Michael Struwe

Some existence results for superlinear elliptic boundary value problems involving critical exponents

The existence of solutions to the problem −Δu − λu = u¦u¦2∗ − 2 in Ωu¦∂Ω = 0 is studied. For an arbitrary domain Ω ⊂Rn, if λ ϵ ]0, λ1[ and n ⩾ 6, the existence of solutions of changing sign is obtained. If Ω = BR(0) ⊂ Rn, λ ϵ ]0, λ1[, and n ⩾ 7, infinitely many radial solutions to this problem are exhibited, characterized by the number of nodes they possess.

Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents

Abstract In this paper we study the existence of nontrivial solutions for the boundary value problem { − Δ u − λ u − u | u | 2 ⁎ − 2 = 0 in Ω u = 0 on ∂ Ω when Ω⊂Rn is a bounded domain, n ⩾ 3, 2 ⁎ = 2 n ( n − 2 ) is the critical exponent for the Sobolev embedding H 0 1 ( Ω ) ⊂ L p ( Ω ) , λ is a real parameter. We prove that there is bifurcation from any eigenvalue λj of − Δ and we give an estimate of the left neighbourhoods ] λ j ⁎ , λj] of λj, j∈N, in which the bifurcation branch can be extended. Moreover we prove that, if λ ∈ ] λ j ⁎ , λj[, the number of nontrivial solutions is at least twice the multiplicity of λj. The same kind of results holds also when Ω is a compact Riemannian manifold of dimension n ⩾ 3, without boundary and Δ is the relative Laplace-Beltrami operator.

Critical points of embeddings of H01,n into Orlicz spaces

Abstract For a domain Ω ⊂ ℝn embeddings u → exp(α(|u|/∥u∥1, n)n/n − 1) of H 0 1 , n ( Ω ) into Orlicz spaces are considered. At the critical exponent α = αn a loss of compactness reminiscent of the Yamabe problem is encountered; however by a result of Carlesson and Chang, if Ω is a ball the best constant for the above embedding is attained. In dimension n = 2 we identify the “limiting problem” responsible for the lack of compactness at the critical exponent α2 = 4π in the radially symmetric case and establish the existence of extremal functions also for nonsymmetric domains Ω. Moreover, we establish the existence of two “branches” of critical points of this embedding beyond the critical exponent α2 = 4π.

Existence of periodic solutions of Hamiltonian systems on almost every energy surface

We estab lish the existence of periodic solutions of Hamiltonian systems on almost every smooth, compact energy surface in the sense of Lebesgue measure.

On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems

SummaryThe Hölder continuity of bounded weak solutions of quasilinear parabolic systems with main part in diagonal form is proved via a “parabolic hole-filling technique”.

The Yang-Mills flow in four dimensions

Global existence and uniqueness is established for the Yang-Mills heat flow in a vector bundle over a compact Riemannian four-manifold for given initial connection of finite energy. Our results are analogous to those valid for the evolution of harmonic maps of Riemannian surfaces.

A note on the problem −Δu=λu+u|u|2*−2

The dual approach to the problem −Δu=λu+u|u|2*−2, uþe01(gW), permits a simple proof of a recent existence result [5] and allows extensions of this result to similar problems also with asymmetric mnonlinearities.

Quasilinear elliptic eigenvalue problems.

SummaryThe generalized Palais-Smale condition introduced in [26] is applied to obtain multiple solutions of variational eigen-value problems with quasilinear principal part, thereby extending some well-known existence results for semilinear elliptic problems.

On a free boundary problem for minimal surfaces

ForC4-embedded manifoldsS ⊂ ℝ3 which are differmorphic to the standard sphere in ℝ3 the existence of non-constant minimal surfaces bounded byS and intersectingS orthogonally along their boundaries is deduced.

Concentration phenomena for Liouville's equation in dimension four

Let Ω be a bounded domain of R and let uk be solutions to the equation (1) ∆uk = Vke in Ω, where (2) Vk → 1 uniformly in Ω, as k → ∞. Throughout the paper we denote as ∆ = − ∑ i( ∂ ∂xi ) 2 the Laplacian with the geometers’ sign convention. Continuing the analysis of [18], here we study the compactness properties of equation (1). Equation (1) is the fourth order analogue of Liouville’s equation. Thus, for problem (1), (2) we may expect similar results to hold as have been obtained by Brezis-Merle [3] in the two-dimensional case. Recall the following result from [3] (we also refer to Li-Shafrir [11]). Theorem 1.1. Let Σ be a bounded domain of R and let (uk)k∈N be a sequence of solutions to the equation (3) ∆uk = Vke in Σ, where Vk → 1 uniformly in Σ as k →∞, and satisfying the uniform bound