Clark's Theorem Research Articles

Multiplicity of solutions for a class of nonhomogeneous quasilinear elliptic system with locally symmetric condition in ℝN

This paper is concerned with a class of nonhomogeneous quasilinear elliptic system driven by the locally symmetric potential and the small continuous perturbations in the whole‐space . By a variant of Clark's theorem without the global symmetric condition and Moser's iteration technique, we obtain the existence of multiple solutions when the nonlinear term satisfies some growth conditions only in a circle with center 0, and the perturbation term is any continuous function with a small parameter and no any growth hypothesis.

Infinitely many solutions for a Schrödinger–Kirchhoff equations with concave and convex nonlinearities

This paper is concerned with the following Schrödinger–Kirchhoff equation − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δu + V ( x ) u = k ( x ) | u | q − 2 u − h ( x ) | u | p − 2 u , u ∈ H 1 ( R 3 ) , where a and b are positive constants, 1 < q < 2 < p < + ∞ . Under some suitable assumptions on V ( x ) , k ( x ) and h ( x ) , we obtain the existence of multiple solutions for this problem via a variant of Clark's theorem.

Existence and multiplicity of solutions involving the $ p(x) $-Laplacian equations: On the effect of two nonlocal terms

&lt;p style='text-indent:20px;'&gt;We study a class of &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ p(x) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-Kirchhoff problems which is seldom studied because the nonlinearity has nonstandard growth and contains a bi-nonlocal term. Based on variational methods, especially the Mountain pass theorem and Ekeland's variational principle, we obtain the existence of two nontrivial solutions for the problem under certain assumptions. We also apply the Symmetric mountain pass theorem and Clarke's theorem to establish the existence of infinitely many solutions. Our results generalize and extend several existing results.&lt;/p&gt;

Existence of infinitely many solutions for an anisotropic equation using genus theory

Using genus theory, the existence of infinitely many solutions for an anisotropic equation involving the subcritical growth is proved. Also, by using Krasnoselskii genus and Clark's theorem, the existence of ‐pairs of distinct solutions is proved. Finally, the existence of infinitely many solutions for an anisotropic equation involving the critical growth is studied.

Infinitely many homoclinic solutions for a class of damped vibration problems

In this paper, we consider the multiplicity of homoclinic solutions for the following damped vibration problems x ¨ ( t ) + B x ˙ ( t ) − A ( t ) x ( t ) + H x ( t , x ( t ) ) = 0 , where A ( t ) ∈ ( R , R N ) is a symmetric matrix for all t ∈ R , B = [ b i j ] is an antisymmetric N × N constant matrix, and H ( t , x ) ∈ C 1 ( R × B δ , R ) is only locally defined near the origin in x for some δ &gt; 0 . With the nonlinearity H ( t , x ) being partially sub-quadratic at zero, we obtain infinitely many homoclinic solutions near the origin by using a Clark's theorem.

On the nonlocal Schrödinger‐poisson type system in the Heisenberg group

This paper is concerned with a class of nonlocal Schrödinger‐Poisson type system with sublinear term. With the aid of the Ekeland's variational principle and the mountain pass theorem, the existence of negative energy solution, positive energy solution, and positive ground state solution is obtained, respectively. Moreover, we also obtain the multiplicity of solutions by using the Clark theorem. The novelty of this paper is that the equation has not only nonlocal term but also nonlocal coefficient.

Existence of critical points for noncoercive functionals with critical Sobolev exponent

We consider the following noncoercive functional with the critical Sobolev exponent where , , , . We prove the existence of a nontrivial critical point via the mountain pass theorem and the multiplicity via Clark's theorem.

Multiple Solution Results for Perturbed Fractional Differential Equations with Impulses

The multiplicity of classical solutions for impulsive fractional differential equations has been studied by many scholars. Using Morse theory, Brezis and Nirenberg’s Linking Theorem, and Clark theorem, we aim to solve this kind of problems. By this way, we obtain the existence of at least three classical solutions and k distinct pairs of classical solutions. Finally, an example is presented to illustrate the feasibility of the main results in this paper.

Existence and multiplicity of solutions for a class of damped-like fractional differential system

In this paper, we obtain some results about the existence and multiplicity of weak solutions for a class of damped-like fractional differential system with a parameter $\lambda$. When the nonlinear term is subquadratic only near the origin, we obtain that system has a ground state weak solution $u_\lambda$ if $\lambda$ is in some given interval, and when the nonlinear term is also even near the origin, then for each $\lambda>0$, system has infinitely many weak solutions $\{u_n^\lambda\}$ with $\|u_n^\lambda\|\to 0$ as $n\to \infty$. We mainly use Ekeland's variational principle and a variant of Clark's theorem together with a cut-off technique to prove our results.

INFINITELY MANY LOW- AND HIGH-ENERGY SOLUTIONS FOR A CLASS OF ELLIPTIC EQUATIONS WITH VARIABLE EXPONENT

This paper is concerned with the $ p(x) $-Laplacian equation of the form $\left\{ \begin{array}{l} - {\Delta _{p(x)}}u = Q(x)|u{|^{r(x) - 2}}u,{\rm{ in}}\;\Omega ,\\ {\rm{u = 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; on}}\;\partial \Omega , \end{array} \right.$ where $ \Omega\subset{\mathbb{R}}^N $ is a smooth bounded domain, $ 1 p^+ $ and $ Q: \overline{\Omega}\to{\mathbb{R}} $ is a nonnegative continuous function. We prove that (0.1) has infinitely many small solutions and infinitely many large solutions by using the Clark's theorem and the symmetric mountain pass lemma.

Infinitely many solutions for Kirchhoff-type variable-order fractional Laplacian problems involving variable exponents

ABSTRACT In this paper, we show the existence of infinitely many solutions for Kirchhoff-type variable-order fractional Laplacian problems involving variable exponents. More precisely, we consider where is a continuous function, Ω is a bounded domain in with for all is the variable-order fractional Laplace operator, and are two continuous functions, are two parameters and In addition to using the new version of Clark's theorem due to Liu and Wang to prove the existence of infinitely many solutions for the above problem, we also apply the symmetric mountain pass theorem, fountain theorem and dual fountain theorem to obtain the same conclusion. The main feature, as well as the main difficulty, of our problem is the fact that the Kirchhoff term M could be zero at zero.

Infinitely many solutions to a quasilinear Schrödinger equation with a local sublinear term

This paper discusses the quasilinear Schrödinger equation −Δu+V(x)u−Δ[(1+u2)12]u2(1+u2)12=K(x)f(u),x∈RN,where N⩾3. Under appropriate assumptions on the potentials V and K and local sublinear growth assumptions on the nonlinear term f, we get the existence of infinitely many nontrivial solutions by using a revised Clark theorem and a priori estimate of the solution.

Multiplicity results for a class of fractional differential equations with impulse

In this paper, we apply Morse theory, local linking arguments and the Clark theorem to a study of the multiplicity of nontrivial solutions for a class of impulsive fractional differential equations with Dirichlet boundary conditions.

Critical point theory to isotropic discrete boundary value problems on weighted finite graphs

In this paper, we use the Linking theorem, Clark theorem and Fountain theorem to obtain the existence of solutions to discrete boundary value problem involving the p-Laplacian on simple, connected, undirected and weighted graphs. We obtain three solutions, infinitely many solutions or a number of solutions depending on a parameter and the number of vertices. The eigenvalues of the nonlinear operator are concerned in the study.

Multiple periodic solutions for two classes of nonlinear difference systems involving classical (ϕ1,ϕ2)-Laplacian

In this paper, we investigate the existence of multiple periodic solutions for two classes of nonlinear difference systems involving $(\phi_1,\phi_2)$-Laplacian. First, by using an important critical point theorem due to B. Ricceri, we establish an existence theorem of three periodic solutions for the first nonlinear difference system with $(\phi_1,\phi_2)$-Laplacian and two parameters. Moreover, for the second nonlinear difference system with $(\phi_1,\phi_2)$-Laplacian, by using the Clark's Theorem, we obtain a multiplicity result of periodic solutions under a symmetric condition. Finally, two examples are given to verify our theorems.

A Variant of Clark's Theorem and Its Applications for Nonsmooth Functionals without the Palais--Smale Condition

By introducing a new notion of the genus with respect to the weak topology in Banach spaces, we prove a variant of Clark's theorem for nonsmooth functionals without the Palais--Smale condition. In this new theorem, the Palais--Smale condition is replaced by a weaker assumption, and a sequence of critical points converging weakly to zero with nonpositive energy is obtained. As applications, we obtain infinitely many solutions for a quasi-linear elliptic equation which is very degenerate and lacks strict convexity, and we also prove the existence of infinitely many homoclinic orbits for a second-order Hamiltonian system for which the functional is not in $C^1$ and does not satisfy the Palais--Smale condition. These solutions cannot be obtained via existing abstract theory.

Existence and multiplicity of systems of Kirchhoff‐type equations with general potentials

This paper is concerned with the following systems of Kirchhoff‐type equations: urn:x-wiley:mma:media:mma4007:mma4007-math-0001 Under more relaxed assumptions on V(x) and F(x,u,v), we first prove the existence of at least two nontrivial solutions for the aforementioned system by using Morse theory in combination with local linking arguments. Then by using the Clark theorem, the existence results of at least 2k distinct pairs of solutions are obtained. Some recent results from the literature are extended. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Existence and Multiplicity of Periodic Solutions to One-dimensional p -Laplacian

The paper deals with the existence and multiplicity of periodic solutions to one-dimensional p-Laplacian equation. Variational method using minimization and extended Clark's theorem are applied. An impulsive problem was also considered. Let p>1 be a real number and ...

Remarks on multiplicity of solutions for a subquadratic elliptic equation

We consider a subquadratic elliptic equation in a bounded domain Ω⊂RN (N⩾1). By the Clark Theorem, we obtain the existence and multiplicity of its nontrivial solutions, and we show that this result has a great relationship with Ω itself. The above argument can be extended to biharmonic equations.

On Clark's theorem and its applications to partially sublinear problems

In critical point theory, Clark's theorem asserts the existence of a sequence of negative critical values tending to 0 for even coercive functionals. We improve Clark's theorem, showing that such a functional has a sequence of critical points tending to 0. Our result also gives more detailed structure of the set of critical points near the origin. An extension of Clark's theorem is also given. Our abstract results are powerful in applications, and thus lead to much stronger results than those in the literature on existence of infinitely many solutions for partially sublinear problems such as elliptic equations and Hamiltonian systems.