We consider the following Hénon equation with critical growth... Furthermore, we prove that equation (*) has infinitely many nonradial solutions, whose energy can be made arbitrarily large.

The study of the Schrödinger equation is one of the main objects of quantum physics. The evolution of a group of identical particles interacting with each other in ultra-cold states, in particular, Bose-Einstein condensates, is described via Hartree approximation, to an excellent degree of accuracy by nonlinear Schrödinger equations. The equation also arises in many fields of physics. For instance, when we describe the propagation of light in some nonlinear optical materials, the nonlinear Schrödinger equations in nonlinear optics are reduced from Maxwell's equations.

We study positive solutions of the equation... where n = 3, 4.5 and ∊ > 0 is small, with Neumann boundary condition in a unit ball B. We prove the existence of solutions with an interior bubble at the center and a boundary layer at the boundary ∂ B. To cite this article: J. Wei, S. Yan, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

We show how a change of variable and peak solution methods can be used to prove that a number of nonlinear elliptic partial differential equations have many solutions.

The aim of this paper is to study the existence of various types of peak solutions for an elliptic system of FitzHugh-Nagumo type. We prove that the system has a single peak solution, which concentrates near the boundary of the domain. Under some extra assumptions, we also construct multi-peak solutions with all the peaks near the boundary, and a single peak solution with its peak near an interior point of the domain.

We show that the critical problem {-∆υ + λυ = υ²*⁻¹ + αυq⁻¹, υ > 0 in Ω, ∂υ∕∂ν = 0 on ∂Ω, 2 < q < 2* = 2N/(N − 2), has no positive solutions concentrating, as λ → ∞, at interior points of Ω if a = 0, but for a class of symmetric domains Ω, the problem with α > 0 has solutions concentrating at interior points of Ω.

We consider an elliptic problem of Ambrosetti-Prodi type involving critical Sobolev exponent on a bounded smooth domain six or higher. By constructing solutions with many sharp peaks near the boundary of the domain, but not on the boundary, we prove that the number of solutions for this problem is unbounded as the parameter tends to infinity, thereby proving the Lazer-McKenna conjecture in the critical case.

In this paper, we consider the following problem involving fractional Laplacian operator:(−Δ)αu=|u|2∗α-2-εu+λu in Ω, u = 0 on ∂Ω,(1) where Ω is a smooth bounded domain in RN, ε ∈[0, 2∗α−2), 0 < α < 1, 2∗α = 2N N-2α, and (−Δ)α is either the spectral fractional Laplacian or the restricted fractional Laplacian. We show for problem (1) with the spectral fractional Laplacian that for any sequence of solutions un of (1) corresponding to εn ∈ [0, 2∗α - 2), satisfying ǁunǁH ≤ C in the Sobolev space H defined in (1.2), un converges strongly in H provided that N > 6α and λ > 0. The same argument can also be used to obtain the same result for the restricted fractional Laplacian. An application of this compactness result is that problem (1) possesses infinitely many solutions under the same assumptions.

In this paper, we consider the planar vortex patch problem in an incompressible steady flow in a bounded domain Ω of R2...

We prove that certain weakly nonlinear elliptic equations have many solutions when a paragraph is large. The nonlinearity grows superlinearly for y positive but grows linearly for y negative.