Nonsmooth Critical Point Theory Research Articles

Infinitely many periodic solutions to a Lorentz force equation with singular electromagnetic potential

We consider the Lorentz force equationddt(mx˙1−|x˙|2/c2)=q(E(t,x)+x˙×B(t,x)),x∈R3, in the physically relevant case of a singular electric field E. Assuming that E and B are T-periodic in time and satisfy suitable further conditions, we prove the existence of infinitely many T-periodic solutions. The proof is based on a min-max principle of Lusternik-Schnirelmann type, in the framework of non-smooth critical point theory. Applications are given to the problem of the motion of a charged particle under the action of a Liénard-Wiechert potential and to the relativistic forced Kepler problem.

Superlinear elliptic hemivariational inequalities

We study a nonlinear nonhomogeneous Dirichlet problem with a nonsmooth potential which is superlinear but without satisfying the Ambrosetti-Rabinowitz condition. Using the nonsmooth critical point theory and critical groups we prove two multiplicity theorems producing three and five solutions respectively. In the second multiplicity theorem, we provide sign information for all the solutions and the solutions are ordered.

Multiple solutions for a class of [formula omitted]-biharmonic problems with potential boundary conditions

We are concerned with multiplicity of solutions for differential inclusions involving the p-biharmonic operator, submitted to a general potential multivalued boundary condition which, among others, covers many particular boundary conditions that are frequently invoked in the literature. Our approach is a variational one and relies on non-smooth critical point theory.

Existence results for Dirichlet double phase differential inclusions

"In this paper we consider a class of double phase differential inclusions of the type $$\left\{ \begin{array}{ll} -{\rm div\;}\left(|\nabla u|^{p-2}\nabla u+\mu(x)|\nabla u|^{q-2}\nabla u\right) \in \partial_C^2 f(x,u) , & \mbox{ in }\Omega,\\ u=0, & \mbox{ on }\partial\Omega, \end{array} \right.$$ where $\Omega \subset \mathbb{R}^N$, with $N\ge 2$, is a bounded domain with Lipschitz boundary, $f(x,t)$ is measurable w.r.t. the first variable on $\Omega$ and locally Lipschitz w.r.t. the second variable and $\partial_C^2 f(x,\cdot)$ stands for the Clarke subdifferential of $t\mapsto f(x,t)$. The variational formulation of the problem gives rise to a so-called hemivariational inequality and the corresponding energy functional is not differentiable, but only locally Lipschitz. We use nonsmooth critical point theory to prove the existence of at least one weak solution, provided the $\partial_C^2 f(x,\cdot)$ satisfies an appropriate growth condition."

On Nonsmooth Global Implicit Function Theorems for Locally Lipschitz Functions from Banach Spaces to Euclidean Spaces

In this paper, we establish a generalization of the Galewski-Rădulescu nonsmooth global implicit function theorem to locally Lipschitz functions defined from infinite dimensional Banach spaces into Euclidean spaces. Moreover, we derive, under suitable conditions, a series of results on the existence, uniqueness, and possible continuity of global implicit functions that parametrize the set of zeros of locally Lipschitz functions. Our methods rely on a nonsmooth critical point theory based on a generalization of the Ekeland variational principle.

Existence and concentration of ground state solutions for alogarithmic Schrödinger equation with the critical exponent

We study a class of logarithmic Schrödinger equations involving competing for weight potentials and critical nonlinearities on an unbounded domain. The corresponding energy functional has no continuous first derivative in its natural Sobolev space. Firstly, using the non-smooth critical point theory, we show that the 问题 admits one ground state positive 解. Then, we relate the concentration of solutions to a global minimum set of a suitable ground energy function via a Nehari manifold method. Finally, we give a sufficient condition forthe nonexistence of ground state solutions.

Existence and subharmonicity of solutions for nonsmooth $ p $-Laplacian systems

<abstract><p>In this paper we study nonlinear periodic systems driven by the vectorial $ p $-Laplacian with a nonsmooth locally Lipschitz potential function. Using variational methods based on nonsmooth critical point theory, some existence of periodic and subharmonic results are obtained, which improve and extend related works.</p></abstract>

Periodic solutions for a differential inclusion problem involving the p(t)-Laplacian

Abstract In the present paper, we consider the nonlinear periodic systems involving variable exponent driven by p(t)-Laplacian with a locally Lipschitz nonlinearity. Our arguments combine the variational principle for locally Lipschitz functions with the properties of the generalized Lebesgue-Sobolev space. Applying the non-smooth critical point theory, we establish some new existence results.

Existence and concentration of positive solutions for a class of discontinuous quasilinear Schr\xf6dinger problems in mathbb{R}^{N}

In this paper, a class of quasilinear Schrödinger equations with discontinuous nonlinearity is considered. After changing variables, by using nonsmooth critical point theory, we obtain the existence and concentration of positive solutions for this problem under suitable conditions. Our results cover and extend some results for these differentiable quasilinear Schrödinger problems.

Nonsmooth critical point theory and applications to the spectral graph theory

Existing critical point theories including metric and topological critical point theories are difficult to be applied directly to some concrete problems in particular polyhedral settings, because the notions of critical sets could be either very vague or too large. To overcome these difficulties, we develop the critical point theory for nonsmooth but Lipschitzian functions defined on convex polyhedrons. This yields natural extensions of classical results in the critical point theory, such as the Liusternik-Schnirelmann multiplicity theorem. More importantly, eigenvectors for some eigenvalue problems involving graph 1-Laplacian coincide with critical points of the corresponding functions on polytopes, which indicates that the critical point theory proposed in the present paper can be applied to study the nonlinear spectral graph theory.

Existence results for mixed hemivariational-like inequalities involving set-valued maps

A hemivariational-like inequality involving a set-valued map and a (nonlinear) bifunction is considered in this paper. The presence of the set-valued map ensures that not one, but two types of solution can be defined, while the presence of the bifunction does not allow the inequality to be written equivalently as an inclusion, making nonsmooth critical point theory unavailable. Using topological methods we are able to prove that our inequality possesses at least one solution (strong solution) provided the set-valued map is upper semicontinuous (lower semicontinuous, respectively). One of the main points of interest in our approach is that we are able to prove the existence of at least one solution even if the KKM technique fails, therefore some of our results are new even in the particular case when the problem reduces to a classical hemivariational, or variational inequality. In the last section we provide a nontrivial example for which our theoretical results are valid. More precisely, we consider a differential inclusion involving the Φ-Laplacian and mixed boundary conditions whose variational formulation leads to a hemivariational-like inequality.

A Nontrivial Solution of a Quasilinear Elliptic Equation Via Dual Approach

In this article, we are concerned with the existence of solutions of a quasilinear elliptic equation in ℝN which includes the so-called modified nonlinear Schrodinger equation as a special case. Combining the dual approach and the nonsmooth critical point theory, we obtain the existence of a nontrivial solution.

The jumping problem for nonlocal singular problems

We consider a jumping problem for nonlocal singular problems. We apply a recent variational approach for nonlocal singular problem, together with a minimax method in the framework of nonsmooth critical point theory.

Periodic solution for second-order impulsive differential inclusions with relativistic operator

In this paper, the boundary value problem of a second-order impulsive differential inclusion involving a relativistic operator is studied. First, the singular problem is reduced to an equivalent non-singular problem in order to better apply the variational methods. Then the existence of a periodic solution is obtained by nonsmooth critical point theory. Moreover, the boundedness and nonnegativity of solutions are obtained by restricting the discontinuous nonlinear term.

A variational inequality of Kirchhoff-type in mathbb{R}^{N}

In this paper, we investigate the existence of nontrivial radial solutions for a kind of variational inequalities in mathbb{R}^{N}. Our main technique is the non-smooth critical point theory, based on the Szulkin-type functionals.

Multiple orthogonal geodesic chords in nonconvex Riemannian disks using obstacles

We use nonsmooth critical point theory and the theory of geodesics with obstacle to show a multiplicity result about orthogonal geodesic chords in a Riemannian manifold (with boundary) which is homeomorphic to an N-disk. This applies to brake orbits in a potential well of a natural Hamiltonian system, providing a further step towards the proof of a celebrated conjecture by Seifert (Math Z 51:197–216, 1948).

On a global implicit function theorem for locally Lipschitz maps via non-smooth critical point theory

We prove a non-smooth generalization of the global implicit function theorem. More precisely we use the non-smooth local implicit function theorem and the non-smooth critical point theory in order to prove a non-smooth global implicit function theorem for locally Lipschitz functions. A comparison between several global inversion theorems is discussed. Applications to algebraic equations are given.

Existence and Multiplicity of Solutions for a Quasilinear Elliptic Inclusion with a Nonsmooth Potential

This paper is concerned with a nonlinear elliptic inclusion driven by a multivalued subdifferential of nonsmooth potential and a nonlinear inhomogeneous differential operator. We obtain two multiplicity theorems in the Orlicz-Sobolev space. In the first multiplicity theorem, we produce three nontrivial smooth solutions. Two of these solutions have constant sign (one is positive, the other is negative). In the second multiplicity theorem, we derive an unbounded sequence of critical points for the problem. Our approach is variational, based on the nonsmooth critical point theory. We also show that $C^1$-local minimizers are also local minimizers in the Orlicz-Sobolev space for a large class of locally Lipschitz functions.

Multiple solutions to a class of p(x)-biharmonic differential inclusion problem with no-flux boundary condition

In this paper, we obtain the existence of at least two nontrivial solutions for a p(x)-biharmonic differential inclusion problem with no-flux boundary condition. Our approach is variational and it is based on the nonsmooth critical point theory for locally Lipschitz functions.

Nonhomogeneous Hemivariational Inequalities with Indefinite Potential and Robin Boundary Condition

We consider a nonlinear, nonhomogeneous Robin problem with an indefinite potential and a nonsmooth primitive in the reaction term. In fact, the right-hand side of the problem (reaction term) is the Clarke subdifferential of a locally Lipschitz integrand. We assume that asymptotically this term is resonant with respect the principal eigenvalue (from the left). We prove the existence of three nontrivial smooth solutions, two of constant sign and the third nodal. We also show the existence of extremal constant sign solutions. The tools come from nonsmooth critical point theory and from global optimization (direct method).