In this paper, we improve some results on the existence of at least three weak solutions for elliptic systems involving the $$(p_1(x),\ldots ,p_n(x))$$
 -Laplacian via an abstract result recently obtained by Ricceri.

Multiple solutions for a class of Neumann doubly eigenvalue boundary value systems involving the $$(p_1(x),\ldots ,p_n(x))$$-Laplacian

In this paper, by exploiting the Nehari manifold method and the Ljusternik–Schnirelman category theory, we establish the existence and multiplicity results of the nonlinear Schrodinger systems of the form 

$$\begin{aligned} \left\{ \begin{array}{l} -\epsilon ^{p\left( x\right) }div\left( \left| \nabla u\right| ^{p\left( x\right) -2}\nabla u\right) +V\left( z\right) \left| u\right| ^{p\left( x\right) -2}u=Q_{u}\left( u,v\right) +\gamma H_{u}\left( u,v\right) \text { in } {\mathbf {R}}^N, \\ -\epsilon ^{q\left( x\right) }div\left( \left| \nabla v\right| ^{q\left( x\right) -2}\nabla v\right) +W\left( z\right) \left| v\right| ^{q\left( x\right) -2}v=Q_{v}\left( u,v\right) +\gamma H_{v}\left( u,v\right) \text { in } {\mathbf {R}}^N, \\ \left( u,v\right) \in W^{1,p\left( x\right) }\left( {\mathbf {R}}^{N}\right) \times W^{1,q\left( x\right) }\left( {\mathbf {R}}^{N}\right) ,\text { }u\left( z\right) ,v\left( z\right) >0\text { for all } z\in {\mathbf {R}}^{N}. \end{array} \right. \end{aligned}$$

.

Multiplicity of solutions for Schrödinger systems with critical exponent via Ljusternik–Schnirelman theory

In this paper, we deal with the Schrodinger’s problems, in the first part we study the theoretical side, we show the existence of at least three weak solutions, our main tools are based on variational inequalities, more precisely, using the three critical points theorem due to Ricceri, existence and multiplicity results are established In the second part, we are interested in the application side, more exactly, we examine some computational problems on the discretization of finite elements of the p(x)-Laplacian, we propose a quasi-Newton minimization approach for the solution, our numerical tests show that these algorithms are able to resolve the problems with p(x)-Laplacian, for different values of p(x)

Existence, multiplicity and numerical examples for Schrödinger systems with nonstandard p ( x )-growth conditions

Abstract This paper deals with a class of the stationary nonlinear Schrödinger Systems in ℝ N {\mathbb{R}^{N}} . Our main tools are based on variational methods. More precisely, by using Ekeland’s Variational Principle, the Mountain Pass Theorem and the Nehari Manifold Method, an existence result is established.

Existence of solutions for nonlinear Schrödinger systems with periodic data perturbations

The existence of at least three weak solutions is established for a class of quasilinear elliptic systems involving the p(x)-Laplace operator with Neumann boundary condition. The technical approach is mainly based on a three critical points theorem due to Ricceri. MSC: 35D05; 35J60; 58E05.

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Three solutions for a class of quasilinear elliptic systems involving the p(x)-Laplace operator

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Three solutions for a class of quasilinear elliptic systems involving the p(x)-Laplace operator