Wolfgang Reichel

Radial solutions of equations and inequalities involving the -Laplacian.

the initial value problem with given data u(ro) uo u’(ro) u and counterexamples to uniqueness are given. For the case where g is increasing in u, a sharp comparison theorem is established; it leads to maximal solutions, nonuniqueness anduniqueness results, among others. Using these results, a strong comparison principle for the boundary value problem and a number of properties of blow-up solutions are proved under weak assumptions

Boundary blowup type sub-solutions to semilinear elliptic equations with Hardy potential

Semilinear elliptic equations which give rise to solutions blowing up at the boundary are perturbed by a Hardy potential. The size of this potential effects the existence of a certain type of solutions (large solutions): if the potential is too small, then no large solution exists. The presence of the Hardy potential requires a new definition of large solutions, following the pattern of the associated linear problem. Nonexistence and existence results for different types of solutions will be given. Our considerations are based on a Phragmen-Lindelof type theorem which enables us to classify the solutions and sub-solutions according to their behavior near the boundary. Nonexistence follows from this principle together with the Keller-Osserman upper bound. The existence proofs rely on sub- and super-solution techniques and on estimates for the Hardy constant derived in Marcus, Mizel and Pinchover.

Positivity and anti-maximum principles for elliptic operators with mixed boundary conditions

We consider linear elliptic equations 1u + q(x)u = u + f in bounded Lipschitz domainsD R N with mixed boundary conditions@u/@n =(x)u +g on@D. The main feature of this boundary value problem is the appearance of both in the equation and in the boundary condition. In general we make no assumption on the sign of the coefficient (x) . We study positivity principles and anti-maximum principles. One of our main results states that if is somewhere negative, q 0 and R D q(x)dx > 0 then there exist two eigenvalues 1, 1 such the positivity principle holds for 2 ( 1, 1) and the anti-maximum principle holds if 2 ( 1, 1 + ) or 2 ( 1 , 1). A similar, but more complicated result holds if q 0. This is due to the fact that 0 = 0 becomes an eigenvalue in this case and that 1() as a function of connects to 1() when the mean value of crosses the value 0 = | D|/|@D|. In dimension N = 1 we determine the optimal -interval such that the anti-maximum principles holds uniformly for all right-hand sides f,g 0. Finally, we apply our result to the problem 1u +q(x)u = u +f inD,@u/@n =u +g on@D with constant coefficients , 2 R.

Computing eigenvalues and Fucik-spectrum of the radially symmetric p -Laplacian

Eigenvalue problems for the radially symmetric p-Laplacian are discussed. We present algorithms which compute a given number of eigenvalues and Fucik-curves together with the corresponding eigenfunctions. The second-order p-Laplacian equation is transformed into a first-order system by a generalized Prufer-transformation. To the first-order system we apply shooting algorithms, Newtons method and in case of the Fucik-curves a predictor-corrector method. Our approach requires analytical and numerical treatment of generalized sine-functions. Singular as well as regular problems are treated, and a detailed error analysis for the approximation of singular problems by regular ones are given. Numerical results are presented.

Uniqueness theorems for variational problems by the method of transformation groups

Introduction.- Uniqueness of Critical Points (I).- Uniqueness of Citical Pints (II).- Variational Problems on Riemannian Manifolds.- Scalar Problems in Euclidean Space.- Vector Problems in Euclidean Space.- Frechet-Differentiability.- Lipschitz-Properties of ge and omegae.

CHAPTER 1 – Solutions of Quasilinear Second-Order Elliptic Boundary Value Problems via Degree Theory

This chapter presents a powerful tool for proving existence of solutions of linear and nonlinear second-order elliptic boundary value problems and some of the most interesting properties and applications. Rather than describing more recent topological developments of the notion of degree and its properties, the chapter discusses different classes of boundary value problems for which variational methods do not apply.

Ground states of a nonlinear curl-curl problem in cylindrically symmetric media

We consider the nonlinear curl-curl problem

Uniqueness for degenerate elliptic equations via Serrin's sweeping principle

{\nabla\times\nabla\times U + V(x) U= \Gamma(x)|U|^{p-1}U}