arXiv subjects

S. Prashanth

Publications and source records attributed to S. Prashanth.

3 recordsLinked to original sources

arXiv2018

Radial singular solutions for the N-Laplace Equation with exponential nonlinearities

In this paper, we consider radial distributional solutions of the quasilinear equation $-Δ_N u=f(u)$ in the punctured open ball $ B_R\backslash\{0\}\subset \RR^N$, $N \geq 2$. We obtain sharp conditions on the nonlinearity $f$ for extending such solutions to the whole domain $B_R$ by preserving the regularity. For a certain class of noninearity $f$ we obtain the existence of singular solutions and deduce upper and lower estimates on the growth rate near the singularity.

math.AP↗

arXiv2016

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

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.

math.AP↗

arXiv2015

Biharmonic equation with singular nonlinearity

We consider the following problem: \begin{eqnarray*} ( P)\qquad \displaystyle\left\{\begin{array} {ll} & Δ^2 u = K(x)u^{-α} \quad \mbox{ in }\,Ω, \\ &u> 0\quad \mbox{ in }\,Ω, \;\;u\vert_{\partialΩ}=0, \,Δu\vert_{\partialΩ} = 0. \end{array}\right. \end{eqnarray*} We prove the main existence result: Assume that $α+β<2$. Then there exists a unique solution $u$ to $(P)$. Furthermore, there exist $c_1, c_2>0$ such that \begin{eqnarray}\label{behaviour-bound} c_1 ρ(x)\leq u(x)\leq c_2 ρ(x) \end{eqnarray} where $ρ(x)=d(x,\partialΩ)$. This result is sharp: Assume that $α+β\geq 2$. Then, there is no solution to $(P)$.

math.AP↗