Necessary and Sufficient Conditions for the Solvability of the L p Dirichlet Problem on Lipschitz Domains

We study the homogeneous elliptic systems of order $2\ellWe study the homogeneous elliptic systems of order with real constant coefficients on Lipschitz domains in, . For any fixed p > 2, we show that a reverse H?lder condition with exponent p is necessary and sufficient for the solvability of the Dirichlet problem with boundary data in L ^p . We also obtain a simple sufficient condition. As a consequence, we establish the solvability of the L ^p Dirichlet problem for and . The range of p is known to be sharp if and . For the polyharmonic equation, the sharp range of p is also found in the case n = 6, 7 if , and if .Research supported in part by the NSF.     

Homogenization of singular numbers for a non self-adjoint elliptic problem in a perforated domain

We consider the spectral problem for a non self-adjoint Dirichlet problem for a higher-order elliptic operator in a sequence of perforated domains. We establish the convergence of the singular numbers generated by the problem to the corresponding singular numbers generated by a limit problem of the same type but containing an additional term of capacity type.Research supported by the National Research Foundation of South Africa.     

Symbolic calculus for boundary value problems on manifolds with edges

Boundary value problems for (pseudo-) differential operators on a manifold with edges can be characterised by a hierarchy of symbols. The symbolic structure is responsible for ellipticity and for the nature of parametrices within an algebra of “edge-degenerate” pseudo-differential operators. The edge symbolic component of that hierarchy takes values in boundary value problems on an infinite model cone, with edge variables and covariables as parameters. Edge symbols play a crucial role in this theory, in particular, the contribution with holomorphic operator-valued Mellin symbols. We establish a calculus in a framework of “twisted homogeneity” that refers to strongly continuous groups of isomorphisms on weighted cone Sobolev spaces. We then derive an equivalent representation with a particularly transparent composition behaviour.     

Multiple solutions for a Hénon-like equation on the annulus

For the equation −Δu=||xα|−2|up−1, 1<|x|<3, we prove the existence of two solutions for α large, and of two additional solutions when p is close to the critical Sobolev exponent ^2∗=2N/(N−2). A symmetry-breaking phenomenon appears, showing that the least-energy solutions cannot be radial functions.     

Some Applications to Variational–Hemivariational Inequalities of the Principle of Symmetric Criticality for Motreanu–Panagiotopoulos Type Functionals

Using the principle of symmetric criticality for Motreanu–Panagiotopoulos type functionals we give some existence and multiplicity results for a class of variational–hemivariational inequalities on ^L+M .This work was partially supported by MEdC-ANCS, research project CEEX 2983/11.10.2005.     

Solutions with transition layer and spike in an inhomogeneous phase transition model

We consider the following singularly perturbed elliptic problem

     

Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity

We investigate entire radial solutions of the semilinear biharmonic equation Δ²u=λexp(u) in Rn, n?5, λ>0 being a parameter. We show that singular radial solutions of the corresponding Dirichlet problem in the unit ball cannot be extended as solutions of the equation to the whole of Rn. In particular, they cannot be expanded as power series in the natural variable s=log|x|. Next, we prove the existence of infinitely many entire regular radial solutions. They all diverge to −∞ as |x|→∞ and we specify their asymptotic behaviour. As in the case with power-type nonlinearities [F. Gazzola, H.-Ch. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann. 334 (2006) 905-936], the entire singular solution x?−4log|x| plays the role of a separatrix in the bifurcation picture. Finally, a technique for the computer assisted study of a broad class of equations is developed. It is applied to obtain a computer assisted proof of the underlying dynamical behaviour for the bifurcation diagram of a corresponding autonomous system of ODEs, in the case n=5.     

Single-point condensation phenomena for a four-dimensional biharmonic Ren–Wei problem

We continue to study the asymptotic behavior of least energy solutions to the following fourth order elliptic problem (E _p ): as p gets large, where Ω is a smooth bounded domain in R ⁴ . In our earlier paper (Takahashi in Osaka J. Math., 2006), we have shown that the least energy solutions remain bounded uniformly in p and they have one or two “peaks” away form the boundary. In this note, following the arguments in Adimurthi and Grossi (Proc. AMS 132(4):1013–1019, 2003) and Lin and Wei (Comm. Pure Appl. Math. 56:784–809, 2003), we will obtain more sharper estimates of the upper bound of the least energy solutions and prove that the least energy solutions must develop single-point spiky pattern, under the assumption that the domain is convex.     

The boundary blow-up rate of large solutions

In this paper we ascertain the blow-up rate of the large solutions of a class of sublinear elliptic boundary value problems with a weight function in front of the nonlinearity that vanishes on the boundary of the underlying domain, Ω, at different rates according to the point of the boundary, . All previous results in the literature assumed the decay rate of the underlying weight function to be the same at any point of ∂Ω. This hypothesis substantially simplified the mathematical analysis of the problem, as it allowed constructing global sub and supersolutions in an open neighborhood of ∂Ω. Obtaining general results requires localizing at each particular point of the boundary, making particularly involved the mathematical analysis of the problem.     

Positive solutions of semilinear biharmonic equations with critical Sobolev exponents

In this paper we study the critical growth biharmonic problem with a parameter λ and establish uniform lower bounds for Λ, which is the supremum of the set of λ, related to the existence of positive solutions of the biharmonic problem.     

Existence and regularity of nonnegative solution of a singular quasi-linear anisotropic elliptic boundary value problem with gradient terms

In this paper, we consider the singular quasi-linear anisotropic elliptic boundary value problem

(P)     

Existence, uniqueness and blow-up rate of large solutions for a canonical class of one-dimensional problems on the half-line

This paper shows the existence and the uniqueness of the positive solution ?(t) of the singular boundary value problem

     

Optimal uniqueness theorems and exact blow-up rates of large solutions

In this paper, we prove some optimal uniqueness results for large solutions of a canonical class of semilinear equations under minimal regularity conditions on the weight function in front of the non-linearity and combine these results with the localization method introduced in [López-Gómez, The boundary blow-up rate of large solutions, J. Differential Equations 195 (2003) 25-45] to prove that any large solution L of Δu=a(x)u^p, p>1, a>0, must satisfy

     

Properties of positive harmonic functions on the half-space with a nonlinear boundary condition

In this paper we study existence and properties of solutions of the problem Δw=0 on the half-space with nonlinear boundary condition ∂w/∂η+w=|w|^p−2w where 2<p<2(N−1)/(N−2) and N?3. We obtain a ground state solution w=w(x₁,…,xN−1,t) which is radial and has exponential decay in the first N−1 variables. Moreover, w has sharp polynomial decay in the variable t.