In this paper, we are concerned with the following nonlinear magnetic Schrödinger equation with critical growth: \(\left\{ {\matrix{ {{{\left( {{\varepsilon \over i}\nabla - A\left( x \right)} \right)}^2}u + V\left( x \right)u = f\left( {{{\left| u \right|}^2}} \right)u + {{\left| u \right|}^{{2^*} - 2}}u\,\,\,\,\,\,{\rm{in}}\,{\mathbb{R}^N},} \hfill \cr {u \in {H^1}\left( {{\mathbb{R}^N},\mathbb{C}} \right),} \hfill \cr } } \right.\) where ∊ > 0 is a parameter, N ≥ 3 and \({2^*} = {{2N} \over {N - 2}}\) is the Sobolev critical exponent, V: ℝ^N → ℝ and A: ℝ^N → ℝ^N are continuous potentials, f: ℝ → ℝ is a subcritical nonlinear term. Under a local assumption on the potential V, by the variational methods, the penalization technique and the Ljusternic—Schnirelmann theory, we prove the multiplicity and concentration of nontrivial solutions of the above problem for ε small. For the problem, the function f is only continuous, which allows to consider larger classes of nonlinearities in the reaction.

在本文中，我们关注具有临界增长的以下非线性磁性Schrödinger方程：\（\ left \ {{\ matrix {{{{\ left（{{\ varepsilon \ over i} \ nabla-A \ left（x \ right）} \ right）} ^ 2} u + V \ left（x \ right）u = f \ left（{{{\ left | u \ right |} ^ 2}} \ right）u + {{\ left | u \ right |} ^ {{{2 ^ *}-2}} u \，\，\，\，\，\，{\ rm {in}} \，{\ mathbb {R} ^ N}， } \ hfill \ cr {u \ in {H ^ 1} \ left（{{\ mathbb {R} ^ N}，\ mathbb {C}} \ right），} \ hfill \ cr}} \ right。\）其中ε> 0是参数，ñ ≥3和\（{2 ^ *} = {{2N} \过度{N - 2}} \）是Sobolev临界指数，V：ℝ ^ñ →ℝ和阿：ℝ ^ñ →ℝ ^ñ是连续的电位，˚F：ℝ→ℝ是亚临界非线性项。在势V的局部假设下，通过变分方法，惩罚技术和Ljusternic-Schnirelmann理论，证明了上述问题对于ε小的非平凡解的多重性和集中性。对于该问题，函数f仅是连续的，这允许考虑反应中更大的非线性类别。