AI 中文总结
本文针对具p-拉普拉斯算子与不定势或临界非线性项的双调和方程，利用局部环绕和莫尔斯理论得到非平凡解，对奇非线性项得大能量解，对有界正势与临界/次临界指数非线性项得多个解。
AI 中文摘要
本文研究形如$$ \begin{cases} \triangle^2 u - \triangle_p u + V(x)u = f(x,u),\thinspace u \thinspace \text{∈} \thinspace H^2(\thinspace \text{ℝ}^N \thinspace) \thinspace \text{,} \thinspace \text{其中} \thinspace p \thinspace \text{≥} \thinspace 2 \thinspace \text{，势函数} \thinspace V(x) \thinspace \text{可以是不定的。} \thinspace \text{利用局部环绕和莫尔斯理论，我们得到了非平凡解。} \thinspace \text{当非线性项} \thinspace f(x,\thinspace \text{·}) \thinspace \text{为奇函数时，我们得到一列大能量解。} \thinspace \text{在本文第二部分，对于有界正势函数，当} \thinspace f(x,u)=\thinspace \text{λ} \thinspace g(x)|\thinspace u \thinspace|^{q-2}u \thinspace + \thinspace |\thinspace u \thinspace|^{m-2}u \thinspace \text{且指数} \thinspace m \thinspace \text{为临界或次临界时，我们得到多个解。
英文摘要
In this paper we consider nonlinear biharmonic equations with $p$-Laplacian ($p\ge2$) of the form $$ \left\{ \begin{array}{l} Δ^2 u - Δ_p u + V (x) u = f (x, u) \text{,} u \in H^2 (\mathbb{R}^N) \text{,} \end{array} \right. $$ where the potential $V(x)$ may be indefinite. Using local linking and Morse theory, nontrivial solutions are obtained. In case the nonlinearity $f(x,\cdot)$ is odd, we obtain a sequence of large energy solutions. In the second part of the paper, for bounded positive potential, we get multiple solutions for the case that $$f(x,u)=λg (x) | u |^{q - 2} u + | u |^{m - 2} u$$ with exponent $m$ critical or subcritical.
