p-LAPLACE EQUATIONS WITH MULTIPLE CRITICAL EXPONENTS AND SINGULAR CYLINDRICAL POTENTIAL

doi:10.1016/S0252-9602(13)60066-1

Abstract

Abstract:

In this paper, we deal with the following problem:
{−Δ_pu − λ|y|^−p|u|^p−2u = |y|^−s|u|p*^(s)−2u + |u|^p*−2u in R^N, y ≠ 0,
u ≥ 0,
where u(x) = u(y, z) : R^m × R^N−m −→ R, N ≥ 3, 2 < m < N, 1 < p < m, λ <( (m−p)/p )^pand 0 < s < p, p*(s) = p(N−s)/N−p , p* = pN/N−p . By variational method, we prove the existence of a nontrivial weak solution when 0 < λ <( (m−p/ p ))^p and the existence of a cylindrical weak solution when λ < 0.

Key words: p-Laplace equation, cylindrical potential, critical exponents

CLC Number:

35J20

SUN Xiao-Mei. p-LAPLACE EQUATIONS WITH MULTIPLE CRITICAL EXPONENTS AND SINGULAR CYLINDRICAL POTENTIAL[J].Acta mathematica scientia,Series B, 2013, 33(4): 1099-1112.

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