具超线性增长非线性项的拟线性椭圆型方程共振问题

Resonance Problem of a Quasilinear Elliptic Quation with Superlinear Nonlinearities

在非线性项具有超线性增长条件下,研究了拟线性椭圆型方程的共振问题.通过建立拟线性算子与线性算子的一种关系,依据Shapiro在加权Sobolev空间中建立的紧嵌入定理和推广的Brouwer定理,运用截断方法证明了近似方程的解存在;借助Sobolev理论、Fatou引理和Lebesgue控制收敛定理证明了上述近似解一致有界;利用投影技巧和Galerkin方法得到共振问题的非平凡解的存在性.

The resonance problem of a quasilinear elliptic equation with superlinear nonlinearities were focused.By establishing the relationship between the quasilinear operators and linear operators,according to the Shapiro-type compact embedding theorem and Brouwer＇s theorem,the existence of solutions of the approximate equation was revealed.With the help of the Sobolev theory,Fatou＇s Lemma and Lebesgue dominated convergence theorem,the uniform boundness of the approximate solutions was proved.By using the projection technique and the Galerkin method,the existence of nontrivial solutions of the resonance problem was revealed.