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In this paper we establish some results concerning existence and multiplicity of solution for a elliptic problem under integral Dirichlet boundary condition. We mainly use a lower and upper solution method and a result due to Rabinowitz [11] related to the existence of a continuum of solutions of an abstract nonlinear eigenvalue problem.

The purpose of this paper is to study an inhomogeneous Dirichlet problem of a superquadratic nonlinear elliptic system with VMO coefficients in Reifenberg flat domains. We derive pointwise potential estimates for weak solution to the Dirichlet problem. As a consequence, we derive global integrability estimates for solutions with respect to Lorentz and Morrey norms.

In this paper we study the Dirichlet problem of a class of nonlinear elliptic equations which includes Monge–Ampere equation, K-Hessian equation and the usual linear elliptic equation. This problem becomes singular near the boundary of the domain. By carefully constructing sub-solutions we obtain the boundary Holder estimate for the convex solution to the problem when the domain satisfies exterior sphere condition.

In this paper, we are interested in studying the Dirichlet problem for the complex Monge-Ampere operator. We provide necessary and sufficient conditions for the problem to have Holder continuous solutions.

In this paper, we consider a variational formulation for the Dirichlet problem of the wave equation with zero boundary and initial conditions, where we use integration by parts in space and time. To prove unique solvability in a subspace of H 1 ( Q ) with Q being the space–time domain, the classical assumption is to consider the right–hand side f in L 2 ( Q ) . Here, we analyze a generalized setting of this variational formulation, which allows us to prove unique solvability also for f being in the dual space of the test space, i.e., the solution operator is an isomorphism between the ansatz space and the dual of the test space. This new approach is based on a suitable extension of the ansatz space to include the information of the differential operator of the wave equation at the initial time t = 0 . These results are of utmost importance for the formulation and numerical analysis of unconditionally stable space–time finite element methods, and for the numerical analysis of boundary element methods to overcome the well–known norm gap in the analysis of boundary integral operators.

A formula for the magnetostatic energy of a finite magnet is proven. In contrast to common approaches, the new energy identity does not rely on evaluation of a nonlocal boundary integral inside the magnet or the solution of an equivalent Dirichlet problem. The formula is therefore computationally efficient, which is also shown numerically. Algorithms for the simulation of magnetic materials could benefit from incorporating the presented representation of the energy. In addition, a natural analogue for the energy via the magnetic induction is given. Proofs are carried out within a setting which is suitable for common discretizations in computational micromagnetics.

We study the homogeneous Dirichlet problem for the equation u t = div ( | ∇ u | p ( x , t ) − 2 ∇ u ) + f ( x , t , u ) in the cylinder Q T = Ω × ( 0 , T ) , Ω ⊂ R d , d ≥ 2 . It is assumed that p ( x , t ) ∈ ( 2 d d + 2 , 2 ) and | ∇ p | , | p t | are bounded a.e. in Q T . We find conditions on p ( x , t ) , f ( x , t , u ) and u ( x , 0 ) sufficient for the existence of strong solutions, local or global in time. It is proven that the strong solutions possess the property of global higher regularity: u t ∈ L 2 ( Q T ) , | ∇ u | ∈ L ∞ ( 0 , T ; L 2 ( Ω ) ) , | D i j 2 u | p ( x , t ) ∈ L 1 ( Q T ) .

A new proof is given of a simple probabilistic lemma which implies the solution of the Dirichlet problem for simply connected domains in the plane. This proof uses the conformal invariance of planar Brownian motion in place of the previously existing measure-theoretic arguments.

Let W be a three-dimensional wedge, and K be the double layer potential operator associated to W and the Laplacian. We show that 1 2 ± K are isomorphisms between suitable weighted Sobolev spaces, which implies a solvability result in weighted Sobolev spaces for the Dirichlet problem on W . Furthermore, we show that the double layer potential operator K is an element in C ⁎ ( G ) ⊗ M 2 ( C ) , where G is the action (transformation) groupoid M ⋊ G , with G = { ( 1 0 a b ) : a ∈ R , b ∈ R + } , which is a Lie group, and M is a kind of compactification of G. This result can be used to prove the Fredholmness of 1 2 + K Ω , where Ω is “a domain with edge singularities” and K Ω the double layer potential operator associated to the Laplacian and Ω.

In this paper, we are concerned with the sublinear problem (0.1) { − Δ u = | u | p − 2 u in Ω , u ν = 0 on ∂ Ω , where Ω ⊂ R N is a bounded domain, and 1 ≤ p 2 . For p = 1 , the nonlinearity | u | p − 2 u will be identified by sgn ( u ) . In contrast to previous work on the Dirichlet problem, some difficulties arise due to the fact that the associated energy functional is not bounded from below. Complementing recent work by Parini and Weth in [15] on least energy solutions, we prove that (0.1) has infinitely many solutions with small negative energy.

We study a nonlinear singular boundary value problem and prove that, depending on a relationship between exponents of power terms, the problem has either solutions of Dirichlet type or homoclinic solutions. We make use of shooting techniques and lower and upper solutions.

In this paper we investigate the global behavior of positive rotationally symmetric solutions for Dirichlet problem involving the k-th mean curvature operator in Minkowski space L n + 1 , which is not elliptic for k > 1 . The proofs of our main results are based upon bifurcation techniques.

We study Dirichlet problem for a nonlinear equation with a drift term. Despite the presence of the singular convection term, we establish existence and uniqueness of a solution in spaces larger than the natural one.

Denote by C α ( D ) the space of the functions f on the unit disk D which are Holder continuous with the exponent α, and denote by C 1 , α ( D ) the space which consists of differentiable functions f such that their derivatives are in the space C α ( D ) . Let C be the Cauchy transform of Dirichlet problem. In this paper, we obtain the norm estimates of ‖ C ‖ L p → L q , where 3 / 2 p 2 and q = p / ( p − 1 ) . Suppose g ∈ L p ( D ) and f = G [ g ] is the Green potential of g. By using Sobolev embedding theorem, we show that if 1 p ≤ 2 , then f ∈ C μ ( D ) , where μ = 2 − 2 / p . We also show that if 2 p ∞ , then f ∈ C 1 , ν ( D ) , where ν = 1 − 2 / p . Finally, for the case p = ∞ , we show that f is not necessarily in C 1 , 1 ( D ) , but its gradient, i.e., | ∇ f | is Lipschitz continuous with respect to the pseudo-hyperbolic metric. This paper is inspired by [2, Chapter 4] and [9] .

We study the homogeneous Dirichlet problem for the class of singular parabolic equations u t − div ( | ∇ u | p [ ∇ u ] − 2 ∇ u ) = f in Ω × ( 0 , T ) , where Ω ⊂ R d , d ≥ 2 , is a smooth domain. The exponent p nonlocally depends on the gradient of the solution: p is a given function defined by p [ ∇ u ] ≡ p ( l ( | ∇ u | ) ) , l ( | s | ) = ∫ Ω | s | α d x with a constant α ∈ ( 1 , 2 ] . We find sufficient conditions on the data that guarantee global in time existence and uniqueness of a strong solution of the problem. It is shown that the problem has a solution if either u 0 and f, or p ′ ( s ) are sufficiently small.

We study the behavior near the boundary conical point of weak solutions to the Dirichlet problem for elliptic quasi-linear second-order equation with the p -Laplacian and the strong nonlinearity on the right side.

We study exact multiplicity and bifurcation curves of positive solutions for a multiparameter Dirichlet problem { u ″ ( x ) + λ [ a − b u + u p 1 + u p ] = 0 , − 1 x 1 , u ( − 1 ) = u ( 1 ) = 0 , where p > 1 , a , b are positive dimensionless parameters, and λ > 0 is a bifurcation parameter. For fixed p > 1 , assume that either 0 a ≤ a 1 , p , b > 0 and ( a , b ) lies below the curve Γ 1 = { ( a , b ) : a ( t ) = t p [ p − 1 − t p ] ( t p + 1 ) 2 , b ( t ) = p t p − 1 ( t p + 1 ) 2 , 0 t p − 1 p + 1 p } , or 0 a ≤ a 2 , p and ( a , b ) lies on or above the curve Γ 1 for some positive constants a 1 , p and a 2 , p . Then on the ( λ , ‖ u ‖ ∞ ) -plane, we give a classification of three qualitatively different bifurcation curves: an S-shaped curve, a broken S-shaped curve, and a monotone increasing curve. Hence we are able to determine the exact multiplicity of positive solutions by the values of a, b and λ.

A complex harmonic function of finite Dirichlet energy on a Jordan domain has boundary values in a certain conformally invariant sense, by a construction of H. Osborn. We call the set of such boundary values the Douglas–Osborn space. One may then attempt to solve the Dirichlet problem on the complement for these boundary values. This defines a reflection of harmonic functions. We show that quasicircles are precisely those Jordan curves for which this reflection is defined and bounded. We then use a limiting Cauchy integral along level curves of Green's function to show that the Plemelj–Sokhotski jump formula holds on quasicircles with boundary data in the Douglas–Osborn space. This enables us to prove the well-posedness of a Riemann–Hilbert problem with boundary data in the Douglas–Osborn space on quasicircles.

In the paper, we consider the Hessian equation σ k ( λ ( D 2 u ) ) = f ( x ) where f is a positive function outside a bounded domain of R n , n ≥ 3 and f ( x ) = 1 + O ( | x | − β ) for some β > 2 at infinity. Using the Perron's method we prove the existence and uniqueness for viscosity solutions of exterior Dirichlet problem with prescribed asymptotic behavior at infinity. There are examples to show that the result is optimal. This is an extension of the theorems given by Bao–Li–Li in [2] for f ≡ 1 and Bao–Li–Zhang in [3] for k = n .

In this paper, we study the Dirichlet problem for degenerate parabolic equations of the form ∂ u ∂ t − div ( a ( x , t , u , ∇ u ) ) − div ϕ ( u ) = f − div g , where the main operator a ( x , t , u , ∇ u ) is allowed to degenerate with respect to the unknown u. Under suitable conditions on a and ϕ, we prove that there exists an unique solution u of this problem.

In this paper we investigate the existence of rotationally symmetric entire graphs (resp. entire spacelike graphs) with prescribed k-th mean curvature function in Euclidean space R n + 1 (resp. Minkowski spacetime L n + 1 ). As a previous step, we analyze the associated homogeneous Dirichlet problem on a ball, which is not elliptic for k > 1 , and then we prove that it is possible to extend the solutions. Moreover, a sufficient condition for uniqueness is given in both cases.

Spacelike surfaces in the Lorentz–Minkowski space L 3 can be endowed with two different Riemannian metrics, the metric inherited from L 3 and the one induced by the Euclidean metric of R 3 . It is well known that the only surfaces with zero mean curvature with respect to both metrics are open pieces of the helicoid and of spacelike planes. We consider the general case of spacelike surfaces with the same mean curvature with respect to both metrics. One of our main results states that those surfaces have non-positive Gaussian curvature in R 3 . As an application of this result, jointly with a general argument on the existence of elliptic points, we present several geometric consequences for the surfaces we are considering. Finally, as any spacelike surface in L 3 is locally a graph, our surfaces are locally determined by the solutions to the H R = H L surface equation. Some uniqueness results for the Dirichlet problem associated to this equation are given.

The existence of a solution to a Dirichlet problem, for a class of nonlinear elliptic equations, with a convection term, is established. The main novelties of the paper stand on general growth conditions on the gradient variable, and on minimal assumptions on Ω. The approach is based on the method of sub and supersolutions. The solution is a zero of an auxiliary pseudomonotone operator build via truncation techniques. We present also some examples in which we highlight the generality of our growth conditions.

In this paper, we prove the existence and investigate the Holder continuity of solutions to the Dirichlet problem of the complex Monge-Ampere type equation on a bounded strictly pseudoconvex domain Ω in C n .

In this paper, we consider the Dirichlet problem for hypersurfaces M = graph u of anisotropic prescribed mean curvature H = H ( x , u , N ) on unbounded domain Ω, where N is the unit normal to M at ( x , u ) . As a corollary of the result, we obtain the existence of translating solutions to the mean curvature flow with a forcing term on unbounded domains. The approach used here is a modified version of classical Perron's method, where the solutions to minimal surface equation are used as supersolutions and a family of auxiliary functions is constructed as local subsolutions.

The following Dirichlet problem { Δ u − u + | u | p − 1 u = 0 in Ω , u = 0 on ∂ Ω , is considered, where Ω is either an annulus or a ball in R N and p > 1 . The uniqueness of radial solutions having exactly k − 1 nodes is shown for the following cases: Ω is a sufficiently thin annulus; Ω is a certain small ball, N ≥ 4 and 1 p N / ( N − 2 ) ; Ω is the unit ball, N = 3 and 1 p ≤ 3 ; Ω is any annulus or any ball, but p > 1 is sufficiently close to 1 and N = 3 , 5 or 7.

In this paper the Dirichlet problem for a class of standard weighted Laplace operators in the upper half plane is solved by means of a counterpart of the classical Poisson integral formula. Boundary limits and representations of the associated solutions are studied within a framework of weighted spaces of distributions. Special attention is given to the development of a suitable uniqueness theory for the Dirichlet problem under appropriate growth constraints at infinity.

We investigate the existence and uniqueness of variational solutions for the Dirichlet problem associated with variational nonlinear elliptic equations in Musielak spaces. We assume neither Δ 2 nor ∇ 2 -condition on the Φ-functions but suppose that they satisfy some structural assumptions that allow to define the variational framework and to obtain Poincare-type inequalities. Within the approach we use, the Galerkin method is coupled with an abstract existence result by F.E. Browder, 1966.

We present a new approach to studying a parametric nonlinear Dirichlet problem driven by a nonhomogeneous differential operator and with a reaction which is concave (i.e., ( p − 1 )-sublinear) near zero and convex (i.e., ( p − 1 )-superlinear) near ±∞. The reaction term is not assumed to be odd. We show that for all small values of the parameter λ > 0 , the problem has infinitely many nodal solutions.

In this article, we are study the following Dirichlet problem with Choquard type non linearity − Δ H u = a u + ( ∫ Ω | u ( η ) | Q λ ⁎ | η − 1 ξ | λ d η ) | u | Q λ ⁎ − 2 u in Ω , u = 0 on ∂ Ω , where Ω is a smooth bounded subset of the Heisenberg group H N , N ∈ N with C 2 boundary and Δ H is the Kohn Laplacian on the Heisenberg group H N . Here, Q λ ⁎ = 2 Q − λ Q − 2 , Q = 2 N + 2 and a is a positive real parameter. We derive the Brezis-Nirenberg type result for the above problem. Moreover, we also prove the regularity of solutions and nonexistence of solutions depending on the range of a.

We study the Dirichlet problem for the degenerate nonlocal parabolic equation u t − div ( a ( ‖ ∇ u ‖ L 2 ( Ω ) 2 ) ∇ u ) = b ( ‖ u ‖ L 2 ( Ω ) 2 ) | u | q ( x , t ) − 2 u + f in Q T , where Q T : = Ω × ( 0 , T ) , T > 0 , Ω ⊂ R n , n ≥ 2 , is a bounded domain with a sufficiently smooth boundary, a, b are real-valued functions defined on R + and q ( x , t ) is a measurable function in Q T with values in an interval [ q − , q + ] ⊂ ( 1 , ∞ ) . We show that under appropriate conditions on a, b, q the problem has a global in time solution and find conditions of uniqueness. Sufficient conditions for extinction in finite time or strict positivity of ‖ u ( t ) ‖ L 2 ( Ω ) for all t > 0 are found, in the case f = 0 the decay rates for ‖ u ( t ) ‖ L 2 ( Ω ) are derived.

This paper deals with the following variational problem with variable exponents { − div ( ( 1 + | ∇ u | p ( x ) 1 + | ∇ u | 2 p ( x ) ) | ∇ u | p ( x ) − 2 ∇ u ) = f ( x , u ) , in Ω , u = 0 , on ∂ Ω , where Ω ⊆ R N ( N ≥ 2 ) be a bounded domain with the smooth boundary ∂Ω, p : Ω ‾ → R is a Lipschitz continuous function with 1 p − : = ess inf x ∈ Ω ⁡ p ( x ) ≤ p ( x ) ≤ p + : = ess sup x ∈ Ω ⁡ p ( x ) N and f ∈ C ( Ω × R , R ) is superlinear but does not satisfy the usual Ambrosetti-Rabinowitz type condition. Under three different superlinear conditions on f at infinity, we prove that the above equation has at least a nontrivial solution. Moreover, the existence of infinite many solutions is proved for odd nonlinearity. Especially, some new tricks are introduced to show the boundedness of Cerami sequences.

In this article, we give a series of results on the improved integrability and boundedness of solutions to several anisotropic problems. In particular, we consider the nonhomogeneous Dirichlet problem for anisotropic elliptic second-order equations. Moreover, we study anisotropic variational inequalities with unilateral and bilateral obstacles as well as gradient constraints. Finally, variational problems for integral functionals with anisotropic integrands are considered.

Systems of differential inclusions of the form − ( ϕ ( u ′ ) ) ′ ∈ ∂ F ( t , u ) , t ∈ [ 0 , T ] , where ϕ = ∇ Φ , with Φ strictly convex, is a homeomorphism of the ball B a ⊂ R N onto R N , are considered under Dirichlet, periodic and Neumann boundary conditions. Here, ∂ F ( t , x ) stands for the generalized Clarke gradient of F ( t , ⋅ ) at x ∈ R N . Using nonsmooth critical point theory, we obtain existence results under some appropriate conditions on the potential F. Examples of applications concerning Filippov type solutions are also provided.

In this study, we propose a construction of the Green kernel for the matrix Dirichlet problem Y ″ − A 2 Y = F and Y ( 0 ) = 0 = Y ( 1 ) , where A ∈ C n × n and F ∈ L 1 ( [ 0 , 1 ] , C n ) . We apply our result to the beam differential equation D y = y ⁗ − ( α 2 + β 2 ) y ″ + α 2 β 2 y = f ′ where α 2 ≠ β 2 , are defined on [ 0 , 1 ] with the Navier boundary conditions y ( 0 ) = y ( 1 ) = y ″ ( 0 ) = y ″ ( 1 ) = 0 . In conclusion, we show that the pointwise maximum principle for the hinged beam equation holds if both α and β are real or β ¯ = α = k ( l + i ) , with | k | ≤ ϰ ( | l | ) 2 , where ϰ ( l ) ∈ ( π , 3 π 2 ) is the first positive root of the equation l tan ⁡ t = tanh ⁡ ( l t ) .

We consider the elliptic equation Δ u + K ( | x | ) e u = 0 in R n \ { 0 } with n > 2 when for l > − 2 , K ( r ) behaves like r l near 0 or ∞. The asymptotic behavior of radial solutions at ∞ is described by − ( 2 + l ) log ⁡ r for l > − 2 and − log ⁡ log ⁡ r for l = − 2 . When r − l K ( r ) → c > 0 as r → ∞ and r → 0 , regular radial solutions at ∞ and singular radial solutions at 0 exhibit self-similarity at ∞ and 0, respectively. Singular solutions with the asymptotic self-similarity exist uniquely in the radial class. Moreover, for n ≥ 10 + 4 l , separation of any two radial solutions with the asymptotic self-similarity may happen, while intersection of two solutions may occur for 2 n 10 + 4 l . In particular, for n ≥ 10 + 4 l with l > − 2 , if K ( ≢ 0 ) satisfies that r 2 K ( r ) → 0 as r → 0 and 0 ≤ k ( r ) = r − l K ( r ) ≤ n − 2 4 ( 2 + l ) inf 0 s ≤ r ⁡ k ( s ) for r > 0 , then any two radial solutions do not intersect each other and each radial solution is linearly stable. When n ≥ 10 + 4 l , we apply the global results to prove the uniqueness of positive radial solutions for the Dirichlet problem with zero data on a ball.

In this paper, we mainly study the second order expansion of classical solutions in a neighborhood of ∂Ω to the singular Dirichlet problem − Δ u = b ( x ) g ( u ) + λ a ( x ) f ( u ) , u > 0 , x ∈ Ω , u | ∂ Ω = 0 , where Ω is a bounded domain with smooth boundary in R N , λ ≥ 0 . The weight functions b , a ∈ C loc α ( Ω ) are positive in Ω and both may be vanishing or be singular on the boundary. The function g ∈ C 1 ( ( 0 , ∞ ) , ( 0 , ∞ ) ) satisfies lim t → 0 + ⁡ g ( t ) = ∞ , and f ∈ C ( [ 0 , ∞ ) , [ 0 , ∞ ) ) . We show that the nonlinear term λ a ( x ) f ( u ) does not affect the second order expansion of solutions in a neighborhood of ∂Ω to the problem for some kinds of functions b and a.

Using the sub-supersolution method with Karamata regular variation theory, we study the existence and asymptotic behavior of a classical solution to the following boundary blow-up semilinear Dirichlet problem {Δu=a(x)f(u),x∈Ω,u>0in Ω;limδ(x)→0⁡u(x)=∞, where Ω is a C1,1C1,1-bounded domain in RnRn, n≥2n≥2 and the function a belongs to Clocα(Ω), (0diam(Ω)η>diam(Ω), c1>0c1>0, c2>0c2>0, δ(x)=dist(x,∂Ω)δ(x)=dist(x,∂Ω), λ1≤λ2≤2λ1≤λ2≤2 and for i∈{1,2}i∈{1,2}, zizi is a continuous function on [0,η][0,η] with zi(0)=0zi(0)=0. For the function f , we assume that there exist constants k1k1, k2k2, p1p1, p2p2 with 00andf(t)≥k1tp1for t≥1.

We provide a new method to study the classical Dirichlet problem for constant coefficient second order elliptic PDEs on convex polyhedrons. Our approach is heavily motivated by Fokas' unified method for boundary value problems, and can be interpreted as the Fourier analogue to the classical boundary integral equations. The central object in this approach is the global relation: an integral equation which couples the known boundary data and the unknown boundary values. This integral equation depends holomorphically on two complex parameters, and the resulting analysis takes place on a Banach space of complex analytic functions closely related to the classical Paley–Wiener space. We write the global relation in the form of an operator equation and show that the analysis can be reduced to the case of Laplace's equation, from which the more general problem turns out to be a compact perturbation. We give a new integral representation to the solution to the underlying boundary value problem which serves as a concrete realisation of the fundamental principle of Ehrenpreis for all constant coefficient elliptic PDEs on convex polyhedra.

In this paper, by constructing suitable comparison functions, we mainly analyze the exact boundary behavior for the unique solution near the boundary to the singular Dirichlet problem − Δ ∞ u = b ( x ) g ( u ) , u > 0 , x ∈ Ω , u | ∂ Ω = 0 , where Ω is a bounded domain with smooth boundary in R N , g ∈ C 1 ( ( 0 , ∞ ) , ( 0 , ∞ ) ) , g is decreasing on ( 0 , ∞ ) with lim s → 0 + ⁡ g ( s ) = ∞ , g is normalized regularly varying at zero with index −γ ( γ > 1 ) and b ∈ C ( Ω ¯ ) which is positive in Ω and may be vanishing on the boundary and rapidly varying near the boundary.

We consider a semilinear system of the form ∂ u i ( t , x ) / ∂ t = k ( t ) A u i ( t , x ) + u i ′ β i ( t , x ) , with Dirichlet boundary conditions on a bounded open set D ⊂ R d , where k : [ 0 , ∞ ) → [ 0 , ∞ ) is continuous, A is the infinitesimal generator of a symmetric Levy process Z ≡ { Z ( t ) } t ≥ 0 , β i > 1 , i ∈ { 1 , 2 } and i ′ = 3 − i . We give conditions on D and on the Levy measure of Z under which our system possesses global positive solutions or exhibits blow-up in finite time. Our approach is based on the intrinsic ultracontractivity property of the semigroup generated by the process Z killed on leaving D.

In this article we address the question whether the complex Monge-Ampere equation is solvable for measures with large singular part. We prove that under some conditions there is no solution when th ...

We consider a nonlinear Dirichlet problem with a Caratheodory reaction which has arbitrary growth from below. We show that the problem has at least three nontrivial smooth solutions, two of constant sign and the third nodal. In the semilinear case (i.e., p = 2 ), with the reaction f ( z , . ) being C 1 and with subcritical growth, we show that there is a second nodal solution, for a total of four nontrivial smooth solutions. Finally, when the reaction has concave terms and is subcritical and for the nonlinear problem (i.e., 1 p ∞ ) we show that again we can have the existence of three nontrivial smooth solutions, two of constant sign and a third nodal.

We show existence and uniqueness theorems of global strong solutions of Cauchy–Dirichlet problem, for second order fully nonlinear parabolic systems, that satisfy the Campanato's condition of ellipticity. We use the Campanato's near operators theory .

We investigate the Dirichlet problem {−diva(x,∇v)=finΩv=0on∂Ω for a quasilinear elliptic equation in a planar domain Ω, when f belongs to the Zygmund space L(logL)12(loglogL)ϵ(Ω), 00. Namely, if the mapping a:Ω×R2→R2 satisfies the Leray–Lions type conditions, then we prove the estimates ‖∇φ‖L2(loglogL)−β(Ω;R2)⩽C(β)‖χ‖L2(loglogL)−β(Ω;R2) by applying a method recently suggested by L. Greco et al., which is based on the uniform estimates ‖∇φ‖L2−σ(Ω;R2)⩽C‖χ‖L2−σ(Ω;R2) available for |σ|⩽σ0 provided that χ∈L2−σ(Ω;R2).

We study the p-harmonic flow from the unit disk D 2 to the unit sphere S 2 under rotational symmetry. We show that the Dirichlet problem with constant boundary condition is locally well-posed in the class of classical solutions and we also give a sufficient criterion, in terms of the boundary condition, for the derivative of the solutions to blow-up in finite time.

We take up the existence and the asymptotic behavior of a classical solution to the following semilinear Dirichlet problem { − Δ u = a ( x ) g ( u ) , x ∈ Ω , u > 0 in Ω , u | ∂ Ω = 0 , where Ω is a C 1 , 1 -bounded domain in R N , N ≥ 2 and the function a belongs to C l o c γ ( Ω ) , ( 0 γ 1 ) such that there exist c 1 , c 2 > 0 satisfying for each x ∈ Ω , c 1 δ ( x ) − λ 1 exp ( ∫ δ ( x ) η z 1 ( s ) s d s ) ≤ a ( x ) ≤ c 2 δ ( x ) − λ 2 exp ( ∫ δ ( x ) η z 2 ( s ) s d s ) , where η > d i a m ( Ω ) , δ ( x ) = d i s t ( x , ∂ Ω ) , λ 1 ≤ λ 2 ≤ 2 and for i ∈ { 1 , 2 } , z i is a continuous function on [ 0 , η ] with z i ( 0 ) = 0 . Our arguments are based on the sub–supersolution method with Karamata regular variation theory.

We study the Dirichlet problem for a functional differential equation containing shifted and contracted argument under the Laplacian sign. We establish conditions for the unique solvability and demonstrate also that the problem may have an infinite dimensional solution manifold.

In this paper, we use the Perron method to prove the existence of viscosity solutions to a class of Monge–Ampere equations on exterior domains in R n ( n ≥ 2 ) with prescribed asymptotic behavior at infinity. This extends the Caffarelli–Li’s result on det ( D 2 u ) = 1 . We also obtain the existence of entire convex viscosity solutions to the Monge–Ampere equations with prescribed asymptotic behavior at infinity.

We prove the existence of multiple positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space \begin{equation*} \begin{cases} -{\rm div}\Big( \nabla u /\sqrt{1 - |\nabla u|^2}\Big)= f(x,u, \nabla u) & \hbox{ in } \Omega, \\ u=0& \hbox{ on } \partial \Omega. \end{cases} \end{equation*} Here $\Omega$ is a bounded regular domain in $\RR^N$ and the function $f=f(x,s,\xi)$ is either sublinear, or superlinear, or sub-superlinear near $s=0$. The proof combines topological and variational methods.