论文标题
Riemannian歧管上的Schrödinger方程的多个解决方案通过$ \ nabla $ -Theorems
Multiple solutions for Schrödinger equations on Riemannian manifolds via $\nabla$-theorems
论文作者
论文摘要
我们认为尺寸$ d \ geq 3 $的平滑，完整和非紧密的Riemannian歧管$（\ Mathcal {m}，g）$，我们为半线性椭圆方程寻找积极的解决方案 $$ -Δ_Gw + v w =αf（w） +λw\ quad \ hbox {in $ \ mathcal {m} $}。 $$ 潜在的$ v \ colon \ mathcal {m} \ to \ mathbb {r} $是一个连续的功能，从适当的意义上讲是强制性的，而非线性$ f $在Sobolev嵌入的意义上具有亚临界增长。通过Marino和Saccon引入的$ \ nabla $ - 理论，我们证明，一旦参数$λ$就足够接近操作员$-Δ_G$的特征值，至少存在三个解决方案。
We consider a smooth, complete and non-compact Riemannian manifold $(\mathcal{M},g)$ of dimension $d \geq 3$, and we look for positive solutions to the semilinear elliptic equation $$ -Δ_g w + V w = αf(w) + λw \quad\hbox{in $\mathcal{M}$}. $$ The potential $V \colon \mathcal{M} \to \mathbb{R}$ is a continuous function which is coercive in a suitable sense, while the nonlinearity $f$ has a subcritical growth in the sense of Sobolev embeddings. By means of $\nabla$-Theorems introduced by Marino and Saccon, we prove that at least three solution exists as soon as the parameter $λ$ is sufficiently close to an eigenvalue of the operator $-Δ_g$.
