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Quasilinear Equations Involving Critical Exponent and Concave Nonlinearity at the Origin
Milan Journal of Mathematics ( IF 1.2 ) Pub Date : 2020-07-15 , DOI: 10.1007/s00032-020-00315-6
Giovany M. Figueiredo , R. Ruviaro , J.C. Oliveira Junior

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。

更新日期:2020-07-15
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