Multiple Solutions for Nonlinear Coercive Problems with a Nonhomogeneous Differential Operator and a Nonsmooth Potential

We consider a nonlinear elliptic problem driven by a nonlinear nonhomogeneous differential operator and a nonsmooth potential. We prove two multiplicity theorems for problems with coercive energy functional. In both theorems we produce three nontrivial smooth solutions. In the second multiplicity theorem, we provide precise sign information for all three solutions (the first positive, the second negative and the third nodal). Out approach is variational, based on the nonsmooth critical point theory. We also prove an auxiliary result relating smooth and Sobolev local minimizer for a large class of locally Lipschitz functionals.     

Multiple solutions for strongly resonant Robin problems

We consider nonlinear (driven by the p‐Laplacian) and semilinear Robin problems with indefinite potential and strong resonance with respect to the principal eigenvalue. Using variational methods and critical groups, we prove four multiplicity theorems producing up to four nontrivial smooth solutions.     

Cerami条件下脉冲边值问题古典解的存在性

在非线性项不满足Ambrosetti-Rabinowitz条件时研究脉冲微分方程边值问题,在原来的变分结构下,利用Cerami条件下成立的临界点理论来研究脉冲微分方程边值问题古典解的存在性和多重性.     

Landesman–Lazer Type Conditions and Multiplicity Results for Nonlinear Elliptic Problems with Neumann Boundary Values

We establish the existence and multiplicity of solutions for Steklov problems under non- resonance or resonance conditions using variational methods. In our main theorems, we consider a weighted eigenvalue problem of Steklov type.     

Existence and multiplicity of periodic solutions for some second order differential systems with p(t)-Laplacian

In this paper, we deal with the existence and multiplicity of periodic solutions for the p(t)-Laplacian Hamiltonian system. Some new existence theorems are obtained by using the least action principle and minmax methods in critical point theory, and our results generalize and improve some existence theorems.     

非线性项含一阶导数广义Sturm-Liouville边值问题的多个正解

By using fixed point theorems, we consider multiplicity of positive solutions for second-order generalized Sturm-Liouville boundary value problem, where the first order derivative is involved in the nonlinear term explicitly. We show the existence of multiple positive solutions for the problems. Example is given to illustrate the main results of the article.     

Multiplicity Results for Second-order Generalized Sturm-Liouville Boundary Value Problems with Dependence on the First Order Derivative

By using fixed point theorems,we consider multiplicity of positive solutions for second-order generalized Sturm-Liouville boundary value problem,where the first order derivative is involved in the nonlinear term explicitly.We show the existence of multiple positive solutions for the problems.Example is given to illustrate the main results of the article.     

Geodesics in stationary spacetimes and classical Lagrangian systems

We state a fundamental correspondence between geodesics on stationary spacetimes and the equations of classical particles on Riemannian manifolds, accelerated by a potential and a magnetic field. By variational methods, we prove some existence and multiplicity theorems for fixed energy solutions (joining two points or periodic) of the above described Riemannian equation. As a consequence, we obtain existence and multiplicity results for geodesics with fixed energy, connecting a point to a line or periodic trajectories, in (standard) stationary spacetimes.     

Existence and multiplicity of positive solutions for singular quasilinear problems

In this work we combine perturbation arguments and variational methods to study the existence and multiplicity of positive solutions for a class of singular p-Laplacian problems. In the first two theorems we prove the existence of solutions in the sense of distributions. By strengthening the hypotheses, in the third and last result, we establish the existence of two ordered positive weak solutions.     

Multiple Solutions for the Eigenvalue Problem of Nonlinear Fractional Differential Equations

In this article, multiple solutions for the eigenvalue problem of nonlinear fractional differential equation is considered. We obtain the existence and multiplicity results of positive solutions by using some fixed point theorems.     

Existence of solutions for a system of diffusion equations with spectrum point zero

In this paper, we consider the existence and multiplicity of homoclinic type solutions to a system of diffusion equations with spectrum point zero. By using some recent critical point theorems for strongly indefinite problems, we obtain at least one nontrivial solution and also infinitely many solutions.     

Periodic solutions of second order nonlinear difference equations with discrete -Laplacian

We use Brouwer degree to prove existence and multiplicity results for the periodic solutions of some nonlinear second order difference equations involving discrete -Laplacian. We obtain in particular upper and lower solutions theorems, Ambrosetti–Prodi type multiplicity results, sharp existence conditions for nonlinearities which are bounded from below or from above and necessary and sufficient conditions for the existence of positive periodic solutions when the nonlinearity is singular at 0.     

Sign-changing solutions for some fourth-order nonlinear elliptic problems

In this paper, we consider the existence and multiplicity of sign-changing solutions for some fourth-order nonlinear elliptic problems and some existence and multiple are obtained. The weak solutions are sought by means of sign-changing critical theorems.     

Singular Sturmian separation theorems for nonoscillatory symplectic difference systems

ABSTRACT

In this paper, we derive new singular Sturmian separation theorems for nonoscillatory symplectic difference systems on unbounded intervals. The novelty of the presented theory resides in two aspects. We introduce the multiplicity of a focal point at infinity for conjoined bases, which we incorporate into our new singular Sturmian separation theorems. At the same time we do not impose any controllability assumption on the symplectic system. The presented results naturally extend and complete the known Sturmian separation theorems on bounded intervals by J. V. Elyseeva [Comparative index for solutions of symplectic difference systems, Differential Equations 45(3) (2009), pp. 445–459, translated from Differencial'nyje Uravnenija 45 (2009), no. 3, 431–444], as well as the singular Sturmian separation theorems for eventually controllable symplectic systems on unbounded intervals by O. Do?lý and J. Elyseeva [Singular comparison theorems for discrete symplectic systems, J. Difference Equ. Appl. 20(8) (2014), pp. 1268–1288]. Our approach is based on developing the theory of comparative index on unbounded intervals and on the recent theory of recessive and dominant solutions at infinity for possibly uncontrollable symplectic systems by the authors [P. ?epitka and R. ?imon Hilscher, Recessive solutions for nonoscillatory discrete symplectic systems, Linear Algebra Appl. 469 (2015), pp. 243–275; P. ?epitka and R. ?imon Hilscher, Dominant and recessive solutions at infinity and genera of conjoined bases for discrete symplectic systems, J. Difference Equ. Appl. 23(4) (2017), pp. 657–698]. Some of our results, including the notion of the multiplicity of a focal point at infinity, are new even for an eventually controllable symplectic difference system.     

Existence and multiplicity results for periodic solutions of nonlinear difference equations

We use Brouwer degree to prove existence and multiplicity results for the periodic solutions of some nonlinear second-order and first-order difference equations. We obtain, in particular upper and lower solutions theorems, Ambrosetti–Prodi type results and sharp existence conditions for nonlinearities which are bounded from below or above.     

Existence and multiplicity of solutions for nonlocal systems involving fractional Laplacian with non-differentiable terms

In this paper, we devote to the study of the existence and multiplicity of solutions of nonlocal systems involving fractional Laplacian with non-differentiable terms using some extended critical point theorems for locally Lipschitz function on product spaces.     

Multiple solutions for certain nonlinear second-order systems

In this paper, we prove the existence of multiple solutions for Neumann and periodic problems. Our main tools are recent general multiplicity theorems proposed by B. Ricceri.     

Qualitative Phenomena for Some Classes of Quasilinear Elliptic Equations with Multiple Resonance

We consider nonlinear nonhomogeneous Dirichlet problems driven by the sum of a p-Laplacian and a Laplacian. The hypotheses on the reaction term incorporate problems resonant at both ±∞ and zero. We consider both cases p>2 and 1<p<2 (singular case) and we prove four multiplicity theorems producing three or four nontrivial solutions. For the case p>2 we provide precise sign information for all the solutions. Our approach uses critical point theory, truncation and comparison techniques, Morse theory and the Lyapunoff-Schmidt reduction method.     

Ljusternik-Schnirelman theory in partially ordered Hilbert spaces

We present several variants of Ljusternik-Schnirelman type theorems in partially ordered Hilbert spaces, which assert the locations of the critical points constructed by the minimax method in terms of the order structures. These results are applied to nonlinear Dirichlet boundary value problems to obtain the multiplicity of sign-changing solutions.

     

Existence and Multiplicity Results for the p-Laplacian with a p-Gradient Term

We study the existence and multiplicity of positive solutions to p-Laplace equations where the nonlinear term depends on a p-power of the gradient. For this purpose we combine Picone’s identity, blow-up arguments, the strong maximum principle and Liouville-type theorems to obtain a priori estimates. Sebastián Lorca: Supported by FONDECYT N ^o 1080500.