Generalized n-Laplacian: boundedness of weak solutions to the Dirichlet problem with nonlinearity in the critical growth range

作者：Cerny Robert^*

来源：Central European Journal of Mathematics, 2014, 12(1): 114-127.

DOI：10.2478/s11533-013-0329-2

• Orlicz-Sobolev spaces
• Trudinger embedding
• Moser-Trudinger inequality
• Moser iteration

Let n >= 2 and let Omega subset of R-n be an open set. We prove the boundedness of weak solutions to the problem
u is an element of(W0L phi)-L-1(Omega) and -dtv(phi(vertical bar del u vertical bar)del u/vertical bar del u vertical bar) + V(x)phi'(vertical bar u vertical bar)u/vertical bar u vertical bar = f(x,u) + mu h(x) in Omega,
where Omega is a Young function such that the space (W0L phi)-L-1(Omega) is embedded into an exponential or multiple exponential Orlicz space, the nonlinearity f(x, t) has the corresponding critical growth, V(x) is a continuous potential, h is an element of L-phi(Omega) is a non-trivial continuous function and A mu a parts per thousand yen 0 is a small parameter. We consider two classical cases: the case of Omega being an open bounded set and the case of Omega = R-n .

• 出版日期2014-1