By establishing a new variational constraint, we prove that for $a_1 > 0$, $a_2 > 0$; $b_1\geq 0$, $b_2\geq 0$, $d\geq 0$ and $b_1 + b_2 + d \neq 0$; $\alpha > 1$, $\beta > 1$ with $ \alpha + \beta=p \in (2, 6)$; and for suitable $\lambda$, the following system $$ \left\{ \begin{aligned} - (a_1 + b_1\int_{\mathbb{R}^3} |\nabla u|^2 + d\int_{\mathbb{R}^3} |\nabla v|^2 ) \Delta u + u & = |u|^{p-2}u + \frac{\alpha\lambda}{p}|v|^\beta |u|^{\alpha-2}u , \\ -(a_2 + b_2\int_{\mathbb{R}^3} {|\nabla v|^2} + d\int_{\mathbb{R}^3} |\nabla u|^2) \Delta v + v & = |v|^{p-2}v + \frac{\beta\lambda}{p}|u|^\alpha |v|^{\beta-2}v \end{aligned} \right. $$ admits a least energy solution $(u,\ v) \in H^1(\mathbb{R}^3)\times H^1(\mathbb{R}^3)$ with $u > 0$ and $v > 0$. In the case of $a_1=a_2$ and $b_1=b_2$, several existence and nonexistence of solutions with special forms are also investigated.

This paper deals with the existence of nontrivial solutions for a class of nonlinear elliptic equations driven by an anisotropic Laplacian operator. In particular, the existence of two nontrivial solutions is obtained, adapting a two critical point results to a suitable functional framework that involves the anisotropic Sobolev spaces.

The aim of this paper is to establish the existence of nontrivial solutions for $p(x)$-Laplacian equations without any growth and Ambrosetti-Rabinowitz conditions. Employing the cutoff function approach, we show that auxiliary problem has at least one nontrivial solution. Furthermore, we obtain nontrivial solutions for original problems using De Giorgi iteration. The results presented here extend some recent contributions obtained for problems driven by the $p(x)$-Laplacian or even to more general differential operators.