| With the development of elastic mechanics and some other physics problems, the emergence of more nonlinear problems in engineering made people begein to pay more attention to a class of nonlinear problems with variable exponential growth conditions gradually. The mathematical models that the practical problems depend on are generally a class of Partial Differential Equation with variable exponential growth conditions, solving this kind of equations has more important theoretical basis after the establishment of variable exponential function spaces and the consummation of the variable exponential function spaces theory. At the same time, quantum mechanics develops very fast, people begin to be fond of solving a class of p - Laplace equations with Hardy potential, solving this kind of equations need to use Hardy inequality, the establishment of Hardy inequality in variable exponential Sobolev space plays an important role.Based on the theory of variable exponential Sobolev space, This paper will discusses the following quasilinear elliptic equation with Hardy potential and variable exponential growth conditions in bounded domain Ω. where p(x) is Lipschitz continuous in Ω and satisfies 1< p_≤p(x)≤p+<N, υ is a positive constant,f is a Caratheodry function and satisfies some appropriate conditions, δ:=dist(x,(?)Ω) is a distance function.Firstly, due to the existence of the distance function in this equation, when the point in Ω gets closed to the boundary, the equation will have strangeness, so in this paper we will use the Hardy inequality to solve this problem. Then we categorized this equation by the structural conditions which the nonlinear term needs to satisfy, Finally, we take advantage of the mountain pass lemma, symmetric mountain pass lemma, Hardy inequality and the critical point theory to get the existence of the weak solution, when f satisfies the mixed type condition, we will get the multiplicity of weak solutions. |
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