arXiv subjects

P. N. Srikanth

Publications and source records attributed to P. N. Srikanth.

4 recordsLinked to original sources

arXiv2016

On a problem of resonance with exponential non linearity

We have considered the following semi linear elliptic problem on the unit disk $B$ $-Δu = λ_1 u+e^u+f $ in $B$ with the Dirichlet boundary condition and $f$ satisfying the following condition : $f\in L^r(B)$, for some $r>2$ and $-\int_B fϕ_1<4π$. Where $ϕ_1$ is the eigen function of $(-Δ)$ corresponding to the first eigenvalue $λ_1$ in $H_0^1(B)$. We shall find the existence of a radial solution of this PDE. We shall use degree theory to get the existence starting from a suitable with known solution with its degree. Connecting those two PDE's by homotopy and getting the uniform estimate for the connecting PDE's we shall achieve our result.

math.AP↗

arXiv2013

On the solutions of a singular elliptic equation concentrating on a circle

Let $A=\{x\in \R^{2N+2} : 0< a< |x| 0 &\mbox{\qquad in} A \frac{\partial u}{\partialν} = 0 &\mbox{\qquad on} \partial A \end{array} %\label{a1} \end{equation} $1<p<2^*-1$. We shall show that there exists a positive solution $u_\eps$ concentrating on an $S^1$ orbit as $\eps\to 0$. We prove this by reducing the problem to a lower dimensional one and analyzing a single point concentrating solution in the lower dimensional space. We make precise how the single peak concentration depends on the parameter $α$.

math.AP↗

arXiv2013

On the solutions of a singular elliptic equation concentrating on two orthogonal spheres

Let $A=\{x\in \R^{2m} : 0< a< |x| 0 &\mbox{\qquad in} A u = 0 &\mbox{\qquad on} \partial A \end{array} %\label{a1} \end{equation} $1<p<2^*-1$. We shall prove the existence of a positive solution $u_\eps$ which concentrates on two different orthogonal spheres of dimension $(m-1)$ as $\eps\to 0$. We achieve this by studying a reduced problem on an annular domain in $\R^{m+1}$ and analyzing the profile of a two point concentrating solution in this domain.

math.AP↗

arXiv2012

A Reduction Method for Semilinear Elliptic Equations and Solutions Concentrating on Spheres

We show that any general semilinear elliptic problem with Dirichlet or Neumann boundary conditions in an annulus A in R^2m ;m >1, invariant by the action of a certain symmetry group can be reduced to a nonhomogenous similar problem in an annulus D in R^(m+1), invariant by another related symmetry. We apply this result to prove the existence of positive and sign changing solutions of a singularly perturbed elliptic problem in A which concentrate on one or two (m-1) dimensional spheres. We also prove that the Morse indices of these solutions tend to infinity as the parameter of concentration tends to infinity.

math.AP↗