacian type operator with unbounded coefficient and exhibiting a convection term composed with an intrinsic operator. The existence of a bounded weak solution is established. Applications address equations with convolution, truncation, and inverse of p-Laplacian. [ABSTRACT FROM AUTHOR]

Using Mountain Pass Theorem, we consider the existence of weak solutions of weighted Robin problem involving p. -biharmonic operator a x Δ p x 2 u = λ b (x) u q (x) - 2 u , i n Ω a (x) Δ u p (x) - 2 ∂ u ∂ υ + β (x) u p (x) - 2 u = 0 , o n ∂ Ω under some conditions in the space W a , b 2 , p (.) Ω. [ABSTRACT FROM AUTHOR]

We prove sharp upper and lower estimates for the parabolic kernel of the singular elliptic operator L = Tr A D 2 + v , ∇ y , in the half-space R + N + 1 = { (x , y) : x ∈ R N , y > 0 } under Neumann or oblique derivative boundary conditions at y = 0 . [ABSTRACT FROM AUTHOR]

The Dirichlet problem of elliptic equation from two phase problems, b (u) − div (| ∇ u) | p (x) − 2 ∇ u + μ (x) | ∇ u | q (x) − 2 ∇ u) = f (x) in Ω, u = 0 on ∂Ω is considered, where Ω is a bounded domain with a smooth boundary in R N , b is a continuous and nondecreasing function. By the theory of the weighted variable Sobolev space, the existence and uniqueness of an entropy solution for L 1 -data f are proved. [ABSTRACT FROM AUTHOR]

Suppose Ω 0 , Ω 1 are two bounded strongly C -convex domains in C n , with n ≥ 2 and Ω 1 ⊃ Ω 0 ¯ . Let R = Ω 1 \ Ω 0 ¯ . We call R a C -convex ring. We will show that for a solution Φ to the homogenous complex Monge–Ampère equation in R , with Φ = 1 on ∂ Ω 1 and Φ = 0 on ∂ Ω 0 , - 1 ∂ ∂ ¯ Φ has rank n - 1 and the level sets of Φ are strongly C -convex. [ABSTRACT FROM AUTHOR]

Within this article, the maximal function method is used to establish the Calderón-Zygmund estimates for the weak solutions of a class of non-uniformly elliptic equations - div A (x , D u) = - div F (x , f) i n R n , where A (x , D u) ≈ | D u | p 1 - 2 + μ (x) | D u | p 2 - 2 , F (x , f) ≈ | f | p 1 - 2 + μ (x) | f | p 2 - 2 and 1 < p 1 < p 2 , 0 ≤ μ (·) ∈ C 0 , α (R n) , α ∈ (0 , 1 ] . The aforementioned problems arise as Euler-Lagrange equations for variational functionals that were originally presented and studied within the context of Homogenization and the Lavrentiev phenomenon by Marcellini (Arch Ration Mech Anal 105:267–284, 1989. ) and Zhikov (Izv Akad Nauk SSSR Ser Mat 29:33–66, 1987. ). They are distinctive in that they exhibit that the growth and ellipticity change between two distinct types of polynomial depending on the position. This feature is characteristic of strongly anisotropic materials. The contribution of this paper is closely tied to the significant advancements made by Colombo and Mingione (J Funct Anal 270:1416–1478, 2016. ) in the qualitative analysis of double phase problems, as well as the related techniques used by Zhang et al. (Ann Polon Math, 114:45–65, 2015. ). [ABSTRACT FROM AUTHOR]

We study a conormal boundary value problem for a class of quasilinear elliptic equations in bounded domain Ω whose coefficients can be degenerate or singular of the type dist (x , ∂ Ω) α , where ∂ Ω is the boundary of Ω and α ∈ (- 1 , ∞) is a given number. We establish weighted Sobolev type estimates for weak solutions under a smallness assumption on the weighted mean oscillations of the coefficients in small balls. Our approach relies on a perturbative method and several new Lipschitz estimates for weak solutions to a class of singular-degenerate quasilinear equations. [ABSTRACT FROM AUTHOR]

Let X = { X 0 , … , X m } be a family of smooth vector fields on an open set Ω ⊆ R N . Motivated by applications to the PDE theory of Hörmander operators, for a suitable class of open sets Ω , we find necessary and sufficient conditions on X for the existence of a Lie group (Ω , ∗) such that the operator L = ∑ i = 1 m X i 2 + X 0 is left-invariant with respect to the operation ∗ . Our approach is constructive, as the group law is constructed by means of the solution of a suitable ODE naturally associated to vector fields in X. We provide an application to a partial differential operator appearing in the Finance. [ABSTRACT FROM AUTHOR]

In this paper we develop a comprehensive study on principal eigenvalues and both the (weak and strong) maximum and comparison principles related to an important class of nonlinear systems involving fractional m-Laplacian operators. Explicit lower bounds for principal eigenvalues of this system in terms of the diameter of bounded domain Ω ⊂ R N are also proved. As application, we measure explicitly how small has to be diam (Ω) so that weak and strong maximum principles associated to this problem hold in Ω . [ABSTRACT FROM AUTHOR]

We study the numerical solutions for a class of quasilinear Schrödinger equations arising from the self-channeling of high-power ultra short lasers in matter, which are associated with energy functionals with nonlinear principle parts so that the classical algorithm cannot directly be used. By the method of variable replacement to transform the quasilinear Schrödinger equation into an semilinear elliptic equation, the numerical mountain pass algorithm is then applied. Some numerical experiments are also performed, including zero potential, nonzero constant potential and singular problem. [ABSTRACT FROM AUTHOR]

In this paper, we establish a positive lower bound estimate for the second smallest eigenvalue of the complex Hessian of solutions to a degenerate complex Monge–Ampère equation. As a consequence, we find that in the space of Kähler potentials any two points close to each other in C 2 norm can be connected by a geodesic along which the associated metrics do not degenerate. [ABSTRACT FROM AUTHOR]

For each k = 0 , ⋯ , n we construct a continuous phase f k , with f k (0) = (n - 2 k) π 2 , and viscosity sub- and supersolutions v k , u k , of the elliptic PDE ∑ i = 1 n arctan (λ i (H w)) = f k (x) such that v k - u k has an isolated maximum at the origin. It has been an open question whether the comparison principle would hold in this second order equation for arbitrary continuous phases f : R n ⊇ Ω → (- n π / 2 , n π / 2) . Our examples show it does not. [ABSTRACT FROM AUTHOR]

In this paper, we study the unique continuation property for the fractional p-elliptic equations in a semigroup form with variable coefficients. By employing an extension procedure, we derive a monotonicity formula for an extended frequency function. Utilizing this monotonicity together with a blow-up analysis, we establish the unique continuation property. [ABSTRACT FROM AUTHOR]

We consider viscosity solutions to two-phase free boundary problems for the p(x)-Laplacian with non-zero right hand side. We prove that flat free boundaries are C 1 , γ . No assumption on the Lipschitz continuity of solutions is made. These regularity results are the first ones in literature for two-phase free boundary problems for the p(x)-Laplacian and also for two-phase problems for singular/degenerate operators with non-zero right hand side. They are new even when p (x) ≡ p , i.e., for the p-Laplacian. The fact that our results hold for merely viscosity solutions allows a wide applicability. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of infinitely many nontrivial solutions to the following semilinear degenerate elliptic equation - Δ λ u + V (x) u = f (x , u) in R N , N ≥ 3 , where V : R N → R is a potential function and allowed to be vanishing at infinitely, f : R N × R → R is a given function and Δ λ is the strongly degenerate elliptic operator. Under suitable assumptions on the potential V and the nonlinearity f, some results on the multiplicity of solutions are proved. The proofs are based on variational methods, in particular, on the well-known mountain pass lemma of Ambrosetti–Rabinowitz. Due to the vanishing potentials and degeneracy of the operator, some new compact embedding theorems are used in the proof. Our results extend and generalize some existing results (Alves and Souto in J Differ Equ 254:1977–1991, 2013; Hamdani in Asia-Eur J Math 13:2050131, , 2020; Luyen in Commun Math Anal 22:61–75, 2019; Luyen and Tri in J Math Anal Appl 461:1271–1286, 2018; Tang in J Math Anal Appl 401:407–415, 2013; Toon and Ubilla in Discrete Contin Dyn Syst 40:5831–5843, 2020). [ABSTRACT FROM AUTHOR]

In this paper, we investigate sharp singular Trudinger–Moser inequalities involving the anisotropic Dirichlet norm ∫ R N F N (∇ u) d x 1 N in Sobolev-type space D N , q (R N) , N ≥ 2 , q ≥ 1 . Here F : R N → [ 0 , + ∞) is a convex function of class C 2 (R N \ { 0 }) , which is even and positively homogeneous of degree 1. Combing with the connection between convex symmetrization and Schwarz symmetrization, we will establish anisotropic singular Trudinger–Moser inequalities and discuss their sharpness under different situations, including the case ‖ F (∇ u) ‖ N ≤ 1 , the case ‖ F (∇ u) ‖ N a + ‖ u ‖ q b ≤ 1 , and whether they are associated with exact growth. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Bianchi-Egnell inequality on the Heisenberg group, which quantifies the stability of the Sobolev inequality. We generalize results of K o ¨ nig in the Euclidean case. [ABSTRACT FROM AUTHOR]

In this article, we consider a class of degenerate singular problems. The degeneracy is captured by the presence of a class of p-admissible weights, which may vanish or blow up near the origin. Further, the singularity is allowed to vary inside the domain. We provide sufficient conditions on the weight function, on the singular exponent and the source function to establish regularity and existence results. [ABSTRACT FROM AUTHOR]

As a first result we prove higher order Schauder estimates for solutions to singular/degenerate elliptic equations of type - div ρ a A ∇ w = ρ a f + div ρ a F in Ω for exponents a > - 1 , where the weight ρ vanishes with non zero gradient on a regular hypersurface Γ , which can be either a part of the boundary of Ω or mostly contained in its interior. As an application, we extend such estimates to the ratio v/u of two solutions to a second order elliptic equation in divergence form when the zero set of v includes the zero set of u which is not singular in the domain (in this case ρ = u , a = 2 and w = v / u ). We prove first the C k , α -regularity of the ratio from one side of the regular part of the nodal set of u in the spirit of the higher order boundary Harnack principle in Savin (Discrete Contin Dyn Syst 35–12:6155–6163, 2015). Then, by a gluing Lemma, the estimates extend across the regular part of the nodal set. Finally, using conformal mapping in dimension n = 2 , we provide local gradient estimates for the ratio, which hold also across the singular set. [ABSTRACT FROM AUTHOR]

We obtain weighted Poincaré inequalities in bounded domains, where the weight is given by a symmetric nonnegative definite matrix, which can degenerate on submanifolds. Furthermore, we investigate uniqueness and nonuniqueness of solutions to degenerate elliptic and parabolic problems, where the diffusion matrix can degenerate on subsets of the boundary of the domain. Both the results are obtained by means of the distance function from the degeneracy set, which is used to construct suitable local sub– and supersolutions. [ABSTRACT FROM AUTHOR]

This paper investigates the existence, nonexistence, and qualitative properties of p-harmonic functions in the upper half-space R + N (N ≥ 3) satisfying nonlinear boundary conditions for 1 < p < N . Moreover, the symmetry of positive solutions is shown by using the method of moving planes. [ABSTRACT FROM AUTHOR]

In this paper we will establish a new Pólya–Szegö type inequality for a weighted gradient of a function on R 2 with respect to a weighted area. In order to do that we need to study an isoperimetric problem for the weighted area. We then apply the inequality to prove embedding theorems for weighted Sobolev spaces and to calculate the best constant in the Sobolev imbedding theorems. In our upcoming manuscript the obtained results in this note will be used to study boundary value problems for semilinear degenerate elliptic equations, see Luyen et al. (). [ABSTRACT FROM AUTHOR]

This paper is devoted to study a class of p(x)-Kirchhoff equation with concave–convex nonlinearities. By means of perturbation technique and the variational method, the multiplicity of nonnegative nontrivial solutions to this problem is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study the following fractional Schrödinger equation with electromagnetic fields and critical or supercritical growth (- Δ) A s u + V (x) u = λ | u | p - 2 u + f (x , | u | 2) u , x ∈ R N , where (- Δ) A s is the fractional magnetic operator with 0 < s < 1 , N > 2 s , 2 s ∗ = 2 N N - 2 s , λ > 0 , V ∈ C (R N , R) and A ∈ C (R N , R N) are the electric and magnetic potentials, respectively. When V and f are asymptotically periodic in x, and f is a continuous function and there exists 2 < q < 2 s ∗ such that | f (x , t) | ≤ C (1 + | t | q - 2 2) for all (x, t), for 2 s ∗ ≤ p < 22 s ∗ - q . For any D > 0 fixed, if λ ∈ (0 , D ] we prove that the equation has a nontrivial solution by the truncation method. Our method can provide a prior L ∞ -estimate. [ABSTRACT FROM AUTHOR]

This study investigates the double-phase system with infinite potentials that vanish and convex–concave nonlinearity in Sobolev spaces with variable exponents on complete manifolds. Using the Nehari manifold and variational techniques, we aim to demonstrate the existence of at least two nonnegative nontrivial solutions. Our findings shed light on the behavior of this complex system and contribute to the mathematical analysis community's comprehension of nonlinear dynamics. [ABSTRACT FROM AUTHOR]

Here, considering - ∞ < a < N - p p , a ≤ e ≤ a + 1 , d = 1 + a - e and p ∗ : = p ∗ (a , e) = Np N - d p , the existence of positive solution of a weighted p-Laplace equation involving vanishing potentials - Δ ap u + V (x) | x | - e p ∗ | u | p - 2 u = | x | - e p ∗ f (u) in R N is proved, where the potential V can vanish at infinity with exponential decay and f is a function with subcritical growth of class C 1 . We use Del Pino & Felmer’s arguments to overcome the lack of compactness and the Moser iteration method with Caffarelli–Kohn–Nirenberg inequality to obtain estimates of the solution in L ∞ (R N). [ABSTRACT FROM AUTHOR]

We study stochastic homogenization for convex integral functionals u ↦ ∫ D W (ω , x ε , ∇ u) d x , where u : D ⊂ R d → R m , defined on Sobolev spaces. Assuming only stochastic integrability of the map ω ↦ W (ω , 0 , ξ) , we prove homogenization results under two different sets of assumptions, namely ∙ 1 W satisfies superlinear growth quantified by the stochastic integrability of the Fenchel conjugate W ∗ (· , 0 , ξ) and a certain monotonicity condition that ensures that the functional does not increase too much by componentwise truncation of u, ∙ 2 W is p-coercive in the sense | ξ | p ≤ W (ω , x , ξ) for some p > d - 1 . Condition ∙ 2 directly improves upon earlier results, where p-coercivity with p > d is assumed and ∙ 1 provides an alternative condition under very weak coercivity assumptions and additional structure conditions on the integrand. We also study the corresponding Euler–Lagrange equations in the setting of Sobolev-Orlicz spaces. In particular, if W (ω , x , ξ) is comparable to W (ω , x , - ξ) in a suitable sense, we show that the homogenized integrand is differentiable. [ABSTRACT FROM AUTHOR]

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]

In this paper, we study the existence and regularity results for some parabolic equations with degenerate coercivity, and a singular right-hand side. The model problem is 0.1 { ∂ u ∂ t − div ( (1 + | ∇ u | − Λ) | ∇ u | p − 2 ∇ u (1 + | u |) θ) = f (e u − 1) γ in Q T , u (x , 0) = 0 on Ω , u = 0 on ∂ Q T , where Ω is a bounded open subset of R N N ≥ 2 , T > 0 , Λ ∈ [ 0 , p − 1) , f is a non-negative function belonging to L m (Q T) , Q T = Ω × (0 , T) , ∂ Q T = ∂ Ω × (0 , T) , 0 ≤ θ < p − 1 + p N + γ (1 + p N) and 0 ≤ γ < p − 1 . [ABSTRACT FROM AUTHOR]

Let F : ℝ N → [ 0 , + ∞) be a convex function of class C 2 (ℝ N ∖ { 0 }) , which is even and positively homogeneous of degree 1. Let Ω ⊂ ℝ N (N ≥ 2) be a smooth bounded domain, we denote γ 1 = inf u ∈ W 0 1 , N (Ω) ∖ { 0 } ∫ Ω F N (∇ u) d x ∥ u ∥ p N and define ∥ u ∥ N , F , γ , p = ∫ Ω F N (∇ u) d x − γ ∥ u ∥ p N 1 N . Then for p > 1 and 0 ≤ γ < γ1, we have sup u ∈ W 0 1 , N (Ω) , ∥ u ∥ N , F , γ , p ≤ 1 ∫ Ω e λ N | u | N N − 1 d x < + ∞ , where λ N = N N N − 1 κ N 1 N − 1 and κN is the volume of a unit Wulff ball in ℝ N . Moreover, by using blow-up analysis and capacity technique, we prove that the supremum can be attained for any 0 ≤ γ < γ1. [ABSTRACT FROM AUTHOR]

By exploiting an old idea first used by Pizzetti for the classical Laplacian, we introduce a notion of asymptotic average solutions making pointwise solvable every Poisson equation ℒ u (x) = − f (x) with continuous data f, where ℒ is a hypoelliptic linear partial differential operator with positive semidefinite characteristic form. [ABSTRACT FROM AUTHOR]

Motivated by the seminal paper due to Brézis–Nirenberg (in: Commun Pure Appl Math 36:437–477, 1983), we will establish the existence of solutions for the following class of degenerate elliptic equations with critical nonlinearity: - Δ γ v = λ | v | q - 2 v + v 2 γ ∗ - 2 v in Ω , v = 0 on ∂ Ω , where Δ γ : = Δ x + (1 + γ) 2 x 2 γ Δ y is the Grushin operator, z : = (x , y) ∈ R N , N = m + n , m , n ≥ 1 , Ω ⊂ R N is a smooth bounded domain, λ > 0 , q ∈ [ 2 , 2 γ ∗) and 2 γ ∗ = 2 N γ N γ - 2 is the critical Sobolev exponent in this context, where N γ = m + (1 + γ) n is the so-called homogeneous dimension attached to the Grushin operator Δ γ . In order to prove our main results it was necessary to do a careful study involving the best constant S γ (m , n) of the Sobolev embedding for the spaces associated with Δ γ . In order to do that, we prove a version of the Lions’ Concentration-Compactness Principle for the Grushin operator. We also provide existence results for a critical problem involving the Grushin operator on the whole space R N . [ABSTRACT FROM AUTHOR]

This paper investigates a P-function associated with solutions to boundary value problems of some generalized Monge-Ampère equations in bounded convex domains. It will be shown that P attains its maximum value either on the boundary or at a critical point of any convex solution. Furthermore, it turns out that such P-function is actually a constant when the underlying domain is a ball. Therefore, our results provide a best possible maximum principle in the sense of L. Payne. As an application, we will use these results to study an overdetermined boundary value problem. More specifically, we will show solvability of this overdetermined boundary value problem forces their P-function to be a constant. [ABSTRACT FROM AUTHOR]

In this paper we prove some integral estimates on the minimal growth of the positive part u + of subsolutions of quasilinear equations div A (x , u , ∇ u) = V | u | p - 2 u on complete Riemannian manifolds M, in the non-trivial case u + ≢ 0 . Here A satisfies the structural assumption | A (x , u , ∇ u) | p / (p - 1) ≤ k ⟨ A (x , u , ∇ u) , ∇ u ⟩ for some constant k > 0 and for p > 1 the same exponent appearing on the RHS of the equation, and V is a continuous positive function, possibly decaying at a controlled rate at infinity. We underline that the equation may be degenerate and that our arguments do not require any geometric assumption on M beyond completeness of the metric. From these results we also deduce a Liouville-type theorem for sufficiently slowly growing solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of renormalized solution for nonlinear multivalued elliptic problems β (u) - d i v (a (x , ∇ u) + F (u)) ∋ f in the framework of Orlicz spaces and L 1 -data. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the existence of ground state solutions for the p (x) -Laplacian equation { − Δ p (x) u + λ | u | p (x) − 2 u = f (x , u) + h (x) in Ω , u = 0 , on ∂ Ω. Using the constraint variational method, quantitative deformation lemma, and strong maximum principle, we proved that the above problem admits three ground state solutions, especially speaking, one solution is sign-changing, one is positive, and one is negative. Our results improve on those existing in the literature. [ABSTRACT FROM AUTHOR]

We give a simple proof of the strong maximum principle for viscosity subsolutions of fully nonlinear nonhomogeneous degenerate elliptic equations on the form F (x , u , D u , D 2 u) = 0 under suitable assumptions allowing for non-Lipschitz growth in the gradient term. In case of smooth boundaries, we also prove a Hopf lemma, a boundary Harnack inequality, and that positive viscosity solutions vanishing on a portion of the boundary are comparable with the distance function near the boundary. Our results apply, e.g., to weak solutions of an eigenvalue problem for the variable exponent p-Laplacian. [ABSTRACT FROM AUTHOR]

We consider a class of fully nonlinear nonlocal degenerate elliptic operators which are modeled on the fractional Laplacian and converge to the truncated Laplacians. We investigate the validity of (strong) maximum and minimum principles, and their relation with suitably defined principal eigenvalues. We also show a Hopf type Lemma, the existence of solutions for the corresponding Dirichlet problem, and representation formulas in some particular cases. [ABSTRACT FROM AUTHOR]

We provide the Alexandroff–Bakelman–Pucci estimate and global C 1 , α -regularity for a class of singular/degenerate fully nonlinear elliptic equations. We also derive the existence of a viscosity solution to the Dirichlet problem with the associated operator. [ABSTRACT FROM AUTHOR]

We prove sharp uniform estimates for strong supersolutions of a large class of fully nonlinear degenerate elliptic complex equations. Our findings rely on ideas of Kuo and Trudinger who dealt with degenerate linear equations in the real setting. We also exploit the pluripotential theory for the complex Monge-Ampère operator as well as suitably tailored theory of Lp-viscosity subsolutions. [ABSTRACT FROM AUTHOR]

A dual variational principle is defined for the nonlinear system of PDE describing the dynamics of dislocations in elastic solids. The dual variational principle accounting for a specified set of initial and boundary conditions for a general class of PDE is also developed. [ABSTRACT FROM AUTHOR]