In this paper we give new results concerning the maximal regularity of the strict solution of an abstract second-order differential equation, with non-homogeneous boundary conditions of Dirichlet type, and set in an unbounded interval. The right-hand term of the equation is a Hölder continuous function.

This paper studies the existence and multiplicity of positive solutions of the following problem:

where Ω⊂RN(N≥3) is a smooth bounded domain, , 1 < p < N, and 0 < α < 1, p - 1 < β < p* - 1 (p* = Np/(N - p)) and 0 < γ < N + ((β + 1)(p - N)/p) are three constants. Also δ(x) = dist(x, ∂Ω), a ∈ Lp and λ < 0 is a real parameter. By using the direct method of the calculus of variations, Ekeland's Variational Principle and an idea of G. Tarantello, it is proved that problem (*) has at least two positive weak solutions if λ is small enough.

In this paper, first, we study the existence of the positive solutions of the nonlinear elliptic equations in unbounded domains. The existence is affected by the properties of the geometry and the topology of the domain. We assert that if there exists a (PS)c-sequence with c belonging to a suitable interval depending by the equation, then a ground state solution and a positive higher energy solution exist, too. Next, we study the upper half strip with a hole. In this case, the ground state solution does not exist, however there exists at least a positive higher energy solution.

In this paper we study the existence and uniqueness of positive solutions of boundary vlue problems for continuous semilinear perturbations, say f: [0, 1) × (0, ∞) → (0, ∞), of class of quasilinear operators which represent, for instance, the radial form of the Dirichlet problem on the unit ball of RN for the operators: p-Laplacian (1 < p < ∞) ad k-Hessian (1 ≤ k ≤ N). As a key feature, f (r, u) is possibly singular at r = 1 or u =0, Our approach exploits fixed point arguments and the Shooting Method.

The problem of scattering of tidal waves by reefs and spits of arbitrary shape is reduced to a skew derivative problem for the two-dimensional Helmholtz equation in the exterior of open arcs in a plane. The resulting boundary-value problem is studied by potential theory and a boundary integral equation method. After some transformations, the skew derivative problem is reduced to a Fredholm integral equation of the second kind, which is uniquely solvable. In this way the solvability theorem is proved and an integral representation of the solution is obtained. A uniqueness theorem is also proved.

A method of finding critical points in a half space is developed. It is then applied to the study of semilinear boundary value problems, and used to determine conditions which lead to multiple non-trivial solutions.

In this paper we complete two tasks. First we extend the nonsmooth critical point theory of Chang to the case where the energy functional satisfies only the weaker nonsmooth Cerami condition and we also relax the boundary conditions. Then we study semilinear and quasilinear equations (involving the p-Laplacian). Using a variational approach we establish the existence of one and of multiple solutions. In simple existence theorems, we allow the right hand side to be discontinuous. In that case in order to have an existence theory, we pass to a multivalued approximation of the original problem by, roughly speaking, filling in the gaps at the discontinuity points.

Let G be a connected Lie group with Lie algebra g and a1, …, ad an algebraic basis of g. Further let Ai denote the generators of left translations, acting on the Lp-spaces Lp(G; dg) formed with left Haar measure dg, in the directions ai. We consider second-order operators in divergence form corresponding to a quadratic form with complex coefficients, bounded Hölder continuous principal coefficients cij and lower order coefficients ci, c′ii, c0 ∈ L∞ such that the matrix C= (cij) of principal coefficients satisfies the subellipticity condition uniformly over G.

We discuss the hierarchy relating smoothness properties of the coefficients of H with smoothness of the kernel and smoothness of the domain of powers of H on the Lρ-spaces. Moreover, we present Gaussian type bounds for the kernel and its derivatives.

Similar theorems are proved for strongly elliptic operators in non-divergence form for which the principal coefficients are at least once differentiable.

We prove two asymptotical estimates for minimizers of a Ginzburg-Landau functional of the form .

We show that the Kato conjecture is true for m-accretive operators with highly singular coefficients. For operators of the form A = *F, where formally corresponds to d/dx + zδ on L2 (R), we prove that Dom (A1/2) = Dom() = e-zHH1(R) where H is the Heavysied function. By adapting recent methods of Auscher and Tchamitchian, we characterize Dom (A) in terms of an unconditional wavelet basis for L2(R).

We prove a new energy or Caccioppoli type estimate for minimisers of the model functional ∫Ω|Du|2 + (det Du)2, where Ω ⊂ 2 and u: Ω → 2. We apply this to establish C∞ regularity for minimisers except on a closed set of measure zero. We also prove a maximum principle and use this to establish everywhere continuity of minimisers.

The existence of positive solutions, vanishing at infinity, for the semilinear eigenvalue problem Lu = λ f(x, y) in RN is obtained, where L is a strictly elliptic operator. The function f is assumed to be of subcritical growth with respect to the variable u.

The existence of positive solutions of some semilinear elliptic equations of the form −Δu = λf(u) is studied. The major results are a nonexistence theorem which gives a λ* = λ*(f,Ω) > 0 below which no positive solutions exist and a lower bound theorem for umax for Ω a ball. As a corollary of the nonexistence theorem that describes the dependence of the number of solutions on λ, two other nonexistence theorems, and an existence theorem are also proved.

We study the existence, uniqueness and regularity of solutions to an exterior elliptic free boundary problem. The solutions model stationary solutions to nonlinear diffusion reaction problems, that is, they have compact support and satisfy both homogeneous Dirichlet and Neumann-type boundary conditions on the free boundary ∂{u > 0}. Then we prove convexity and symmetry properties of the free boundary and of the level sets {u > c} of the solutions. We also establish symmetry properties for the corresponding interior free boundary problem.

Let ui(x, t) be the ith component of a nonnegative solution of a weakly coupled system of second-order, linear, parabolic partial differential equations, for x ∈ Rn and 0 < t < T. We obtain lower estimates, near t = 0, for the Lebesgue measure of the set of x for which exceeds 1. Related results for Poisson integrals on a half-space are also described, some applications are given, and interesting comparisons emerge.

A posteriori error estimates for a class of elliptic unilateral boundary value problems are obtained for functions satisfying only part of the boundary conditions. Next, we give an alternative approach to the a posteriori error estimates for self-adjoint boundary value problems developed by Aubin and Burchard. Further, we are able to construct an alternative estimate with mild additional assumptions. An example of a linear differential operator of order 2k is given.

Consider a point transformation of the biharmonic equation

namely a coordinate transformation

together with a change of dependent variable given by

for some multiplier F(ξ, η).

The purpose of this note is to investigate traces of the functions In(1 +|u|), where u is a solution of a semi-linear partial differential equation of elliptic type, belonging to an appropriate Sobolev space. This article complements the results of Chabrowski and Thompson (1980), and Mihailov (1972), (1976).

We consider general boundary value problems for homogeneous elliptic partial differential operators with constant coefficients. Under natural conditions on the operators, these problems give rise to isomorphisms between the appropriate spaces with homogeneous norms. We also consider operators which are not properly elliptic and boundary systems which do not satisfy the complementing condition and determine when they give rise to left or right invertible operators. A priori inequalities and regularity results for the corresponding boundary value problems in Sobolev spaces are then readily obtained.

The existence of spherically symmetric solutions of the equation ▵u = f(u) is proved for a large class of functions f(z). Among others, functions satisfying an inequality zf(z) < 0 for |z| > A, and in particular the function f(z) = -sinh z, belong to this class.