Nontrivial Solutions of the Asymmetric Beam System with Jumping Nonlinear Terms

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.</p

Critical Point Theory Applied to a Class of the Systems of the Superquadratic Wave Equations

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

Fractional N-Laplacian Problems Defined on the One-Dimensional Subspace

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

Multiplicity results for p-Laplacian boundary value problem with jumping nonlinearities

Abstract 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

Fourth order elliptic system with dirichlet boundary condition

<p>Abstract</p> <p>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 <it>&#955;_{<it>k </it>}&lt; <it>c </it>&lt; &#955;</it> _<it>k</it>+1and &#955;_{<it>k</it>+<it>n</it>}(<it>&#955;</it> _{<it>k</it>+<it>n </it>}- <it>c</it>) <it>&lt; a </it>+ <it>b &lt; &#955;</it> _{<it>k</it>+<it>n</it>+1}(<it>&#955;</it> _{<it>k</it>+<it>n</it>+1}- <it>c</it>). 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 <it>&#955;</it> _{<it>k </it>}&lt; <it>c </it>&lt; <it>&#955;</it> _<it>k</it>+1and <it>&#955;</it> _<it>k</it>(<it>&#955;</it> _{<it>k </it>}- <it>c</it>) &lt; 0, <it>a</it>+<it>b &lt; &#955;</it> _<it>k</it>+1(<it>&#955;</it> _<it>k</it>+1- <it>c</it>). We prove this result by the contraction mapping principle on the Banach space.</p> <p> <b>AMS Mathematics Subject Classification: </b>35J30, 35J48, 35J50</p