Advances in Differential Equations

Existence results for some quasilinear elliptic equations involving critical Sobolev exponents

Hirokazu Ohya

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Abstract

In this paper we study the existence of solutions to zero-Dirichlet-boundary-value problems for the quasilinear elliptic equation ${\rm (QE)_c}$ $- \Delta_p u - p \nabla \theta(x) \cdot \nabla u |\nabla u|^{p-2} = \lambda a(x) |u|^{p-2}u + K(x)|u|^{p^*-2}u$ in an unbounded domain $\Omega \subset {\bf R}^N$ with smooth boundary $\partial \Omega$. By using Brézis-Nirenberg's results, we prove that ${\rm (QE)_c}$ admits at least one nontrivial weak solution for positive $\lambda$ in a suitable interval.

Article information

Source
Adv. Differential Equations Volume 9, Number 11-12 (2004), 1339-1368.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355867905

Mathematical Reviews number (MathSciNet)
MR2099559

Zentralblatt MATH identifier
05054510

Subjects
Primary: 35J60: Nonlinear elliptic equations
Secondary: 35B33: Critical exponents 35D05 35J20: Variational methods for second-order elliptic equations 47J30: Variational methods [See also 58Exx]

Citation

Ohya, Hirokazu. Existence results for some quasilinear elliptic equations involving critical Sobolev exponents. Adv. Differential Equations 9 (2004), no. 11-12, 1339--1368. https://projecteuclid.org/euclid.ade/1355867905.


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