THE EXISTENCE OF MULTIPLE SOLUTIONS OF p-LAPLACIAN ELLIPTIC EQUATION

Abstract

Abstract:

In this paper, we consider the quasilinear elliptic equation(− △p u = |u|m−1u + |u|q−1u, x ∈ ,u ∈ W 1,p0 (),(1) Where −△pu = −div(|▽u|p−2▽u), and 0 < m < p−1 < q < +∞, is a bounded domain in RN(N ≥ 3). is a positive number. Our object is to estimate exactly the magnitute of
such that (1) has at least one positive solution if ∈ (0, ) and no positive solutions if > . Furthermore, (1) has at least one positive solution when = , and at least two positive solutions when ∈ (0, ) and q ≤ Np N−p −1. Finally, we obtain a multiplicity result with positive energy of (1) when 0 < m < p − 1 < q = Np N−p − 1.

Key words: Quasilinear elliptic equation, super- and subsolution method, critical Sobolev exponent, positive solutions, multiple solutions

CLC Number:

35J20

TAN Zhong, YAO Zheng-An. THE EXISTENCE OF MULTIPLE SOLUTIONS OF p-LAPLACIAN ELLIPTIC EQUATION[J].Acta mathematica scientia,Series B, 2001, 21(2): 203-212.

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