On a class of nonhomogeneous elliptic equations on noncompact Riemannian manifolds

ABSTRACT

Let (M, g) be a complete noncompact Riemannian $n-$manifold $(n\ge 2)$. In this paper, a fixed point result and a version of the Trudinger–Moser inequality are employed to establish sufficient conditions for the existence of solutions for quasilinear nonhomogeneous elliptic equation of the type $$-{\text{div}}_g{(|\mathrm{\nabla }u|}^{n-2}{\mathrm{\nabla }u)+V(x)|u|}^{n-2}u=\varphi (x)f(x,u)+\lambda h\text{in}M.$$

Here the function V(x) can change sign, f(x, u) is possibly discontinuous and can enjoy exponential critical growth and h belongs to the dual of an appropriated function space.

KEYWORDS:

• Critical growth
• Trudinger–Moser inequality
• fixed point
• discontinuous nonlinearity

AMS SUBJECT CLASSIFICATIONS:

• 35J92
• 47H10
• 35B33

Acknowledgements

The author would like to thank the referees for the careful review and the valuable comments, which provided insights that helped improve the paper. The author also would like to thank Princeton University, for its hospitality while part of this work was completed, and the Federal University of Paraíba for supporting his long term visit to Princeton University during the academic year 2017–2018.

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

Research supported by CAPES [grant number 88881.119057/2016-01]; CNPq [grant number 06498/2016-2].