Konijeti Sreenadh

Positive solutions of fractional elliptic equation with critical and singular nonlinearity

Abstract In this article, we study the following fractional elliptic equation with critical growth and singular nonlinearity: ( - Δ ) s ⁢ u = u - q + λ ⁢ u 2 s * - 1 , u > 0   in ⁢ Ω , u = 0   in ⁢ ℝ n ∖ Ω , (-\Delta)^{s}u=u^{-q}+\lambda u^{{2^{*}_{s}}-1},\qquad u>0\quad\text{in }% \Omega,\qquad u=0\quad\text{in }\mathbb{R}^{n}\setminus\Omega, where Ω is a bounded domain in ℝ n {\mathbb{R}^{n}} with smooth boundary ∂ ⁡ Ω {\partial\Omega} , n > 2 ⁢ s {n>2s} , s ∈ ( 0 , 1 ) {s\in(0,1)} , λ > 0 {\lambda>0} , q > 0 {q>0} and 2 s * = 2 ⁢ n n - 2 ⁢ s {2^{*}_{s}=\frac{2n}{n-2s}} . We use variational methods to show the existence and multiplicity of positive solutions with respect to the parameter λ.

On Dirichlet problem for fractional p-Laplacian with singular non-linearity

Abstract In this article, we study the following fractional p-Laplacian equation with critical growth and singular non-linearity: ( - Δ p ) s ⁢ u = λ ⁢ u - q + u α , u > 0   in ⁢ Ω , u = 0   in ⁢ ℝ n ∖ Ω , (-\Delta_{p})^{s}u=\lambda u^{-q}+u^{\alpha},\quad u>0\quad\text{in }\Omega,% \qquad u=0\quad\text{in }\mathbb{R}^{n}\setminus\Omega, where Ω is a bounded domain in ℝ n {\mathbb{R}^{n}} with smooth boundary ∂ ⁡ Ω {\partial\Omega} , n > s ⁢ p {n>sp} , s ∈ ( 0 , 1 ) {s\in(0,1)} , λ > 0 {\lambda>0} , 0 < q ≤ 1 {0<q\leq 1} and 1 < p < α + 1 ≤ p s * {1<p<\alpha+1\leq p^{*}_{s}} . We use variational methods to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter λ.

Positive solutions for nonlinear Choquard equation with singular nonlinearity

Abstract In this article, we study the following non-linear Choquard equation with singular non-linearity where is a bounded domain in with smooth boundary , and . Using variational approach and structure of associated Nehari manifold, we show the existence and multiplicity of positive weak solutions of the above problem, if is less than some positive constant. We also study the regularity of these weak solutions.

Variational Inequalities for the Fractional Laplacian

In this paper we study the obstacle problems for the fractional Lapalcian of order s ∈ (0, 1) in a bounded domain Ω⊂ℝn

On W 1, p(x) versus C1 local minimizers of functionals related to p(x)-Laplacian

{\Omega }\subset \mathbb {R}^{n}

On Doubly Nonlocal

, under mild assumptions on the data.

p

Let Ω be a bounded domain in IR N , N ≥ 2 with smooth boundary. We consider the functional J: W 1, p(x)(Ω) → IR defined as where p(x) be a Hölder continuous function satisfying 1 < p(x) ≤ N in and . Under appropriate assumptions on f (x, t) and g(x, t) which include some critical cases, we show that a C 1,γ(Ω) local minimizer u 0 ≢ 0 of J is also a W 1, p(x) local minimizer.

-fractional Coupled Elliptic System

\noi We study the following nonlinear system with perturbations involving p-fractional Laplacian \begin{equation*} (P)\left\{ \begin{split} (-\De)^s_p u+ a_1(x)u|u|^{p-2} &= \alpha(|x|^{-\mu}*|u|^q)|u|^{q-2}u+ \beta (|x|^{-\mu}*|v|^q)|u|^{q-2}u+ f_1(x)\; \text{in}\; \mb R^n,\\ (-\De)^s_p v+ a_2(x)v|v|^{p-2} &= \gamma(|x|^{-\mu}*|v|^q)|v|^{q-2}v+ \beta (|x|^{-\mu}*|u|^q)|v|^{q-2}v+ f_2(x)\; \text{in}\; \mb R^n, \end{split} \right. \end{equation*} where

Existence of Three Positive Solutions for a Nonlocal Singular Dirichlet Boundary Problem

n>sp

On W 1,p versus C 1(Ω) local minimizers for functionals with critical growth

,