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Multiplicity of Solutions for a Class of Elliptic Problem of - Laplacian Type with a - Gradient Term

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  • Zakariya Chaouai
  • Soufiane Maatouk

Abstract

We consider the following problem: where is a bounded set in ( ) with a smooth boundary, , , , and and belong to for some . In this paper, we assume that a.e. in and without sign condition and then we prove the existence of at least two bounded solutions under the condition that and are suitably small. For this purpose, we use the Mountain Pass theorem, on an equivalent problem to with variational structure. Here, the main difficulty is that the nonlinearity term considered does not satisfy Ambrosetti and Rabinowitz condition. The key idea is to replace the former condition by the nonquadraticity condition at infinity .

Suggested Citation

  • Zakariya Chaouai & Soufiane Maatouk, 2019. "Multiplicity of Solutions for a Class of Elliptic Problem of - Laplacian Type with a - Gradient Term," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2019, pages 1-10, February.
    • Handle: RePEc:hin:jijmms:6824502
    DOI: 10.1155/2019/6824502
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