arXiv subjects
Publications and source records attributed to A. Panda.
We prove the existence of infinitely many solutions to an elliptic problem by borrowing the techniques from algebraic topology. The solution(s) thus obtained will also be proved to be bounded.
In this paper, we prove the existence of multiple nontrivial solutions of the following equation. \begin{align*} \begin{split} -\Delta_{p}u & = \frac{\lambda}{u^{\gamma}}+g(u)+\mu~\mbox{in}\,\,\Omega, u & = 0\,\, \mbox{on}\,\, \partial\Omega, u&>0 \,\,\mbox{in}\,\,\Omega, \end{split} \end{align*} where $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain with $N \geq 3$, $1 < p-1 < q$ , $ \lambda>0$, $\gamma>0$, $g$ satisfies certain conditions, $\mu\geq 0$ is a bounded Radon measure.
The aim of this paper is to prove the existence of solution for a partial differential equation involving a singularity with a general nonnegative, Radon measure $\mu$ as its nonhomogenous term which is given as \begin{eqnarray} -\Delta u&=& f(x)h(u)+\mu~\text{in}~\Omega,\nonumber u&=&0~\text{on}~\partial\Omega,\nonumber u&>& 0~\text{on}~\Omega\nonumber, \end{eqnarray} where $\Omega$ is a bounded domain of $\mathbb{R}^N$, $f$ is a nonnegative function over $\Omega$.