Infinitely Many Solutions for a Non-homogeneous Differential Inclusion with Lack of Compactness

In this paper, we consider the following class of differential inclusion problems in {\mathbb{R}^{N}} involving the {p(x)} -Laplacian: -\Delta_{p(x)}u+V(x)\lvert u\rvert^{p(x)-2}u\in a(x)\partial F(x,u)\quad\text{% in}\ \mathbb{R}^{N}. We are concerned with a multiplicity property, and our arguments combine the variational principle for locally Lipschitz functions with the properties of the generalized Lebesgue–Sobolev space. Applying the nonsmooth symmetric mountain pass lemma and the fountain theorem, we establish conditions such that the associated energy functional possesses infinitely many critical points, and then we obtain infinitely many solutions.

Infinitely many solutions for the p-fractional Kirchhoff equations with electromagnetic fields and critical nonlinearity

In this paper, we consider the fractional Kirchhoff equations with electromagnetic fields and critical nonlinearity. By means of the concentration-compactness principle in fractional Sobolev space and the Kajikiya's new version of the symmetric mountain pass lemma, we obtain the existence of infinitely many solutions, which tend to zero for suitable positive parameters.

Infinitely Many Solutions for a Class of Fractional Impulsive Coupled Systems with (p,q)-Laplacian

By using the symmetric mountain pass lemma, we investigate the problem of existence of infinitely many solutions for a class of fractional impulsive coupled systems with (p,q)-Laplacian, which possesses mixed type nonlinearities, and the nonlinearities do not need to satisfy the well-known Ambrosetti-Rabinowitz condition.

Infinitely many solutions for a class of hemivariational inequalities involving p(x)-Laplacian

In this paper hemivariational inequality with nonhomogeneous Neumann boundary condition is investigated. The existence of infinitely many small solutions involving a class of p(x) - Laplacian equation in a smooth bounded domain is established. Our main tool is based on a version of the symmetric mountain pass lemma due to Kajikiya and the principle of symmetric criticality for a locally Lipschitz functional.

Infinitely many periodic solutions of non-autonomous second-order Hamiltonian systems

In this paper, we study the existence of infinitely many periodic solutions for the non-autonomous second-order Hamiltonian systems with symmetry. Based on the minimax methods in critical point theory, in particular, the fountain theorem of Bartsch and the symmetric mountain pass lemma due to Kajikiya, we obtain the existence results for both the superquadratic case and the subquadratic case, which unify and sharply improve some recent results in the literature.

Multiple Solutions for Generalized Asymptotical Linear Hamiltonian Systems Satisfying Bolza Boundary Conditions

This paper is devoted to multiple solutions of generalized asymptotical linear Hamiltonian systems satisfying Bolza boundary conditions. We classify the linear Hamiltonian systems by the index theory and obtain the existence and multiplicity of solutions for the Hamiltonian systems, based on an application of the classical symmetric mountain pass lemma.

Periodic Solutions of a Class of Fourth-Order Superlinear Differential Equations

This paper deals with the periodic solutions of a class of fourth-order superlinear differential equations. By using the classical variational techniques and symmetric mountain pass lemma, the periodic solutions of a single equation in literature are extended to that of equations, and also, the cubic growth of nonlinear term is extended to a general form of superlinear growth.