On a singular elliptic system with quadratic growth in the gradient

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:448

Article

Benrhouma, Mohamed¹ 

[1] Univ Monastir, Math Dept, Fac Sci Monastir, Monastir 5019, Tunisia

关键词: Singular elliptic systems;    Quadratic growth in the gradient;    Sub-supersolutions;    Variational method;    Nehari manifold;    Concentration-compactness principle;   

DOI  :  10.1016/j.jmaa.2016.11.038

来源: Elsevier

【 摘 要 】

In this paper, we deal with the following singular elliptic system: { -Delta u + alpha vertical bar del u vertical bar(2)/u = p/p+q alpha(x)vertical bar v vertical bar q vertical bar u vertical bar(p-2) u + f, x is an element of R-N, -Delta v + beta vertical bar del u vertical bar(2)/v = q/q+p alpha(x)vertical bar u vertical bar p vertical bar v vertical bar(q-2) v + g, x is an element of R-N, where N >= 3, alpha, beta > N+2/4, p, q > 1 and p + q <= N+2/N-2. We show through the sub- and supersolutions method, the existence of a nonnegative solution for an approximated system. The limit of the approximated solution is a positive solution. In the case, alpha= beta = 0, p = q and f = g, we prove the uniqueness of a solution. Among others, we prove some existence and uniqueness results for some auxiliary problems by using the comparison principle, a minimization method and with the help of Nehari manifold. The proofs rely on the concentration-compactness principle. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

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