Volume 11, Issue 4

In the present article we study global existence for a nonlinear parabolic equation having a reaction term and a Radon measure datum: where 1 < p < N, Ω is a bounded open subset of ℝ N (N ≥ 2), Δ p u = div(|∇u| p−2 ∇u) is the so called p-Laplacian operator, sign s ., ϕ(ν 0 ) ∈ L 1 (Ω), μ is a finite Radon measure and f ∈ L ∞ (Ω×(0, T)) for every T > 0. Then we apply this existence result to show wild nonuniqueness for a connected nonlinear parabolic problem having a gradient term with natural growth.

We consider a nonlinear Neumann problem driven by the p-Laplacian and with a reaction which exhibits an asymmetric behaviour near +∞ and near −∞. Namely, it is (p − 1)- superlinear near +∞ (but need not satisfy the Ambrosetti-Rabinowitz condition) and it is (p − 1)-linear near −∞. Combining variational methods with Morse theory, we show that the problem has at least three nontrivial smooth solutions.

We consider a reaction-diffusion system of activator-inhibitor or substrate-depletion type in one space dimension which is subject to diffusion-driven instability. We determine the change of bifurcation when a pure Neumann condition is supplemented with a Signorini condition. We show that this change differs essentially from the known case when also Dirichlet conditions are assumed.

It is well-known that solutions on the stable manifold of a hyperbolic periodic solution of an autonomous system of ordinary differential equations have an asymptotic phase which has the same order of smoothness as the vector field. In this paper we show if the system depends on a parameter that, in general, the asymptotic phase loses one order of smoothness in the parameter.

We study radially symmetric systems with a singularity of repulsive type. In the presence of a radially symmetric periodic forcing, we show the existence of three distinct families of subharmonic solutions: One oscillates radially, one rotates around the origin with small angular momentum, and the third one with both large angular momentum and large amplitude. The proofs are carried out by the use of topological degree theory.

For a class of equations generalizing the model case Δ p u − a(r)u p−1 + b(r)u q = 0 in B, u = 0 on∂B, where B is the unit ball in R n , n ≥ 1, r = |x|, p, q > 1, and Δ p denotes the p-Laplace operator, we give conditions for the existence and uniqueness of positive solution. In case n = 1, we give a more general result.

In this paper we consider a biharmonic equation of the form Δ 2 u+V(x)u = f (u) in the whole four-dimensional space ℜ 4 . Assuming that the potential V satisfies some symmetry conditions and is bounded away from zero and that the nonlinearity f is odd and has subcritical exponential growth (in the sense of an Adams’ type inequality), we prove a multiplicity result. More precisely we prove the existence of infinitely many nonradial sign-changing solutions and infinitely many radial solutions in H 2 (ℜ 4 ). The main difficulty is the lack of compactness due to the unboundedness of the domain ℜ 4 and in this respect the symmetries of the problem play an important role.

For a class of divergence type quasi-linear degenerate parabolic equations with a Radon measure on the right hand side we derive pointwise estimates for solutions via nonlinear Wolff potentials.

We prove that under suitable assumptions, from a sequence of solutions of Globally Modified Navier-Stokes equations with delays we can extract a subsequence which converges in an adequate sense to a weak solution of a three-dimensional Navier-Stokes equation with delays. An additional case with a family of different delays involved in the approximating problems is also discussed.

Let V be a finite-dimensional representation of a compact connected Lie group G and let ϕ : V × [λ 1 , λ 2 ] → ℝ be a family of G-invariant potentials of the class C 2 such that ∇ v ϕ(0, λ) = 0 for every λ ∈ [λ 1 , λ 2 ], where ∇ v ϕ is the gradient of ϕ with respect to the first coordinate. Denote by V′ i a direct sum of eigenspaces of the isomorphism ∇ v 2 ϕ(0, λ i ) corresponding to positive eigenvalues, i = 1, 2. We have proved that if dimV′ 1 ≧ dimV′ 2 and the representation V′ 1 is not equivalent (as a linear representation of G) to the representation V′ 2 ⊕ ℝ[2k], k ∈ {0} ∪ ℕ (where ℝ[2k] is a 2k-dimensional trivial representation of G), then there is a global bifurcation of G-orbits of non-zero solutions of problem ∇ v ϕ(v, λ) = 0 from the interval {0} × [λ 1 , λ 2 ]. We have proved that a global bifurcation and symmetrybreaking of solutions phenomenon occurs simultaneously for elliptic differential equations considered by Smoller and Wasserman in [25], which improves results of [25].