We are interested in quasilinear problems as follows:

$$ \left\{ \begin{array}{ll} -\Delta u -u \Delta (u^2)= -\lambda |u|^{q-2}u+|u|^{22^*-2}u+\mu g(x,u), \quad \mathrm{in}~ \Omega ,\\ u=0,\quad \mathrm{on}~ \partial \Omega , \end{array}\right. $$(p)

where \(\Omega \subset \mathbb {R}^N \)is a bounded domain with regular boundary \(\partial \Omega , N\ge 3, \lambda , \mu > 0,1<q<4,22^*:=4N/(N-2)\) and g has a subcritical growth and possesses a condition of monotonicity. We prove a regularity result for all weak solutions for a modified problem associated to (p) and, introducing a new type of constraint, we demonstrate a multiplicity result for solutions, including a ground state.

中文翻译：

---

在原点涉及临界指数和凹非线性的拟线性方程

我们对准线性问题感兴趣，如下所示：

$$ \ left \ {\ begin {array} {ll}-\ Delta u -u \ Delta（u ^ 2）=-\ lambda | u | ^ {q-2} u + | u | ^ {22 ^ *- 2} u + \ mu g（x，u），\ quad \ mathrm {in}〜\ Omega，\\ u = 0，\ quad \ mathrm {on}〜\ partial \ Omega，\ end {array} \ right。$$（p）

其中\（\ Omega \ subset \ mathbb {R} ^ N \）是具有规则边界\（\ partial \ Omega，N \ ge 3，\ lambda，\ mu> 0,1 <q <4,22 ^ *：= 4N /（N-2）\）， 并且g具有亚临界增长并且具有单调性的条件。我们证明了与（p）相关的修正问题的所有弱解的正则结果，并引入一种新型约束，我们证明了包括基态在内的解的多重性结果。