Abstract We consider the existence of solutions of the following weighted problem: { L : = - d i v ( ρ ( x ) | ∇ u | N - 2 ∇ u ) + ξ ( x ) | u | N - 2 u = f ( x , u ) i n B u > 0 i n B u = 0 o n ∂ B , \left\{ {\matrix{{L: = - div\left( {\rho \left( x \right){{\left| {\nabla u} \right|}^{N - 2}}\nabla u} \right) + \xi \left( x \right){{\left| u \right|}^{N - 2}}} \hfill & {u = f\left( {x,u} \right)} \hfill & {in} \hfill & B \hfill \cr {} \hfill & {u > 0} \hfill & {in} \hfill & B \hfill \cr {} \hfill & {u = 0} \hfill & {on} \hfill & {\partial B,} \hfill \cr } } \right. where B is the unit ball of ℝ N , N #62; 2, ρ ( x ) = ( log e | x | ) N - 1 \rho \left( x \right) = {\left( {\log {e \over {\left| x \right|}}} \right)^{N - 1}} the singular logarithm weight with the limiting exponent N − 1 in the Trudinger-Moser embedding, and ξ(x) is a positif continuous potential. The nonlinearities are critical or subcritical growth in view of Trudinger-Moser inequalities of double exponential type. We prove the existence of positive solution by using Mountain Pass theorem. In the critical case, the function of Euler Lagrange does not fulfil the requirements of Palais-Smale conditions at all levels. We dodge this problem by using adapted test functions to identify this level of compactness.

中文翻译：

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双指数增长的非自治加权椭圆方程

| ∇ 你 | ñ | 你| N - 2 \left\{ {\matrix{{L: = - div\left( {\rho \left( x \right){{\left| {\nabla u} \right|}^{N - 2}}\nabla u} \right) + \xi \left( x \right){{\left| u \right|}^{N - 2}}} \hfill & {u = f\left( {x,u} \right)} \hfill & {in} \hfill & B \hfill \cr {} \hfill & {u > 0} \hfill & {in} \hfill & B \hfill \cr {} \hfill & {u = 0} \hfill & {on} \hfill & {\partial B,} \hfill \cr } } \正确的。其中 B 是 ℝ N , N #62 的单位球；2、x | ) N - 1 \rho \left( x \right) = {\left( {\log {e \over {\left| x \right|}}} \right)^{N - 1}} 奇异对数 在 Trudinger-Moser 嵌入中具有极限指数 N - 1 的权重，而 ξ(x) 是一个正连续势。鉴于双指数类型的 Trudinger-Moser 不等式，非线性是临界或亚临界增长。我们用山口定理证明了正解的存在。在危急情况下，欧拉拉格朗日函数不能满足各级Palais-Smale条件的要求。我们通过使用经过调整的测试函数来识别这种紧凑程度来避免这个问题。在危急情况下，欧拉拉格朗日函数不能满足各级Palais-Smale条件的要求。我们通过使用经过调整的测试函数来识别这种紧凑程度来避免这个问题。在危急情况下，欧拉拉格朗日函数不能满足各级Palais-Smale条件的要求。我们通过使用经过调整的测试函数来识别这种紧凑程度来避免这个问题。