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arXiv · 1606.01370

Elliptic Problems in $\mathbb{R}^N$ with Critical and Singular Discontinuous Nonlinearities

R. Dhanya·S. Prashanth·Sweta Tiwari·K. Sreenadh
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Abstract

Let $Ω$ be a bounded domain in $\mathbb R^{N}$, $N\geq3$ with smooth boundary, $a>0, λ>0$ and $0<δ<3$ be real numbers. Define $2^*:=\displaystyle\frac{2N}{N-2}$ and the characteristic function of a set $A$ by $χ_A$. We consider the following critical problem with singular and discontinuous nonlinearity: \begin{eqnarray*} (P_\la^a)~~~~ \qquad \Biggl\{\begin{array}{rl} -Δu &= λ\left(u^{2^*-1}+ \displaystyle χ_{\{u 0~~\text{in} ~~Ω, \\ u & = 0 ~\text{on}~ \partial Ω. \end{array} \end{eqnarray*} \noindent We study the existence and the global multiplicity of solutions to the above problem.

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R. Dhanya, S. Prashanth, Sweta Tiwari, K. Sreenadh. 2016-06-04. Elliptic Problems in $\mathbb{R}^N$ with Critical and Singular Discontinuous Nonlinearities. https://doi.org/10.1080/17476933.2016.1198787

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