Electron. J. Diff. Eqns.,
Vol. 2007(2007), No. 90, pp. 114.
Infinitely many weak solutions for a
Laplacian equation
with nonlinear boundary conditions
JiHong Zhao, PeiHao Zhao
Abstract:
We study the following quasilinear problem with
nonlinear boundary conditions
where
is a bounded domain in
with smooth
boundary and
is the outer normal
derivative,
is the
pLaplacian with 1<p<N. We consider the above problem under
several conditions on f and g, where f and g are both
Caratheodory functions. If f and g are both superlinear
and subcritical with respect to u, then we prove the existence
of infinitely many solutions of this problem by using "fountain
theorem" and "dual fountain theorem" respectively. In the case,
where g is superlinear but subcritical and f is critical with
a subcritical perturbation, namely
, we show that there
exists at least a nontrivial solution when
and there
exist infinitely many solutions when 1<r<p, by using
"mountain pass theorem" and "concentrationcompactness principle"
respectively.
Submitted March 26, 2007. Published June 15, 2007.
Math Subject Classifications: 35J20, 35J25.
Key Words: pLaplacian; nonlinear boundary conditions; weak solutions;
critical exponent; variational principle.
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JiHong Zhao
Department of Mathematics
Lanzhou University
Lanzhou, 730000, China
email: zhaojihong2007@yahoo.com.cn


PeiHao Zhao
Department of Mathematics
Lanzhou University
Lanzhou, 730000, China
email: zhaoph@lzu.edu.cn

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