Band 6, Heft 4

Using variational methods we establish existence of multi-bump solutions for the following class of quasilinear problems -Δ p u + (λV(x) + Z(x))u p-1 = f(u); u > 0 in ℝ N where Δ p u is the p-Laplacian operator, 2 ≤ p < N, λ ∈ (0, ∞), f is a continuous function with subcritical growth and V,Z : ℝ N → ℝ are continuous functions verifying some hypothesis.

In this paper, we study some fourth order conformal invariants on the standard n-spheres, n ≥ 5. Using topological arguments, we give a variety of classes of functions that can be realized as Paneitz curvature.

In this paper we show the axial symmetry of a particular type of solutions of the following semilinear nonhomogeneous problem where B is the unit ball of ℝ N , N ≥ 3, u+ = 0 if u < 0 and φ 1 (x) > 0 is the eigenfunction of -Δ corresponding to the first eigenvalue λ 1 , with Dirichlet boundary conditions. The solutions considered exhibit one or two peaks and concentrate, as the parameter s goes to +∞.

This article is concerned with the regularity of the entropy solution of where Ω is a smooth bounded domain Ω of ℝ N such that 0 ∈ Ω, 1 < p < N. and γ < (N-p)/p. Assuming f ∈ L q (Ω, |x| α(q-1) dx) for some q ≥ 1 and N γp /(N-p) ≤ α ≤ (γ+1)p, we obtain estimates for the entropy solution u and its weak gradient in Lebesgue spaces with weights. Moreover, we introduce some explicit examples showing the optimality of our results and a relation between our problem and a Hardy-Sobolev type inequality.

In this paper, we will use some Sobolev constants to construct certain classes of nondegenerate potentials for linear differential equations with deviation. As an application of these potentials, we will give some existence results of periodic solutions of semilinear equations with deviation.

In this paper, we consider the problem Δ 2 u = |u| 8/(n-4) u + εf(x, u) in Ω, u = Δu = 0 on ∂Ω, where Ω is a bounded and smooth domain in ℝ n , n ≥ 6, ε is a small positive parameter, and f is a smooth function. Our main purpose is to construct families of solutions which concentrate around some well defined points depending on f.

We consider a wide class of degenerate quasilinear second order elliptic equations with model representative , whose solutions have singularity at a point. There are established sharp point-wise conditions for removable isolated singularity of solutions of such equations. For solutions with non-removable point singularity (source type or fundamental solution), precise upper and lower estimates near the singularity point are obtained.

In this paper, we prove that there exist at least two geometrically distinct symmetric brake orbits in every bounded convex symmetric domain in R n for n ≥ 2.