In this work, we study a class of nonlinear elliptic equations of the form $Au=-\Delta _{p}u+ div \left( Vu\right) =R(x,u)$ in a bounded domain $\Omega \subset \mathbb{R}^n$, where $-\Delta _{p}u=- div \left( \left \vert \nabla u\right \vert ^{p-2}\nabla u\right)$ denotes the p-Laplacian operator, $V(x)$ is a given vector field, and $R(x)$ represents a source term. We establish the existence and uniqueness of weak solutions under suitable assumptions on $V$ and $R$. To establish the existence of weak solutions to the nonlinear elliptic problem we employ a fixed point approach based on the compactness of the associated operator. The existence result is obtained by proving that the composition $A^{-1}N_{R}$ is a compact and continuous mapping that admits at least one fixed point, where $N_{R}$ is the Nemyskii operator corresponding to a Caratheodory function $R$. This ensures the existence of a weak solution $u\in W_{0}^{1,p}\left( \Omega \right)$. For the uniqueness of solutions, we rely on the strict monotonicity of the p-Laplacian operator and appropriate growth and Lipschitz conditions on $R(x,u)$.
| Mualliflar | Benaissa, S., Messelmi, F., Merouani, A. |
|---|---|
| Jurnal | O‘zbekiston matematika jurnali |
| Nashr sanasi | 2026-07-02 |
| Jild | 70 |
| Son | 2 |
| Betlar | 33-41 |
| Til | Ingliz |
| DOI | 10.29229/uzmj.2026-2-4 |
DOI: 10.29229/uzmj.2026-2-4 · Maqolaning asl sahifasi
p-Laplacian, Fixed point, Variational formulation, Caratheodory function, Nemytskii operator
In this paper, an evasion game for an infinite system of ternary differential equations is studied. The game is considered in the Hilbert space $l_{2}$. The control parameters of pursuer and evader are subject to…
Boundary value problems are considered for quasilinear elliptic equations with a principal quasilinear elliptic operator of the second order in the Sobolev space $W_p^2 \left(\Omega\right)$. The main content of the work…
We present fixed point theorem for a new type of nonlinear contractive mappings in $\mathfrak{b}$-metric spaces. Also, we present examples to illustrate the validity of the results obtained in the paper. In addition, by…
In this paper, we introduce generalized grand Sobolev spaces and using the integral representation method, study some properties of functions from these spaces from the point of view of embedding theory.
This work introduces an efficient methodology for approximating solutions to first-order non-linear differential equations. The approach is based on formulating a differential equation with piecewise constant arguments…
This work is devoted to study pursuit problems involving multiple pursuers and a single evader, all moving on the 1-skeleton graph of the dodecahedron. Solutions to the pursuit problems are provided, where the maximum…
Inspired by quantum mechanics, we introduce a weak form of solutions for differential equations. We show that Schrödinger equation is a weak form of the classical Euler-Lagrange equation.
In this paper, the Cauchy problem for a differential equation with a fractional Hilfer derivative $D_t^{\alpha,\beta}u(t)+Au(t)=f(t), \ 0
We find all the eta quotients in the spaces $M_{1}\left(\Gamma_{0}(20), \left(\frac{d}{*}\right)\right)$ with $d = -4, -20$ of modular forms, and we determine their Fourier coefficients, where $\left(\frac{d}{*}\right)$…
In this work, the problem of constructing an optimal interpolation formula involving derivatives is studied. The values of the unknown function are required not only at the nodal points but also the values of its first…