THE EXISTENCE OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS WITH CHANGE OF SIGN

Abstract

Abstract:

This paper considers the following quasilinear elliptic problem( −div(|∇u|p−2∇u) = a(x)g(u) in u = 0 on @ where is a bounded regular domain in RN(N≥3), N > p > 1. When g(u) satisfies suitable conditions and g(u)u − R u 0g(s)ds is unbounded, a(x) is a H¨older continuous function which changes sign on and R − |a(x)|dx is suitably small. The authors prove the existence of a nonnegative nontrivial solution for N > p > 1, in particular, the existence of a positive solution to the problem for N > p≥2. Our main theorem generalizes a recent result of Samia Khanfir and Leila Lassoued (see [1]) concerning the case where p = 2.They prove also that if g(u) = |u|q−2u with p < q < p and + = {x∈|a(x) > 0} is a nonempty open set, then the above problem possesses infinitely many solutions.

Key words: Qasilinear elliptic equation, (PS) condition, mountain-pass Lemma, infinite solution

CLC Number:

35J60

LI Gong-Bao, YU Chun. THE EXISTENCE OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS WITH CHANGE OF SIGN[J].Acta mathematica scientia,Series B, 2001, 21(4): 469-482.

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References

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