Multiplicity of solutions for a fourth order problem with exponential nonlinearity

Let B be the unit ball in RN, N?5 and n be the exterior unit normal vector on the boundary. We consider radial solutions to

     

Morse theory for a fourth order elliptic equation with exponential nonlinearity

Given a Hilbert space (H,á·,·?){(\mathcal H,\langle\cdot,\cdot\rangle)}, and interval L ì (0,+￥){\Lambda\subset(0,+\infty)} and a map K ? C²(H,\mathbb R){K\in C^2(\mathcal H,\mathbb R)} whose gradient is a compact mapping, we consider the family of functionals of the type:

On solutions of second and fourth order elliptic equations with power-type nonlinearities

We study second and fourth order semilinear elliptic equations with a power-type nonlinearity depending on a power p$p$ and a parameter λ>0$\lambda >0$. For both equations we consider Dirichlet boundary conditions in the unit ball B⊂Rⁿ$B\subset {\mathbb{R}}^n$. Regularity of solutions strictly depends on the power p$p$ and the parameter λ$\lambda$. We are particularly interested in the radial solutions of these two problems and many of our proofs are based on an ordinary differential equation approach.     

Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity

We investigate entire radial solutions of the semilinear biharmonic equation Δ²u=λexp(u) in Rn, n?5, λ>0 being a parameter. We show that singular radial solutions of the corresponding Dirichlet problem in the unit ball cannot be extended as solutions of the equation to the whole of Rn. In particular, they cannot be expanded as power series in the natural variable s=log|x|. Next, we prove the existence of infinitely many entire regular radial solutions. They all diverge to −∞ as |x|→∞ and we specify their asymptotic behaviour. As in the case with power-type nonlinearities [F. Gazzola, H.-Ch. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann. 334 (2006) 905-936], the entire singular solution x?−4log|x| plays the role of a separatrix in the bifurcation picture. Finally, a technique for the computer assisted study of a broad class of equations is developed. It is applied to obtain a computer assisted proof of the underlying dynamical behaviour for the bifurcation diagram of a corresponding autonomous system of ODEs, in the case n=5.     

Monotone positive solutions for a fourth order equation with nonlinear boundary conditions

This work is concerned with the existence of monotone positive solutions for a class of beam equations with nonlinear boundary conditions. The results are obtained by using the monotone iteration method and they extend early works on beams with null boundary conditions. Numerical simulations are also presented.     

Positive solutions for a fourth order equation invariant under isometries

Let be a smooth compact Riemannian manifold of dimension . We consider the problem

where , , . We require to be positive and invariant under isometries. We prove existence results for on arbitrary compact manifolds. This includes the case of the geometric Paneitz-Branson operator on the sphere.

     

Two homoclinic solutions for a nonperiodic fourth order differential equation with a perturbation

In this paper, we study multiple homoclinic solutions for a class of fourth order differential equations with a perturbation. By establishing a compactness lemma and using variational methods, the existence result of two homoclinic solutions is obtained under some suitable assumptions, but not requiring the periodicity condition. Some recent results are improved and extended.     

Multiplicity result for a singular elliptic equation with indefinite nonlinearity

Via constraint minimization argument and delicate energy estimates, we show the existence of two positive solutions for a singular elliptic equation with indefinite nonlinearity.     

Quasi-periodic solutions for 1D wave equation with higher order nonlinearity

In this paper, one-dimensional (1D) nonlinear wave equation$u_{tt}?u_{xx}+mu+u^5=0$ on the finite x-interval $[0,\pi ]$ with Dirichlet boundary conditions is considered. It is proved that there are many 2-dimensional elliptic invariant tori, and thus quasi-periodic solutions for the above equation. The proof is based on infinite-dimensional KAM theory and partial Birkhoff normal form.     

Oscillating solutions of a nonlinear fourth order ordinary differential equation

We study the existence of periodic solutions for a nonlinear fourth order ordinary differential equation. Under suitable conditions we prove the existence of at least one solution of the problem applying coincidence degree theory and the method of upper and lower solutions.     

Almost global existence for solutions of a nonlinear fourth order hyperbolic equation

We study the long time existence of the solutions of the Cauchy problem for a class of nonlinear fourth order hyperbolic equations, in which the principal part is given by the composition of two waves operators with different propagation speeds. The presence of these two speeds makes this problem essentially different from the correspondent one for the second order wave equation. In fact, in the present case, we cannot apply the approach of S. Klainerman, based on the invariance of the D'Alembertian operator under the complete Lorentz group. Furthermore, even the alternative method used by F. John and S. Klainerman, only for the wave equation in three space dimensions, does not seem directly applicable. Nevertheless, thanks to the special structure of the equations, we are able to show the almost global existence of the solutions to the related Cauchy problem. Received December 29, 1995     

Multiplicity of positive solutions to a p-Laplacian equation involving critical nonlinearity

In this paper we deal with multiplicity of positive solutions to the p-Laplacian equation of the type

     

Existence and concentration of positive solutions for a super-critical fourth order equation

In this paper we investigate the problem of existence of solutions for a super-critical fourth order Yamabe type equation and we exhibit a family of solutions concentrating at two points, provided the domain contains one hole and we give a multiplicity result if we are given multiple holes.     

Interior estimates for solutions of a fourth order nonlinear partial differential equation

Let be a locally strongly convex hypersurface, given by the graph of a convex function xn+1=f(x₁,…,xn) defined in a convex domain Ω⊂Rn. M is called a α-extremal hypersurface, if f is a solution of

     

Existence of solutions for fourth order differential equation with four-point boundary conditions

In this paper we investigate the existence of solutions of a class of four-point boundary value problems for a fourth order ordinary differential equation. Our analysis relies on a nonlinear alternative of Leray–Schauder type.     

Multiplicity of solutions for a class of fourth elliptic equations

We show that there exist at least three nontrivial solutions for a class of fourth elliptic equations under Navier boundary conditions by linking approaches.     

Supercritical biharmonic equations with power-type nonlinearity

We study two different versions of a supercritical biharmonic equation with a power-type nonlinearity. First, we focus on the equation Δ² u = |u| ^p-1 u over the whole space , where n > 4 and p > (n + 4)/(n − 4). Assuming that p < p _c, where p _c is a further critical exponent, we show that all regular radial solutions oscillate around an explicit singular radial solution. As it was already known, on the other hand, no such oscillations occur in the remaining case p ≥ p _c. We also study the Dirichlet problem for the equation Δ² u = λ (1 + u) ^p over the unit ball in , where λ > 0 is an eigenvalue parameter, while n > 4 and p > (n + 4)/(n − 4) as before. When it comes to the extremal solution associated to this eigenvalue problem, we show that it is regular as long as p < p _c. Finally, we show that a singular solution exists for some appropriate λ > 0.