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K. Sreenadh

Publications and source records attributed to K. Sreenadh.

At least 19 recordsLinked to original sources
arXiv2026

The isocritical regime for mixed local-nonlocal $(p,q)$ Laplacian: existence of ground state, and decay estimates

We study the mixed local-nonlocal operator $\mathcal{L}_{p,q} := -\Delta_p + (-\Delta)_q^s$ in the isocritical regime $p^* = q_s^*$, i.e. $1 - N/p = s - N/q$, under which both operators become critical for the same nonlinearity. We consider \[ -\Delta_p u + (-\Delta)_q^s u = |u|^{p^*-2}u \qquad \text{in } \mathbb{R}^N, \] with $N \geq 2$, $1 < p < N$, $0 < s < 1$, $1 < sq < N$. In this regime the energy space reduces to $\mathcal{D}_0^{1,p}(\mathbb{R}^N)$, and both best Sobolev constants enter the variational structure simultaneously. We prove: $(i)$ existence of a nonnegative radial ground state via Nehari manifold methods and a double-threshold concentration-compactness analysis; $(ii)$ a logarithmic energy estimate, weak comparison principle, and strong maximum principle for all admissible exponents; $(iii)$ a weak Harnack inequality; and $(iv)$ sharp two-sided decay $U(x) \asymp |x|^{-(N-p)/(p-1)}$ for positive radial solutions, matching the fundamental solution of the $p$-Laplacian.

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arXiv2026

Normalized solutions to an exponential growth Choquard equation driven by mixed local-nonlocal operator in $\mathbb{R}^2$

In this article, we study the existence of normalized solutions to the following mixed nonlinear Choquard equation with exponential growth \begin{align*} \left\{ \begin{aligned} \mathcal{L}u+\lambda u \; &=\; \Lambda(I_{\alpha}\ast F(u))F'(u), \quad \text{in }\mathbb{R}^{2}, \int_{\mathbb{R}^{2}}|u|^{2}\,dx \; &=\; a^{2}, \end{aligned} \right. \end{align*} where $\mathcal{L}= -\Delta+(-\Delta)^s$, $0 0$, $I_{\alpha}$ is the Riesz potential of order $\alpha\in (0,2)$, $\Lambda>0$ is a parameter and $\lambda\in \mathbb{R}$ appears as a Lagrange multiplier. Here, the nonlinearity $F$ has exponential growth in $\mathbb{R}^{2}$. Using variational methods, we prove the existence of normalized solution in the Poho\v{z}aev manifold. Moreover, we discuss the regularity result and the construction of the Poho\v{z}aev identity, essential for the existence. \keywords{Normalized solutions; Nonlinear Schr\"odinger equations; Choquard nonlinearity; Critical exponential growth; Trudinger-Moser inequality}

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arXiv2026

Multiplicity of solutions with prescribed mass for a quasilinear critical Choquard equation driven by a local-nonlocal operator

In this paper we study the normalized solutions of the following critical growth Choquard equation with mixed local and non-local operators: \begin{equation*} \begin{array}{rcl} -\Delta_p u +(-\Delta_p)^s u & = & \lambda |u|^{p-2}u +\mu |u|^{q-2}u +(I_{\alpha}*|u|^{p^*_{\alpha}})|u|^{p^*_{\alpha}-2}u \text{ in } \mathbb{R}^N; \left\| u \right\|_p & = & \tau. \end{array} \end{equation*} Here, $N\geq 3$, $2 \le p 0$, $I_{\alpha}$ is the Riesz potential of order $\alpha\in (\max\{0,N-2p\}, N)$, $p^*_{\alpha}=\frac{p}{2}\left(\frac{N+\alpha}{N-p}\right)$ is the critical exponent corresponding to the Hardy Littlewood Sobolev inequality, $(-\Delta_p)^s$ is the non-local fractional p-Laplacian operator with $s\in (0,1)$, $\mu>0$ is a parameter and $\lambda$ appears as a Lagrange multiplier. We show the existence of at least two distinct solutions in the presence of a mass subcritical perturbation, $\mu |u|^{q-2}u$ with $p<q<p+\frac{sp^2}{N}$ under some conditions on $p,N$ and $s$.

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arXiv2025

Poho\v{z}aev identity and the existence of normalized ground state solutions for variable exponent problems

In this article, we investigate normalized solutions for nonlinear problems involving variable exponents. To the best of our knowledge, normalized solutions have not been previously studied in this setting, and our results appear to be new. A key difficulty is that the standard scaling argument, which is important in the classical normalized solution approach, is no longer available in the variable exponent setup. To address this, we work with a constrained variational framework and establish the existence of a ground state solution. We further show that these solutions are $C^{1,\alpha}_{loc}(\mathbb{R}^N)$. Finally, we derive a Poho\v zaev-type identity adapted to the variable exponent structure in $\mathbb{R}^N$, which is used to prove that the solution is a ground state.

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arXiv2025

Asymptotic behaviour and existence of positive solutions for mixed local nonlocal elliptic equations with Hardy potential

We investigate the existence and multiplicity of positive solutions to the following problem driven by the superposition of the Laplacian and the fractional Laplacian with Hardy potential \begin{equation*} \left\{ \begin{aligned} -\Delta u + (-\Delta)^s u - \mu \frac{u}{|x|^2} &= \lambda |u|^{p-2} u + |u|^{2^*-2} u \quad \text{in } \Omega \subset \mathbb{R}^N, u &= 0 \quad \text{in } \mathbb{R}^N \setminus \Omega, \end{aligned} \right. \end{equation*} where $ \Omega \subset \mathbb{R}^N $ is a bounded domain with smooth boundary, $ 0 < s < 1 $, $ 1 < p < 2^* $, with $ 2^* = \frac{2N}{N-2} $, $ \lambda > 0 $, and $ \mu \in (0, \bar{\mu}) $ where $\bar \mu = \left( \frac{N-2}{2} \right)^2$. The aim of this paper is twofold. First, we establish uniform asymptotic estimates for solutions of the problem by means of a suitable transformation. Then, according to the value of the exponent $p$, we analyze three distinct cases and prove the existence of a positive solution. Moreover, in the sublinear regime $1 < p < 2$, we demonstrate the existence of multiple positive solutions for small perturbations of the fractional Laplacian.

math.AP↗
arXiv2025

Existence of multiple normalized solutions to a critical growth Choquard equation involving mixed operator

In this paper we study the normalized solutions of the following critical growth Choquard equation with mixed local and non-local operators: \begin{equation*} \begin{array}{rcl} -\Delta u +(-\Delta)^s u & = & \lambda u +\mu |u|^{p-2}u +(I_{\alpha}*|u|^{2^*_{\alpha}})|u|^{2^*_{\alpha}-2}u \text{ in } \mathbb{R}^N;\;\; \left\| u \right\|_2 & = & \tau, \end{array} \end{equation*} here $N\geq 3$, $\tau>0$, $I_{\alpha}$ is the Riesz potential of order $\alpha\in (0,N)$, $2^*_{\alpha}=\frac{N+\alpha}{N-2}$ is the critical exponent corresponding to the Hardy Littlewood Sobolev inequality, $(-\Delta)^s$ is the non-local fractional Laplacian operator with $s\in (0,1)$, $\mu>0$ is a parameter and $\lambda$ appears as Lagrange multiplier. We have shown the existence of atleast two distinct solutions in the presence of mass subcritical perturbation, $\mu |u|^{p-2}u$ with $2<p<2+\frac{4s}{N}$ under some assumptions on $\tau$.

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arXiv2025

Multiplicity results for mixed local-nonlocal variable exponent problem involving singular and superlinear term

In this paper, we study a class of quasilinear elliptic equations involving both local and nonlocal operators with variable exponents. The problem exhibits singular nonlinearities along with a subcritical superlinear growth term and a parameter $\lambda$. We study the existence of multiple solutions with the help of variational methods by restricting the associated energy functional on appropriate subsets of the Nehari manifold. Using the topological index and the structure of the fibering maps, we analyse a key splitting property of the associated Nehari manifold. This decomposition allows us to establish the existence of two distinct solutions. Additionally, we establish the $L^\infty$-bound for the solutions.

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arXiv2025

On the eigenvalues and Fu\v{c}\'{\i}k spectrum of $p$-Laplace local and nonlocal operator with mixed interpolated Hardy term

In this article, we are concerned with the eigenvalue problem driven by the mixed local and nonlocal $p$-Laplacian operator having the interpolated Hardy term \begin{equation*} \mathcal{T}(u) :=- \Delta_p u + (- \Delta_p)^s u - \mu \frac{|u|^{p-2}u}{|x|^{p \theta}}, \end{equation*} where $0<s<1<p<N$, $\theta \in [s,1]$, and $\mu \in (0,\mu_0(\theta))$. First, we establish a mixed interpolated Hardy inequality and then show the existence of eigenvalues and their properties. We also investigate the Fu\v{c}\'{\i}k spectrum, the existence of the first nontrivial curve in the Fu\v{c}\'{\i}k spectrum, and prove some of its properties. Moreover, we study the shape optimization of the domain with respect to the first two eigenvalues, the regularity of the eigenfunctions, the Faber-Krahn inequality, and a variational characterization of the second eigenvalue.

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arXiv2024

Normalized solutions to a quasilinear equation involving critical Sobolev exponent

In this paper we study the existence and regularity results of normalized solutions to the following quasilinear elliptic Choquard equation with critical Sobolev exponent and mixed diffusion type operators: \begin{equation*} \begin{array}{rcl} -\Delta_p u+(-\Delta_p)^su & = & \lambda |u|^{p-2}u +|u|^{p^*-2}u+ \mu(I_{\alpha}*|u|^q)|u|^{q-2}u\;\;\text{in } \mathbb{R}^N, \int_{\mathbb{R}^N}|u|^pdx & = & \tau, \end{array} \end{equation*} where $N\geq 3$, $\tau>0$, $\frac{p}{2}(\frac{N+\alpha}{N}) 0$ is a parameter, $(-\Delta_p)^s$ is the fractional p-laplacian operator, $p^*=\frac{Np}{N-p}$ is the critical Sobolev exponent and $\lambda$ appears as a Lagrange multiplier.

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arXiv2024

Quasilinear Schr\"{o}dinger Equation involving Critical Hardy Potential and Choquard type Exponential nonlinearity

In this article, we study the following quasilinear Schr\"{o}dinger equation involving Hardy potential and Choquard type exponential nonlinearity with a parameter $\alpha$ \begin{equation*} \left\{ \begin{array}{l} - \Delta_N w - \Delta_N(|w|^{2\alpha}) |w|^{2\alpha - 2} w - \lambda \frac{|w|^{2\alpha N-2}w}{\left( |x| \log\left(\frac{R}{|x|} \right) \right)^N} = \left(\int_{\Omega} \frac{H(y,w(y))}{|x-y|^{\mu}}dy\right) h(x,w(x))\; \mbox{in }\; \Omega, w > 0 \mbox{ in } \Omega \setminus \{ 0\}, \quad \quad w = 0 \mbox{ on } \partial \Omega, \end{array} \right. \end{equation*} where $N\geq 2$, $\alpha>\frac12$, $0\leq \lambda< \left(\frac{N-1}{N}\right)^N$, $0 < \mu < N$, $h : \mathbb R^N \times \mathbb R \rightarrow \mathbb R$ is a continuous function with critical exponential growth in the sense of the Trudinger-Moser inequality and $H(x,t)= \int_{0}^{t} h(x,s) ds$ is the primitive of $h$. With the help of Mountain Pass Theorem and critical level which is obtained by the sequence of Moser functions, we establish the existence of a positive solution for a small range of $\lambda$. Moreover, we also investigate the existence of a positive solution for a non-homogeneous problem for every $0\leq \lambda <\left(\frac{N-1}{N}\right)^N.$ To the best of our knowledge, the results obtained here are new even in case of $N$-Laplace equation with Hardy potential.

math.AP↗
arXiv2024

Existence, symmetry and regularity of ground states of a non linear choquard equation in the hyperbolic space

In this paper, we explore the positive solutions of the following nonlinear Choquard equation involving the green kernel of the fractional operator $(-\Delta_{\mathbb{B}^N})^{-\alpha/2}$ in the hyperbolic space \begin{equation} \begin{aligned} -\Delta_{\mathbb{B}^{N}} u \, - \, \lambda u \, &= \left[(- \Delta_{\mathbb{B}^{N}})^{-\frac{\alpha}{2}}|u|^p\right]|u|^{p-2}u, \end{aligned} \end{equation} where $\Delta_{\mathbb{B}^{N}}$ denotes the Laplace-Beltrami operator on $\mathbb{B}^{N}$, $\lambda \leq \frac{(N-1)^2}{4}$, $1 < p < 2^*_{\alpha} = \frac{N+\alpha}{N-2}$, $0 < \alpha < N$, $N \geq 3$, $2^*_\alpha$ is the critical exponent in the context of the Hardy-Littlewood-Sobolev inequality. This study is analogous to the Choquard equation in the Euclidean space, which involves the non-local Riesz potential operator. We consider the functional setting within the Sobolev space $H^1(\mathbb{B}^N)$, employing advanced harmonic analysis techniques, particularly the Helgason Fourier transform and semigroup approach to fractional Laplacian. Moreover, the Hardy-Littlewood-Sobolev inequality on complete Riemannian manifolds, as developed by Varopoulos, is pivotal in our analysis. We prove an existence result for the above problem in the subcritical case. Moreover, we also demonstrate that solutions exhibit radial symmetry, and establish the regularity properties.

math.AP↗
arXiv2023

High energy solutions for $p$-Kirchhoff elliptic problems with Hardy-Littlewood-Sobolev nonlinearity

This article deals with the study of the following Kirchhoff-Choquard problem: \begin{equation*} \begin{array}{cc} \displaystyle M\left(\, \int\limits_{\mathbb{R}^N}|\nabla u|^p\right) (-\Delta_p) u + V(x)|u|^{p-2}u = \left(\, \int\limits_{\mathbb{R}^N}\frac{F(u)(y)}{|x-y|^{\mu}}\,dy \right) f(u), \;\;\text{in} \; \mathbb{R}^N, u > 0, \;\; \text{in} \; \mathbb{R}^N, \end{array} \end{equation*} where $M$ models Kirchhoff-type nonlinear term of the form $M(t) = a + bt^{\theta-1}$, where $a, b > 0$ are given constants; $1<p<N$, $\Delta_p = \text{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian operator; potential $V \in C^2(\mathbb{R}^N)$; $f$ is monotonic function with suitable growth conditions. We obtain the existence of a positive high energy solution for $\theta \in \left[1, \frac{2N-\mu}{N-p}\right) $ via the Poho\v{z}aev manifold and linking theorem. Apart from this, we also studied the radial symmetry of solutions of the associated limit problem.

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arXiv2023

On a class of elliptic equations with Critical Perturbations in the hyperbolic space

We study the existence and non-existence of positive solutions for the following class of nonlinear elliptic problems in the hyperbolic space $$ -\Delta_{\mathbb{B}^N} u-\lambda u=a(x)u^{p-1} \, + \, \varepsilon u^{2^*-1} \,\;\;\text{in}\;\mathbb{B}^{N}, \quad u \in H^{1}{(\mathbb{B}^{N})}, $$ where $\mathbb{B}^N$ denotes the hyperbolic space, $2<p<2^*:=\frac{2N}{N-2}$, if $N \geqslant 3; 2<p<+\infty$, if $N = 2,\;\lambda < \frac{(N-1)^2}{4}$, and $0< a\in L^\infty(\mathbb{B}^N).$ We first prove the existence of a positive radially symmetric ground-state solution for $a(x) \equiv 1.$ Next, we prove that for $a(x) \geq 1$, there exists a ground-state solution for $\varepsilon$ small. For proof, we employ ``conformal change of metric" which allows us to transform the original equation into a singular equation in a ball in $\mathbb R^N$. Then by carefully analysing the energy level using blow-up arguments, we prove the existence of a ground-state solution. Finally, the case $a(x) \leq 1$ is considered where we first show that there is no ground-state solution, and prove the existence of a \it bound-state solution \rm (high energy solution) for $\varepsilon$ small. We employ variational arguments in the spirit of Bahri-Li to prove the existence of high energy-bound-state solutions in the hyperbolic space.

math.AP↗
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} -\Delta u &= \lambda \alpha(x)u + \left(\int\limits_{\Omega}\frac{u(y)^{2^*_{\mu}}}{|x-y|^{\mu}}\;dy\right)u^{2^*_{\mu}-1}, \;\;\text{in} \; \Omega,\\ \frac{\partial u}{\partial \nu} &= 0\;\; \text{on} \; \partial\Omega, \end{aligned} \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^N$ $(N\geq 4)$, $\nu$ is the unit outer normal to $\partial \Omega$ and $\mu \in (0, N)$. According to the parameter $\lambda$, 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.

math.AP↗
arXiv2023

Fractional Hamiltonian type system on $\mathbb{R}$ with critical growth nonlinearity

This article investigates the existence and properties of ground state solutions to the following nonlocal Hamiltonian elliptic system: \begin{align*} \begin{cases} (-\Delta)^\frac12 u +V_0 u =g(v),~x\in \mathbb{R} (-\Delta)^\frac12 v +V_0 v =f(u),~x\in \mathbb{R}, \end{cases} \end{align*} where $(-\Delta)^\frac12$ is the square root Laplacian operator, $V_0 >0$ and $f,~g$ have critical exponential growth in $\mathbb{R}$. Using minimization technique over some generalized Nehari manifold, we show that the set $\mathcal{S}$ of ground state solutions is non empty. Moreover for $(u,v) \in \mathcal{S}$, $u,~v$ are uniformly bounded in $L^\infty(\mathbb{R})$ and uniformly decaying at infinity. We also show that the set $\mathcal{S}$ is compact in $H^\frac12(\mathbb{R}) \times H^\frac12(\mathbb{R})$ up to translations. Furthermore under locally lipschitz continuity of $f$ and $g$ we obtain a suitable Poho\v{z}aev type identity for any $(u,v) \in \mathcal{S}$. We deduce the existence of semi-classical ground state solutions to the singularly perturbed system \begin{align*} \begin{cases} \epsilon(-\Delta)^\frac12 \varphi +V(x) \varphi =g(\psi),~x\in \mathbb{R} \epsilon (-\Delta)^\frac12 \psi +V(x) \psi =f(\varphi),~x\in \mathbb{R}, \end{cases} \end{align*} where $\epsilon>0$ and $V \in C(\mathbb{R})$ satisfy the assumption $(V)$ given below (see Section 1). Finally as $\epsilon \rightarrow 0$, we prove the existence of minimal energy solutions which concentrate around the closest minima of the potential $V$.

math.AP↗
arXiv2023

Mixed local and nonlocal semilinear elliptic equation with strongly singular and critical Choquard nonlinearity

In this article, we study an elliptic problem of mixed order with both local and nonlocal aspects involving singular nonlinearity in combination with critical Hartree-type nonlinearity. Using variational methods together with the critical point theory of nonsmooth analysis and the geometry of the energy functional, we show the existence and multiplicity of positive solutions with respect to the parameter $\lambda$.

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