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This paper deals with nonlocal fractional p-Laplacian problems with difference. We get a theorem which shows existence of a sequence of weak solutions for a family of nonlocal fractional p-Laplacian problems with difference. We first show that there exists a sequence of weak solutions for these problems on the finite-dimensional subspace. We next show that there exists a limit sequence of a sequence of weak solutions for finite-dimensional problems, and this limit sequence is a sequence of the solutions of our problems. We get this result by the estimate of the energy functional and the compactness property of continuous embedding inclusions between some special spaces.

We investigate the multiplicity of solutions for problems involving the fractional N-Laplacian. We obtain three theorems depending on the source terms in which the nonlinearities cross some eigenvalues. We obtain these results by direct computations with the eigenvalues and the corresponding eigenfunctions for the fractional N-Laplacian eigenvalue problem in the fractional Orlicz–Sobolev spaces, the contraction mapping principle on the fractional Orlicz–Sobolev spaces and Leray–Schauder degree theory.

We consider a boundary value problem for p-Laplacian systems with two singular and subcritical nonlinearities. We obtain one theorem which shows that there exists at least one nontrivial weak solution for these problems under some conditions. We obtain this result by variational method and critical point theory.

We investigate the multiplicity of solutions for one-dimensional p-Laplacian Dirichlet boundary value problem with jumping nonlinearities. We obtain three theorems: The first states that there exists exactly one solution when nonlinearities cross no eigenvalue. The second guarantees that there exist exactly two solutions, exactly one solution and no solution, depending on the source term, when nonlinearities cross just the first eigenvalue. The third claims that there exist at least three solutions, exactly one solution and no solution, depending on the source term, when nonlinearities cross the first and second eigenvalues. We obtain the first and second theorem by considering the eigenvalues and the corresponding normalized eigenfunctions of the p-Laplacian eigenvalue problem, and the contraction mapping principle in the p-Lebesgue space (when p≥2\\(p\\ge 2\\)). We obtain the third result by Leray–Schauder degree theory.

We investigate the multiplicity of the solutions of the fourth order elliptic system with Dirichlet boundary condition. We get two theorems. One theorem is that the fourth order elliptic system has at least two nontrivial solutions when λ k < c < λ k +1 and λ k + n ( λ k + n - c ) < a + b < λ k + n +1 ( λ k + n +1 - c ). We prove this result by the critical point theory and the variation of linking method. The other theorem is that the system has a unique nontrivial solution when λ k < c < λ k +1 and λ k ( λ k - c ) < 0, a + b < λ k +1 ( λ k +1 - c ). We prove this result by the contraction mapping principle on the Banach space. AMS Mathematics Subject Classification: 35J30, 35J48, 35J50

The research of the fractional Orlicz-Sobolev space and the fractional N-Laplacian operators will give the development of nonlinear elasticity theory, electro rheological fluids, non-Newtonian fluid theory in a porous medium as well as Probability and Analysis as they proved to be accurate models to describe different phenomena in Physics, Finance, Image processing and Ecology. We study the number of weak solutions for one-dimensional fractional N-Laplacian systems in the product of the fractional Orlicz-Sobolev spaces, where the corresponding functionals of one-dimensional fractional N-Laplacian systems are even and symmetric. We obtain two results for these problems. One result is that these problems have at least one nontrivial solution under some conditions. The other result is that these problems also have infinitely many weak solutions on the same conditions. We use the variational approach, critical point theory and homology theory on the product of the fractional Orlicz-Sobolev spaces.

We investigate the nontrivial solutions for a class of the systems of the superquadratic nonlinear wave equations with Dirichlet boundary condition and periodic condition with a superquadratic nonlinear terms at infinity which have continuous derivatives. We approach the variational method and use the critical point theory on the manifold, in terms of the limit relative category of the sublevel subsets of the corresponding functional.

We investigate the existence of multiple nontrivial solutions for perturbations and of the beam system with Dirichlet boundary condition in , in , where , and are nonzero constants. Here is the beam operator in , and the nonlinearity crosses the eigenvalues of the beam operator.

We show the existence of four -periodic solutions of the nonlinear Hamiltonian system with some conditions. We prove this problem by investigating the geometry of the sublevels of the functional and two pairs of sphere-torus variational linking inequalities of the functional and applying the critical point theory induced from the limit relative category.

We show the existence of a nontrivial solution for a class of the systems of the superquadratic nonlinear wave equations with Dirichlet boundary conditions and periodic conditions with a superquadratic nonlinear terms at infinity which have continuous derivatives. We approach the variational method and use the critical point theory which is the Linking Theorem for the strongly indefinite corresponding functional.