Semilinear elliptic problems with singular terms on the Heisenberg group

ABSTRACT

We consider the semilinear elliptic problem: $\begin{array}{rl}& -{\mathrm{\Delta }}_{{\mathbb{H}}^n}u-\mu \frac{g\left(\xi \right)u}{(|z{|}^4+t^2{)}^{1/2}}=\frac{\lambda f\left(\xi \right)}{u^{\gamma }}+h\left(\xi \right)u^p\mathrm{in}\mathrm{\Omega },\\ & u>0\mathrm{in}\mathrm{\Omega },\\ & u=0\mathrm{on}\mathrm{\partial \Omega }.\end{array}$ where $\mathrm{\Omega }\subset {\mathbb{H}}^n$ is an open bounded subset, $N\ge 3;0\in \mathrm{\Omega },0\le f,g,h\in L^{\mathrm{\infty }}\left(\mathrm{\Omega }\right).$ Under assumptions $\gamma ,p>0,\lambda >0$ small we show that there exists a solution $u\in H_0^1(\mathrm{\Omega },{\mathbb{H}}^n)$ $\cap L^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ to this problem. The interesting point of the problem under consideration is that it has a strong singularity.

Keywords:

• Nonlinear elliptic equation
• Hardy potential
• positive solution

AMS SUBJECT CLASSIFICATIONS:

• Primary 35J05
• 35J25
• Secondary 46E35

Acknowledgments

The author thanks the referee for useful comments and also thanks Dr Jagmohan Tyagi for suggesting this problem and several helpful detailed discussions. Without his help, I could not have written this paper.

Disclosure statement

No potential conflict of interest was reported by the author.