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K. Sreenadh

Publications and source records attributed to K. Sreenadh.

At least 37 records · Page 2Linked to original sources
arXiv2023

Quasilinear Schrödinger equations with Stein-Weiss type convolution and critical exponential nonlinearity in $\mathbb R^N$

In this article, we investigate the existence of the positive solutions to the following class of quasilinear {Schrödinger} equations involving Stein-Weiss type convolution \begin{align*} -Δ_N u -Δ_N (u^{2})u +V(x)|u|^{N-2}u= \left(\int_{\mathbb R^N}\frac{F(y,u)}{|y|^β|x-y|^μ}~dy\right)\frac{f(x,u)}{|x|^β} \;\; \text{ in}\; \mathbb R^N, \end{align*} where $N\geq 2,\,$ $0<μ<N,\, β\geq 0,$ and $2β+μ\leq N.$ The potential $V:\mathbb R^N\to \mathbb R$ is a continuous function satisfying $0<V_0\leq V(x)$ for all $x\in \mathbb R^N$ and some appropriate assumptions. The nonlinearity $f:\mathbb R^N\times \mathbb R\to \mathbb R$ is a continuous function with critical exponential growth in the sense of the Trudinger-Moser inequality and $F(x,s)=\int_{0}^s f(x,t)dt$ is the primitive of $f$.

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arXiv2023

Critical exponent Neumann problem with Hardy-Littlewood-Sobolev nonlinearity

In this article, we study the Brezis-Nirenberg type problem of nonlinear Choquard equation with Neumann boundary condition \begin{equation*} \begin{aligned} -Δu &= λα(x)u + \left(\int\limits_Ω\frac{u(y)^{2^*_μ}}{|x-y|^μ}\;dy\right)u^{2^*_μ-1}, \;\;\text{in} \; Ω,\\ \frac{\partial u}{\partial ν} &= 0\;\; \text{on} \; \partialΩ, \end{aligned} \end{equation*} where $Ω$ is a bounded domain in $\mathbb{R}^N$ $(N\geq 4)$, $ν$ is the unit outer normal to $\partial Ω$ and $μ\in (0, N)$. According to the parameter $λ$, we prove necessary and sufficient conditions for the existence and non-existence of positive weak solutions to the problem. The proof is based on variational arguments.

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arXiv2022

Critical growth fractional Kirchhoff elliptic problems

This article is concerned with the existence and multiplicity of positive weak solutions for the following fractional Kirchhoff-Choquard problem: \begin{equation*} \begin{array}{cc} \displaystyle M\left( \|u\|^2\right) (-Δ)^s u = \dsλf(x)|u|^{q-2}u + \left( \int\limits_Ω \frac{|u(y)|^{2^{*}_{μ,s}}}{|x-y|^ μ}\, dy\right) |u|^{2^{*}_{μ,s}-2}u \;\text{in} \; Ω, u > 0\quad \text{in} \; Ω, \,\, u = 0\quad \text{in} \; \mathbb{R}^{N}\backslashΩ, \end{array} \end{equation*} where $Ω$ is open bounded domain of $\mathbb{R}^{N}$ with $C^2$ boundary, $N > 2s$ and $s \in (0,1)$, here $M$ models Kirchhoff-type coefficient of the form $M(t) = a + bt^{\te-1}$, where $a, b > 0$ are given constants. $(-Δ)^s$ is fractional Laplace operator, $λ> 0$ is a real parameter. We explore using the variational methods, the existence of solution for ${q} \in (1,2^*_s)$ and $\te \geq 1$. % and we also consider the case when $\te > 2^*_{μ,s}$ for $2< q < 2^*_{s}$. Here $2^*_s = \frac{2N}{N-2s}$ and $2^{*}_{μ,s} = \frac{2N-μ}{N-2s}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality.

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arXiv2022

Existence of positive solutions for a class of quasilinear Schrödinger equations with critical Choquard nonlinearity

This article is concerned with the existence of positive weak solutions for the following quasilinear Schrödinger Choquard equation: \begin{equation*} \begin{array}{cc} \displaystyle -div(g^2(u)\nabla u) + g(u)g'(u)\nabla u + a(x) u = k(x, u) \;\text{in} \; \mathbb{R}^N, \end{array} \end{equation*} where $N \geq 3$, $\displaystyle k(x,u) := h(x,u) + (I_{\vartheta}*|u|^{α\cdot2^*_μ})|u|^{α\cdot2^*_μ-2}u$, $g : \mathbb{R} \to \mathbb{R}^+$ is a differentiable even function with $g(0) = 1$ and $g'(t) \geq 0$ for all $t \geq 0$; $h\in C( \mathbb{R}^N \times\mathbb{R}, \mathbb{R})$ and the potential $a \in C( \mathbb{R}^N, \mathbb{R})$. We establish the existence of a positive solution using the change of variable and variational methods under appropriate assumptions on $g$, $h$ and $a$.

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arXiv2022

Multiplicity of positive solutions for a class of nonhomogeneous elliptic equations in the hyperbolic space

The paper is concerned with positive solutions to problems of the type \begin{equation*} -Δ_{\mathbb{B}^N} u - λu = a(x) |u|^{p-1}\;u \, + \, f \, \;\;\text{in}\;\mathbb{B}^{N}, \quad u \in H^{1}{(\mathbb{B}^{N})}, \end{equation*} where $\mathbb{B}^N$ denotes the hyperbolic space, $1 0.$ Subsequently, we establish the existence of two positive solutions for $a(x) \equiv 1$ and prove asymptotic estimates for positive solutions using barrier-type arguments. The proofs for existence combine variational arguments, key energy estimates involving hyperbolic bubbles.

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arXiv2022

Existence of high energy positive solutions for a class of elliptic equations in the hyperbolic space

We study the existence of positive solutions for the following class of scalar field problem on the hyperbolic space $$ -Δ_{\mathbb{H}^N} u - λu = a(x) |u|^{p-1} \, u\;\;\text{in}\;\mathbb{B}^{N}, \quad u \in H^{1}{(\mathbb{B}^{N})}, $$ where $\mathbb{B}^N$ denotes the hyperbolic space, $1<p<2^*-1:=\frac{N+2}{N-2}$, if $N \geqslant 3; 1<p<+\infty$, if $N = 2,\;λ< \frac{(N-1)^2}{4}$, and $0< a\in L^\infty(\mathbb{B}^N).$ We prove the existence of a positive solution by introducing the min-max procedure in the spirit of Bahri-Li in the hyperbolic space and using a series of new estimates involving interacting hyperbolic bubbles.

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arXiv2022

A note on the global regularity results for strongly nonhomogeneous $p,q$-fractional problems and applications

In this article, we communicate with the glimpse of the proofs of global regularity results for weak solutions to a class of problems involving fractional $(p,q)$-Laplacian, denoted by $(-Δ)^{s_1}_{p}+(-Δ)^{s_2}_{q}$, for $s_2, s_1\in (0,1)$ and $1<p,q<\infty$. We also obtain the boundary Hölder continuity results for the weak solutions to the corresponding problems involving at most critical growth nonlinearities. These results are almost optimal. Moreover, we establish Hopf type maximum principle and strong comparison principle. As an application to these new results, we prove the Sobolev versus Hölder minimizer type result, which provides the multiplicity of solutions in the spirit of seminal work \cite{Brezis-Nirenberg}.

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arXiv2021

Global regularity results for non-homogeneous growth fractional problems

This article concerns with the global Hölder regularity of weak solutions to a class of problems involving the fractional $(p,q)$-Laplacian, denoted by $(-Δ)^{s_1}_{p}+(-Δ)^{s_2}_{q}$, for $1<p,q<\infty$ and $s_1,s_2\in (0,1)$. We use a suitable Caccioppoli inequality and local boundedness result in order to prove the weak Harnack type inequality. Consequently, by employing a suitable iteration process, we establish the interior Hölder regularity for local weak solutions, which need not be assumed bounded. The global Hölder regularity result we prove expands and improves the regularity results of Giacomoni, Kumar and Sreenadh (arXiv: 2102.06080) to the subquadratic case (that is, $q<2$) and more general right hand side, which requires a different and new approach. Moreover, we establish a nonlocal Harnack type inequality for weak solutions, which is of independent interest.

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arXiv2021

Some existence and uniqueness results for logistic Choquard equation

We consider the following doubly nonlocal nonlinear logistic problem driven by the fractional $p$-Laplacian \begin{equation*} \pl u = f(x,u) -\cq ~\text{in}~ Ø, ~u=0 ~\text{in}~ \Rn\setminusØ. \end{equation*} Here $ Ø\subset \Rn (N\geq2)$ is a bounded domain with $ C^{1,1}$ boundary $\partial Ø$, $ s \in (0,1) $, $p \in (1,\infty)$ are such that $ps < N$. Also $p_{s,\a}^\#\leq r<\infty$ , where $p_{s,\a}^\#=(2N-\a)/2N$. Under suitable and general assumptions on the nonlinearity $f$, we study the existence, nonexistence, uniqueness, and regularity of weak solutions. As for applications, we treat cases of subdiffusive type logistic Choquard problem. We also consider in the superdiffusive case the Brezis-Nirenberg type problem with logistic Choquard and show the existence of a nontrivial solution for a suitable choice of $ł$. Finally for a particular choice of $f$ viz. $f(x,t)=łt^{q-1}$ with $1<p<2r<q$, we show the existence of at least one energy nodal solution.

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arXiv2021

Three solutions for a fractional elliptic problem with asymmetric critical Choquard nonlinearity

In this paper we study the existence and multiplicity of weak solutions for the following asymmetric nonlinear Choquard problem on fractional Laplacian: \begin{equation*} \begin{array}{rl} (-Δ)^s u &= \displaystyle-λ|u|^{q-2}u + au + b\left( \int\limits_Ω \frac{(u^{+}(y))^{2^{*}_{μ,s}}}{|x-y|^ μ}\, dy\right) (u^{+})^{2^{*}_{μ,s}-2}u \quad\text{in} \; Ω, u &= 0\quad \text{in} \; \mathbb{R}^{N}\backslashΩ, \end{array} \end{equation*} where $Ω$ is open bounded domain of $\mathbb{R}^{N}$ with $C^2$ boundary, $N > 2s$ and $s \in (0,1)$. Here $(-Δ)^s$ is the fractional Laplace operator, $λ> 0$ is a real parameter, $q \in (1, 2)$, $a > 0$ and $b> 0$ are given constants, and $2^{*}_{μ,s} = \frac{2N-μ}{N-2s}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and the notation $u^{+} = \max \{u, 0\}$. We prove that the above problem has at least three nontrivial solutions using the Mountain pass Lemma and Linking theorem.

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arXiv2021

Multiple positive solutions for degenerate Kirchhoff equations with singular and Choquard nonlinearity

In this paper we study the existence, multiplicity and regularity of positive weak solutions for the following Kirchhoff-Choquard problem: \begin{equation*} \begin{array}{cc} \displaystyle M\left( \iint\limits_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\,dxdy\right) (-Δ)^s u = \fracλ{u^γ} + \left( \int\limits_Ω \frac{|u(y)|^{2^{*}_{μ,s}}}{|x-y|^ μ}\, dy\right) |u|^{2^{*}_{μ,s}-2}u \;\text{in} \; Ω, %\quad \quad u > 0\quad \text{in} \; Ω, \quad \quad u = 0\quad \text{in} \; \mathbb{R}^{N}\backslashΩ, \end{array} \end{equation*} where $Ω$ is open bounded domain of $\mathbb{R}^{N}$ with $C^2$ boundary, $N > 2s$ and $s \in (0,1)$. $M$ models Kirchhoff-type coefficient in particular, the degenerate case where Kirchhoff coefficient M is zero at zero. $(-Δ)^s$ is fractional Laplace operator, $λ> 0$ is a real parameter, $γ\in (0,1)$ and $2^{*}_{μ,s} = \frac{2N-μ}{N-2s}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We prove that each positive weak solution is bounded and satisfy Hölder regularity of order $s$. Furthermore, using the variational methods and truncation arguments we prove the existence of two positive solutions.

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arXiv2021

Sobolev and Hölder regularity results for some singular nonhomogeneous quasilinear problems

This article deals with the study of the following singular quasilinear equation: \begin{equation*} (P) \left\{ \ -Δ_{p}u -Δ_{q}u = f(x) u^{-δ},\; u>0 \text{ in }\; \Om; \; u=0 \text{ on } \pa\Om, \right. \end{equation*} where $\Om$ is a bounded domain in $\mathbb{R}^N$ with $C^2$ boundary $\pa\Om$, $1< q< p<\infty$, $\de>0$ and $f\in L^\infty_{loc}(\Om)$ is a non-negative function which behaves like $\textnormal{dist}(x,\pa\Om)^{-\ba},$ $\ba\ge 0$ near the boundary of $\Om$. We prove the existence of a weak solution in $W^{1,p}_{loc}(\Om)$ and its behaviour near the boundary for $\ba<p$. Consequently, we obtain optimal Sobolev regularity of weak solutions. By establishing the comparison principle, we prove the uniqueness of weak solution for the case $\ba<2-\frac{1}{p}$. Subsequently, for the case $\ba\ge p$, we prove the non-existence result. Moreover, we prove Hölder regularity of the gradient of weak solution to a more general class of quasilinear equations involving singular nonlinearity as well as lower order terms (see \eqref{Prb}). This result is completely new and of independent interest. In addition to this, we prove Hölder regularity of minimal weak solutions of $(P)$ for the case $β+δ\geq 1$ that has not been fully answered in former contributions even for $p$-Laplace operators.

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arXiv2021

Interior and boundary regularity results for strongly nonhomogeneous $p,q$-fractional problems

In this article, we deal with the global regularity of weak solutions to a class of problems involving the fractional $(p,q)$-Laplacian, denoted by $(-Δ)^{s_1}_{p}+(-Δ)^{s_2}_{q}$, for $s_2, s_1\in (0,1)$ and $1<p,q<\infty$. We establish completely new Hölder continuity results, up to the boundary, for the weak solutions to fractional $(p,q)$-problems involving singular as well as regular nonlinearities. Moreover, as applications to boundary estimates, we establish new Hopf type maximum principle and strong comparison principle in both situations.

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arXiv2020

Unbalanced fractional elliptic problems with exponential nonlinearity: subcritical and critical cases

This paper deals with the qualitative analysis of solutions to the following $(p,q)$-fractional equation: \begin{equation*} \begin{array}{rllll} (-Δ)^{s_1}_{p}u+(-Δ)^{s_2}_{q}u+V(x) \big(|u|^{p-2}u+|u|^{q-2}u\big) = K(x)\frac{f(u)}{|x|^\ba} \; \text{ in } \mb R^N, \end{array} \end{equation*} \noi where $1< q< p$, $0<s_2\leq s_1<1$, $ps_1=N$, $\ba\in[0,N)$, and $V,K:\mb R^N\to\mb R$, $f:\mb R\to \mb R$ are continuous functions satisfying some natural hypotheses. We are concerned both with the case when $f$ has a subcritical growth and with the critical framework with respect to the exponential nonlinearity. By combining a Moser-Trudinger type inequality for fractional Sobolev spaces with Schwarz symmetrization techniques and related variational methods, we prove the existence of nonnegative solutions.

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arXiv2020

A qualitative study of (p,q) Singular parabolic equations: local existence, Sobolev regularity and asymptotic behaviour

The purpose of the article is to study the existence, regularity, stabilization and blow up results of weak solution to the following parabolic $(p,q)$-singular equation: \begin{equation*} (P_t)\; \left\{\begin{array}{rllll} u_t-Δ_{p}u -Δ_{q}u & = \vth \; u^{-\de}+ f(x,u), \; u>0 \text{ in } \Om\times (0,T), \\ u&=0 \quad \text{ on } \pa\Om\times (0,T), u(x,0)&= u_0(x) \; \text{ in }\Om, \end{array} \right. \end{equation*} where $\Om$ is a bounded domain in $\mathbb{R}^N$ with $C^2$ boundary $\pa\Om$, $1 0$, $N\ge 2$ and $\vth>0$ is a parameter. Moreover, we assume that $f:\Om\times [0,\infty) \to \mb R$ is a bounded below Carathéodory function, locally Lipschitz with respect to the second variable uniformly in $x\in\Om$ and $u_0\in L^\infty(\Om)\cap W^{1,p}_0(\Om)$. We distinguish the cases as $q$-subhomogeneous and $q$-superhomogeneous depending on the growth of $f$ (hereafter we will drop the term $q$). In the subhomogeneous case, we prove the existence and uniqueness of the weak solution to problem $(P_t)$ for $\de<2+1/(p-1)$. For this, we first study the stationary problems corresponding to $(P_t)$ by using the method of sub and super solutions and subsequently employing implicit Euler method, we obtain the existence of a solution to $(P_t)$. Furthermore, in this case, we prove the stabilization result, that is, the solution $u(t)$ of $(P_t)$ converges to $u_\infty$, the unique solution to the stationary problem, in $L^\infty(\Om)$ as $t\ra\infty$. For the superhomogeneous case, we prove the local existence theorem by taking help of nonlinear semigroup theory. Subsequently, we prove finite time blow up of solution to problem $(P_t)$ for small parameter $\vartheta>0$ in the case $\de\leq 1$ and for all $\vth>0$ in the case $\de>1$.

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