The purpose of the present paper is to study a class of semilinear elliptic Dirichlet boundary value problems in the ball, where the nonlinearities involve the sum of a sublinear variable exponent and a superlinear (may be supercritical) variable exponents of the form \(0\le f(r,u)\le a_1|u|^{p(r)-1}\), if \(u\ge 0\), where \(r = |x|\), \(p(r ) = 2^* + r^{\alpha }\), with \(\alpha > 0\), and \(2^* = 2N/(N - 2)\) is the critical Sobolev embedding exponent. We do not impose the Ambrosetti–Rabinowitz condition on the nonlinearity (or some additional conditions) to obtain Palais–Smale or Cerami compactness condition. We employ techniques based on the Galerkin approximations scheme, combining with a Sobolev type embeddings for radial functions into variable exponent Lebesgue spaces (due to do Ó et al. in Calc Var Partial Differ Equ 55:83, 2016), to establish the existence result.

本文的目的是研究球中的一类半线性椭圆Dirichlet边值问题，其中非线性涉及亚线性变量指数和形式为\（0 \的超线性（可能是超临界）变量指数的和。 le f（r，u）\ le a_1 | u | ^ {p（r）-1} \），如果\（u \ ge 0 \），其中\（r = | x | \），\（p（ r）= 2 ^ * + r ^ {\ alpha} \），其中\（\ alpha> 0 \）和\（2 ^ * = 2N /（N-2）\）是Sobolev嵌入的关键指数。我们没有将Ambrosetti–Rabinowitz条件施加于非线性（或某些其他条件）上以获得Palais–Smale或Cerami压实条件。我们采用基于Galerkin近似方案的技术，结合径向函数的Sobolev型嵌入到可变指数的Lebesgue空间中（由于Ó等人在Calc Var Partial Differ Equ 55:83，2016中进行了建立） 。