Existence of entire explosive positive radial solutions for a class of quasilinear elliptic systems

We show the existence of entire explosive positive radial solutions for quasilinear elliptic systems div(|∇u|^m−2∇u)=p(|x|)g(v), div(|∇v|ⁿ⁻²∇v)=q(|x|)f(u) on , where f and g are positive and non-decreasing functions on (0,∞) satisfying the Keller-Osserman condition.     

A complete classification of bifurcation diagrams of a p-Laplacian Dirichlet problem

We study the bifurcation diagrams of classical positive solutions u with ‖u_‖∞∈(0,∞) of the p-Laplacian Dirichlet problem

     

A Bahri–Lions theorem revisited

In 1988, A. Bahri and P.L. Lions [A. Bahri, P.L. Lions, Morse-index of some min–max critical points. I. Application to multiplicity results, Comm. Pure Appl. Math. 41 (1988) 1027–1037] studied the following elliptic problem:

where Ω is a bounded smooth domain of , 2<p<(2N−2)/(N−2) and f(x,u) is not assumed to be odd in u. They proved the existence of infinitely many solutions under an appropriate growth restriction on f. In the present paper, we improve this result by showing that under the same growth assumption on f the problem admits in fact infinitely many sign-changing solutions. In addition we derive an estimate on the number of their nodal domains. We also deal with the corresponding fourth order equation Δ²u=|u|^p−2u+f(x,u) with both Dirichlet and Navier boundary conditions, as well as with strongly coupled elliptic systems.     

Optimal bounds on the Kuramoto-Sivashinsky equation

In this paper, we consider solutions u(t,x) of the one-dimensional Kuramoto-Sivashinsky equation, i.e.

     

Existence of positive solutions for second-order boundary value problem with one parameter

In the case of not requiring f(t,u) to be nonnegative, by transforming the boundary value problem into the integral equation system, and applying the fixed point index theory, the author studies the following second-order boundary value problem with one parameter

     

Global branch from the second eigenvalue for a semilinear Neumann problem in a ball

Let (n?3) be a ball, and let f∈C³. We are concerned with the Neumann problem

     

Multiplicity of solutions for the plasma problem in two dimensions

Let Ω be a bounded domain in R², u₊=u if u?0, u₊=0 if u<0, u₋=u₊−u. In this paper we study the existence of solutions to the following problem arising in the study of a simple model of a confined plasma

     

Large solutions of semilinear elliptic equations under the Keller-Osserman condition

We consider the equation Δu=p(x)f(u) where p is a nonnegative nontrivial continuous function and f is continuous and nondecreasing on [0,∞), satisfies f(0)=0, f(s)>0 for s>0 and the Keller-Osserman condition where . We establish conditions on the function p that are necessary and sufficient for the existence of positive solutions, bounded and unbounded, of the given equation.     

Existence results for some polyharmonic problems in the half-space

We study the existence of positive solutions of the m-polyharmonic nonlinear elliptic equation m(−Δ)u+f(⋅,u)=0 in the half-space , n?2 and m?1. Our purpose is to give two existence results for the above equation subject to some boundary conditions, where the nonlinear term f(x,t) satisfies some appropriate conditions related to a certain Kato class of functions .     

The Liouville equation in a half-plane

We classify the solutions of the equation Δu+aeu=0 in the half-plane that satisfy the Neumann boundary condition ∂u/∂t=ceu/2 on . An analogous problem in the once punctured disc D^∗⊂R² is also solved.     

Classification of Razor Blades to the filtration equation—the sublinear case

We study nonnegative solutions of the filtration equation u_t=Δ?(u) in , where ? is continuous, increasing and sublinear. More precisely, we study the Razor Blades, i.e., solutions which may be singular at |x|=0 for t>0, and start with zero initial data. We first prove a nonexistence result when ? is too sublinear and we show an axial trace result in the other case: there exist a time τ≡τ(u) and a Radon measure λ on [0,τ) such that

     

The Toda system and multiple-end solutions of autonomous planar elliptic problems

We prove the existence of a new class of entire, positive solutions for the classical elliptic problem Δu−u+up=0 in R², when p>2. The solutions we construct are obtained by perturbing the function

     

Deduction of L. Hörmander's extension of Ásgeirsson's mean value theorem

L. Hörmander's extension of Ásgeirsson's mean value theorem states that if u is a solution of the inhomogeneous ultrahyperbolic equation (Δ_x−Δ_y)u=f, , , then

     

Positive solutions of reaction diffusion equations with super-linear absorption: Universal bounds, uniqueness for the Cauchy problem, boundedness of stationary solutions

Consider classical solutions u∈C²(Rⁿ×(0,∞))∩C(Rⁿ×[0,∞)) to the parabolic reaction diffusion equation

     

A two phase elliptic singular perturbation problem with a forcing term

We study the following two phase elliptic singular perturbation problem:

Δu^ε=β_ε(u^ε)+f^ε,