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arXiv 1909.02822math.AP
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具有有限或无限深势阱的拟线性椭圆问题

On quasilinear elliptic problems with finite or infinite potential wells

Shibo Liu

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21 pages
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摘要：

本文考虑拟线性椭圆问题的形式\[ -\operatorname{div}\big(ϕ(|\nabla u|)\nabla u\big)+V(x)ϕ(|u|)u=f(u)\qquad u\in W^{1,Φ}(\mathbb{R}^{N}), \]其中$ϕ$和$f$满足适当的条件。正势$V\in C(\mathbb{R}^{N})$表现出有限或无限势阱，因为$V(x)$趋向于其上确界$V_{\infty}\le+\infty$为$|x|\to\infty$。通过变分方法得到了非平凡解。当$V_{\infty }=+\infty$时，建立了从$W^{1,Φ}(\mathbb{R}^{N})$的适当子空间到$L^Φ(\mathbb{R}^{N})$的紧嵌入，这使得我们能够在$f$是奇数的情况下得到无穷多个解。对于$V(x)=λa(x) + 1$具有由正参数$λ$控制的陡势阱的情形，我们得到了大$λ$的非平凡解。

英文摘要：

We consider quasilinear elliptic problems of the form \[ -\operatorname{div}\big(ϕ(|\nabla u|)\nabla u\big)+V(x)ϕ(|u|)u=f(u)\qquad u\in W^{1,Φ}(\mathbb{R}^{N}), \] where $ϕ$ and $f$ satisfy suitable conditions. The positive potential $V\in C(\mathbb{R}^{N})$ exhibits a finite or infinite potential well in the sense that $V(x)$ tends to its supremum $V_{\infty}\le+\infty$ as $|x|\to\infty$. Nontrivial solutions are obtained by variational methods. When $V_{\infty }=+\infty$, a compact embedding from a suitable subspace of $W^{1,Φ}(\mathbb{R}^{N})$ into $L^Φ(\mathbb{R}^{N})$ is established, which enables us to get infinitely many solutions for the case that $f$ is odd. For the case that $V(x)=λa(x) + 1$ exhibits a steep potential well controlled by a positive parameter $λ$, we get nontrivial solutions for large $λ$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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