具有不定权和超指数非线性项的Minkowski曲率方程的正解
Positive solutions for a Minkowski-curvature equation with indefinite weight and super-exponential nonlinearity
摘要：
研究了一类具有不定权和非线性项的Minkowski曲率方程正解的存在性，该方程在零点处具有超线性增长，在无穷远处具有超指数增长性.作为例子，对于方程\begin{equation*} \Biggl{(} \dfrac{u'}{\sqrt{1-(u')^{2}}}\Biggr{)}' + a(t) \bigl{(}e^{u^{p}}-1\bigr{)} = 0, \end{equation*}，其中$p > 1$和$a(t)$是一个满足均值条件$\int_{0}^{T} a(t)\,\mathrm{d}t < 0$的变号函数，我们证明了在周期边界条件和Neumann边界条件下正解的存在性.证明依赖于拓扑度技术。
英文摘要：
We investigate the existence of positive solutions for a class of Minkowski-curvature equations with indefinite weight and nonlinear term having superlinear growth at zero and super-exponential growth at infinity. As an example, for the equation \begin{equation*} \Biggl{(} \dfrac{u'}{\sqrt{1-(u')^{2}}}\Biggr{)}' + a(t) \bigl{(}e^{u^{p}}-1\bigr{)} = 0, \end{equation*} where $p > 1$ and $a(t)$ is a sign-changing function satisfying the mean-value condition $\int_{0}^{T} a(t)\,\mathrm{d}t < 0$, we prove the existence of a positive solution for both periodic and Neumann boundary conditions. The proof relies on a topological degree technique.
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