$\mathbb{R}^n$ 上 $p(x)$-Laplace型椭圆问题的无穷多解

讨论了涉及一般散度型椭圆算子($p(x)$-Laplace算子为其特例) 非线性偏微分方程的弱解存在性和多解性问题, 假定非线性项 $f_1,\, f_2$ 其中之一是超线性的, 且满足 Ambrosetti-Rabinowitz 条件, 另一项是次线性的. 所采用的方法依赖于变指数 Sobolev 空间 $W^{1,p(x)}(\mathbb{R}^n)$ 理论. 主要结果的证明基于喷泉定理和对偶喷泉定理.

Infinitely Many Solutions to $p(x)$-Laplace Type Elliptic Problems in $\mathbb{R}^n$

In this paper, the existence and multiplicity of weak solutions to nonlinear partial differential equations involving a general elliptic operator in divergence form (in particular, a $p(x)$-Laplace operator) in $\mathbb{R}^n$ are investigated, assumed that one of the nonlinear terms $f_1$ and $ f_2$ is superlinear and satisfies the Ambrosetti-Rabinowitz type condition and another one is sublinear. Our approach relies on the theory of variable exponent Sobolev space $W^{1,p(x)}(\mathbb{R}^n)$. The proofs of our main results are based on the Fountain theorem and the Dual Fountain theorem.