Quantization for a fourth order equation with critical exponential growth

For concentrating solutions weakly in H ²(Ω) to the equation on a domain with Navier boundary conditions the concentration energy is shown to be strictly quantized in multiples of the number .     

Quantization for an evolution equation with critical exponential growth on a closed Riemann surface

In this paper, we analyze the concentration behavior of a positive solution to an evolution equation with critical exponential growth on a closed Riemann surface, and particularly derive an energy identity for such a solution. This extends a result of Lamm-Robert-Struwe and complements that of Yang.     

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:

Multiple positive solutions for a quasilinear elliptic equation with critical exponential nonlinearity

Let Ω be a bounded domain in R^N,N≥2, with C² boundary. In this work, we study the existence of multiple positive solutions of the following problem:

     

A priori bounds for an indefinite superlinear elliptic equation with exponential growth

This paper considers positive solutions to problem

     

Removability of singularity for an anisotropic elliptic equation of second order with nonstandard growth

Following Moser’s method we shall give a sufficient condition for removability of singularity for an anisotropic elliptic equation of second order with nonstandard growth.     

Nonlinear elliptic equations of order 2m and subdifferentials

LetA be an operator of the calculus of variations of order 2m onW ^m,p (Ω) andj a normal convex integrand. Forf ∈L ^p (Ω), the equation $$\mathcal{A}u + \partial j(x,u) \ni f, in \Omega , u - \phi \in W_0^{m,p} (\Omega ),$$ may have no strong solutions whenm>1, even ifj is independent ofx and φ=0. However, we obtain existence results whenj is everywhere finite and $$\int_\Omega {j(x,\phi ) dx< + \infty ,} $$ by the study of the subdifferential of the function $$\upsilon \mapsto \int_\Omega {j(x,\upsilon + \phi ) dx on W_0^{m,p} (\Omega ).} $$      

On a difference equation of the second order with an exponential coefficient

This paper is devoted to a generalization of some previous results, as we completely solve a linear homogeneous difference equation of the second order with an exponential coefficient.     

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.     

Infinitely many turning points for an elliptic problem with a singular non-linearity

We consider the problem – u = |x|/(1 – u)^p inB, u = 0 on B, 0 < u < 1 in B, where 0, p 1 and Bis the unit ball in ^N (N 2). We show that there exists a _*> 0 such that for < _*, the minimizer is the only positiveradial solution. Furthermore, if , then the branch of positive radial solutions must undergoinfinitely many turning points as the maximums of the radialsolutions on the branch converge to 1. This solves ConjectureB in [N. Ghoussoub and Y. Gun, SIAM J. Math. Anal. 38 (2007)1423–1449]. The key ingredient is the use of monotonicityformula.     

Two dimensional elliptic equation with critical nonlinear growth

We study the asymptotic behavior of solutions to a semilinear elliptic equation associated with the critical nonlinear growth in two dimensions.

where is a unit disk in and denotes a positive parameter. We show that for a radially symmetric solution of (1.1) satisfies

Moreover, by using the Pohozaev identity to the rescaled equation, we show that for any finite energy radially symmetric solutions to (1.1), there is a rescaled asymptotics such as

locally uniformly in . We also show some extensions of the above results for general two dimensional domains.

     

On some semilinear elliptic equation involving exponential growth

In this paper, we establish the existence of at least two nontrivial solutions for some semilinear elliptic equation involving a nonlinearity term having a critical exponential growth condition. Our main argument is critical point theory.