在這篇文章中，我們探討涉及凹凸非線性的不定半線性橢圓方程正解的存在性與多解性，並且我們假設函數f,g和V滿足適當的條件。

In this paper, we study the existence, multiplicity of positive solutions for the indefinite semilinear elliptic equations involving concave-convex nonlinearities. We assume that the functions f,g and V satisfy suitable conditions with the potential V and the weight function g without the assumptions of infinite limits.

1Introduction
2 Preliminaries
2.1 Inequalities
2.1.1 Hölder inequality
2.1.2 Hardy inequality
2.1.3 Sobolev inequality
2.2 Variational methods and Theorems
2.3 Implicit Function Theorem
2.4 Mean Value Theorem
2.5 Brezis-Lieb Lemma
2.6 Variational Settings
3 Proof of Theorem 1.1

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