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In this paper, we study the existence of at least one bounded weak solution for Kohn–Spencer Laplacian with a weight depending on the solution and convection term of the form \[ -div_{\mathbb{H}^n}(\nu(\xi,u) |D_{\mathbb{H}^n}u|^{p-2}_{\mathbb{H}^n}D_{\mathbb{H}^n}u)=f(\xi,u,D_{\mathbb{H}^n} u) \] − di v H n (ν (ξ , u) | D H n u | H n p − 2 D H n u) = f (ξ , u , D H n u) in a bounded domain $ \Omega \subset \mathbb {H}^n $ Ω ⊂ H n . We show the set of solutions is uniformly bounded by a special Moser's iteration. [ABSTRACT FROM AUTHOR]

In this work, we will study the existence of solutions for a class of problems involving the p-Laplacian operator with weight and a nonlinearity that changes sign in a domain $ \Omega \subset \mathbb {R}^N $ Ω ⊂ R N with $ \partial \Omega $ ∂Ω smooth and $ N\geq 3 $ N ≥ 3. For this class of problems with subcritical growth, we can show the existence of two solutions, one nodal and one that does not change sign in Ω. To class with critical case, we prove the existence of nontrivial solution which does not change sign in Ω. [ABSTRACT FROM AUTHOR]

Existence and multiplicity of weak solutions for a weighted (p, q)-Laplacian problem consisting of a singular term are proved, where the weight function satisfies Muckenhoupt's condition. The method is based on the variational methods. [ABSTRACT FROM AUTHOR]

Here, we consider an anisotropic equation \[ -\Delta _{{\mathbb {H}^n},\overrightarrow {p}}u +a(q)|u|^{p^–2}u=\lambda w(q)|u|^{m-2}u-h(q)|u|^{l-2}u, \] in the Heisenberg group {\mathbb {H}^n}, where the operator \Delta _{{\mathbb {H}^n},\overrightarrow {p}} is the horizontal anisotropic p-Laplacian on the Heisenberg group and is defined in the sequel. By the variational methods, we prove the existence of the entire weak solutions. [ABSTRACT FROM AUTHOR]

Here, we study the Fučik spectrum for the $ p, q $-Laplacian$ -\Delta_p u-\Delta_q u = \alpha \left((u^+)^{p-1}+(u^+)^{q-1}\right)-\beta \left((u^-)^{p-1}+(u^-)^{q-1}\right) $in a bounded domain $ \Omega $ in $ \mathbb{R}^N $ ($ N\geq1 $), with Dirichlet boundary condition and $ 1

Here, the anisotropic $ (p, q) $-Laplacian$ - \sum\limits_{i = 1}^N\frac{\partial}{\partial x_i}\left(\left|\frac{\partial u}{\partial x_i}\right|^{p_i-2}\frac{\partial u}{\partial x_i}\right) - \sum\limits_{i = 1}^N\frac{\partial}{\partial x_i}\left(\left|\frac{\partial u}{\partial x_i}\right|^{q_i-2}\frac{\partial u}{\partial x_i}\right) = \lambda u^{\gamma-1} $is considered, where $ \Omega $ is a bounded and regular domain of $ \mathbb{R}^N $, $ q_i\leq p_i $ for $ i = 1, \cdots, N $ and $ \gamma > 1 $. The existence of positive solution is proved via sub-supersolution method. [ABSTRACT FROM AUTHOR]

A variational principle is applied to examine a Muckenhoupt weighted p(·)-Laplacian equation on the Heisenberg groups. The existence of at least one positive radial solution to the problem under the Dirichlet boundary condition belongs to the first order Heisenberg-Sobolev spaces is proved. [ABSTRACT FROM AUTHOR]