Searcharxiv⌕ Search

arXiv subjects

Nidhi Nidhi

Publications and source records attributed to Nidhi Nidhi.

6 recordsLinked to original sources
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+λu \; &=\; Λ(I_α\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 $\mathcal{L}= -Δ+(-Δ)^s$, $0 0$, $I_α$ is the Riesz potential of order $α\in (0,2)$, $Λ>0$ is a parameter and $λ\in \mathbb{R}$ 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 Pohožaev manifold. Moreover, we discuss the regularity result and the construction of the Pohožaev identity, essential for the existence. \keywords{Normalized solutions; Nonlinear Schrödinger equations; Choquard nonlinearity; Critical exponential growth; Trudinger-Moser inequality}

math.AP↗
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} -Δ_p u +(-Δ_p)^s u & = & λ|u|^{p-2}u +μ|u|^{q-2}u +(I_α*|u|^{p^*_α})|u|^{p^*_α-2}u \text{ in } \mathbb{R}^N; \left\| u \right\|_p & = & τ. \end{array} \end{equation*} Here, $N\geq 3$, $2 \le p 0$, $I_α$ is the Riesz potential of order $α\in (\max\{0,N-2p\}, N)$, $p^*_α=\frac{p}{2}\left(\frac{N+α}{N-p}\right)$ is the critical exponent corresponding to the Hardy Littlewood Sobolev inequality, $(-Δ_p)^s$ is the non-local fractional p-Laplacian operator with $s\in (0,1)$, $μ>0$ is a parameter and $λ$ appears as a Lagrange multiplier. We show the existence of at least two distinct solutions in the presence of a mass subcritical perturbation, $μ|u|^{q-2}u$ with $p<q<p+\frac{sp^2}{N}$ under some conditions on $p,N$ and $s$.

math.AP↗
arXiv2026

Ground state solutions of mixed local-nonlolcal equations with Hartree type nonlinearities

We study a class of mixed local-nonlocal equations with Hartree-type nonlinearities of the form \begin{equation}\label{meqnab} -Δu + (-Δ)^s u + u = (I_α* F(u))\,F'(u) \quad \text{in } \mathbb{R}^N, \end{equation} where $N \geq 3$, $s \in (0,1)$, and $F \in C^1(\mathbb{R},\mathbb{R})$ satisfies Berestycki-Lions type assumptions. The equation combines the classical Laplacian with the fractional Laplacian, while the Hartree-type nonlinearity is given by a nonlocal convolution term involving the Riesz potential $I_α$, with $α\in (0,N)$. We prove the existence of ground state solutions. To this end, we establish regularity properties and derive a Pohožaev-type identity for general weak solutions. Moreover, we obtain symmetry properties of ground state solutions via polarization methods.

math.AP↗
arXiv2025

Pohožaev 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 $C^{1,α}_{loc}(\mathbb{R}^N)$. 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.

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} -Δu +(-Δ)^s u & = & λu +μ|u|^{p-2}u +(I_α*|u|^{2^*_α})|u|^{2^*_α-2}u \text{ in } \mathbb{R}^N;\;\; \left\| u \right\|_2 & = & τ, \end{array} \end{equation*} here $N\geq 3$, $τ>0$, $I_α$ is the Riesz potential of order $α\in (0,N)$, $2^*_α=\frac{N+α}{N-2}$ is the critical exponent corresponding to the Hardy Littlewood Sobolev inequality, $(-Δ)^s$ is the non-local fractional Laplacian operator with $s\in (0,1)$, $μ>0$ is a parameter and $λ$ appears as Lagrange multiplier. We have shown the existence of atleast two distinct solutions in the presence of mass subcritical perturbation, $μ|u|^{p-2}u$ with $2<p<2+\frac{4s}{N}$ under some assumptions on $τ$.

math.AP↗