On the existence of multiple solutions of nonhomogeneous elliptic equations involving critical Sobolev exponents
type: Item
instance of: scholarly article
title: On the existence of multiple solutions of nonhomogeneous elliptic equations involving critical Sobolev exponents
zbMATH Open document ID: 0849.35027
DOI: 10.1007/BF00917575
author: Huansong Zhou
author: Daomin Cao
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 1996-11-07T00:00:00Z
review text: This paper is concerned with the existence of at least two positive solutions for the following problem: \[ - \Delta u= c_1 u^{p- 1}+ f(x, u)+ h,\quad u> 0\quad \text{in } \Omega; \quad u= 0\quad \text{in } \partial\Omega, \] where \(c_1> 0\), \(\Omega\subset \mathbb{R}^N\) \((N\geq 3)\) is a bounded smooth domain, \(h\in H^{- 1}(\Omega)\), \(p= 2N/(N- 2)\) is the critical exponent of the embeddings from \(H^1_0(\Omega)\) to \(L^q(\Omega)\) and \(f\) satisfies some conditions. By using the super- and subsolution method, the existence of the minimum positive solution is proved. To obtain the second positive solution, the authors first apply the Mountain Pass Lemma ``around'' the minimum positive solution and get a sequence of approximated solutions, and then find a subsequence converging weakly to a limit which is another positive solution of this problem.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35J20
zbMATH DE Number: 859675
zbMATH Keywords: critical Sobolev exponents
zbMATH Keywords: minimum positive solution
reviewed by: Vasile Iftode
MaRDI profile type: Publication
cites work: A note on the problem \(-\Delta u=\lambda u+u\,| u| ^{2^*-2}\)
cites work: Positive solutions of nonlinear elliptic equations involving critical sobolev exponents
cites work: Monotone methods for semilinear elliptic equations in unbounded domains
cites work: Elliptic partial differential equations of second order
cites work: On nonhomogeneous elliptic equations involving critical Sobolev exponent
cites work: Existence of multiple positive solutions of inhomogeneous semilinear elliptic problems in unbounded domains
description: scientific article; zbMATH DE number 859675