Existence results to a nonlinear p(x)-Laplace equation with degenerate coercivity and zero-order term: renormalized and entropy solutions

Abstract

This paper is devoted to the study of a nonlinear elliptic equation with degenerate coercivity, zero-order term and L¹ datum in the setting of variable exponent Sobolev space. The authors obtain the existence of renormalized solution and entropy solution via strong convergence of truncation. They also get some a priori estimates to the solution and the generalized gradient of the solution in the Marcinkiewicz space with variable exponent.

Keywords:

• p(x)-Laplace equation
• lower order term
• L¹ datum
• truncation method
• Marcinkiewicz estimates with variable exponent

AMS Subject Classifications:

• 35J70
• 35J60
• 46E35

Acknowledgements

We are particularly grateful to the anonymous referees for their valuable comments on and suggestions regarding the original manuscript.

Additional information

Funding

Supported by National Natural Science Foundation of China [grant number 11271154], [grant number 11401252], by Key Lab of Symbolic Computation and Knowledge Engineering of Ministry of Education. The first author is also supported by the Graduate Innovation Fund of Jilin University [grant number 2014084].