Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

Query:
Fill in the form and click »Search«...
Format:
Display: entries per page entries
Zbl 0614.35035
Cerami, G.; Solimini, S.; Struwe, M.
Some existence results for superlinear elliptic boundary value problems involving critical exponents.
(English)
[J] J. Funct. Anal. 69, 289-306 (1986). ISSN 0022-1236

This paper deals with the superlinear elliptic boundary value problem $$- \Delta u-\lambda u=u \vert u\vert\sp{2\sp*-2}\quad in\quad \Omega;\quad u\vert\sb{\partial \Omega}=0,$$ where $\Omega$ is a smoothly bounded domain in $R\sp n$, $n>2$, $\lambda\in R$, $2\sp*=2n/(n-2)$ and $2\sp*$ is the limiting Sobolev exponent for the embedding $H\sp 1\sb 0(\Omega)\to L\sp p(\Omega)$. Solving this problem is equivalent to finding critical points in $H\sp 1\sb 0(\Omega)$ of the energy functional $$I\sb{\lambda}(u)=(1/2)\int\sb{\Omega}(\vert \nabla u\vert\sp 2-\lambda \vert u\vert\sp 2)dx-(1/2)\int\sb{\Omega}\vert u\vert\sp{2\sp*} dx.$$ First, based on the global compactness theorem for the problem (1), the compactness question is discussed. Then some results about the existence and multiplicity of solutions to the above problem are obtained.
[Xu Zhenyuan]
MSC 2000:
*35J65 (Nonlinear) BVP for (non)linear elliptic equations
35J20 Second order elliptic equations, variational methods
35D05 Existence of generalized solutions of PDE
46E35 Sobolev spaces and generalizations

Keywords: superlinear elliptic boundary value problem; limiting Sobolev exponent; critical points; energy functional; global compactness; existence; multiplicity of solutions

Highlights
Master Server