Critical Point Theorem Research Articles

Existence and multiplicity of weak solutions for gradient-type systems wıth oscillatory nonlinearities on the sierpinski gasket

In this paper, we establish the existence and multiplicity results of solutions for parametric quasi-linear systems of the gradient-type on the Sierpinski gasket is proved. Our technical approach is based on variational methods and critical points theory and on certain analytic and geometrical properties of the Sierpinski fractal. Indeed, using a consequence of the local minimum theorem due to Bonanno, the Palais-Smale condition cut off upper at $r$, and the Palais-Smale condition for the Euler functional we investigate the existence of one and two solutions for our problem under algebraic conditions on the nonlinear part. Moreover by applying a different three critical point theorem due to Bonanno and Marano we guarantee the existence of third solution for our problem.

Multiplicity of solutions to discrete inclusions with the p(k)-Laplace type equations

AbstractIn this article, we prove the existence and multiplicity of solutions to discrete inclusions with the p(k)-Laplace type equations. We are interested in the existence of three solutions with the aid of linking arguments and using a three critical points theorem, for locally Lipschitz continuous fonctions.

Existence of Three Positive Solutions for a Nonlocal Singular Dirichlet Boundary Problem

Abstract In this article, we prove the existence of at least three positive solutions for the following nonlocal singular problem: { ( - Δ ) s ⁢ u = λ ⁢ f ⁢ ( u ) u q , u > 0 ⁢ in ⁢ Ω , u = 0 in ⁢ ℝ n ∖ Ω , \left\{\begin{aligned} \displaystyle(-\Delta)^{s}u&\displaystyle=\lambda\frac{% f(u)}{u^{q}},&&\displaystyle u>0\text{ in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{in }\mathbb{R}^{% n}\setminus\Omega,\end{aligned}\right. where ( - Δ ) s {(-\Delta)^{s}} denotes the fractional Laplace operator for s ∈ ( 0 , 1 ) {s\in(0,1)} , n > 2 ⁢ s {n>2s} , q ∈ ( 0 , 1 ) {q\in(0,1)} , λ > 0 {\lambda>0} and Ω is a smooth bounded domain in ℝ n {\mathbb{R}^{n}} . Here f : [ 0 , ∞ ) → [ 0 , ∞ ) {f:[0,\infty)\to[0,\infty)} is a continuous nondecreasing map satisfying lim u → ∞ ⁡ f ⁢ ( u ) u q + 1 = 0 . \lim_{u\to\infty}\frac{f(u)}{u^{q+1}}=0. We show that under certain additional assumptions on f, the above problem possesses at least three distinct solutions for a certain range of λ. We use the method of sub-supersolutions and a critical point theorem by Amann [H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 1976, 4, 620–709] to prove our results. Moreover, we prove a new existence result for a suitable infinite semipositone nonlocal problem which played a crucial role to obtain our main result and is of independent interest.

A critical point theorem in bounded convex sets and localization of Nash-type equilibria of nonvariational systems

The localization of a critical point of minimum type of a smooth functional is obtained in a bounded convex conical set defined by a norm and a concave upper semicontinuous functional. A vector version is also given in order to localize componentwise solutions of variational systems. The technique is then used for the localization and multiplicity of Nash-type positive equilibria of nonvariational systems. Applications are given to periodic problems.

Variational analysis to fourth-order impulsive differential equations

Applying two critical point theorems, we prove the existence of atleast three solutions for a one-dimensional fourth-order impulsive differential equation with two real parameters.

Multiple critical points of saddle geometry functionals

We study the multiplicity of critical points for continuously differentiable functionals on real Banach spaces. We prove that a functional which satisfies the assumptions of the Saddle Point Theorem and moreover is bounded from below has at least three critical points. Apparently, there is a global minimizer and a saddle point and we show the existence of a third critical point. The idea of the proof is based on the minus-gradient flow. This result is closely related to the three critical points theorem of H. Brezis and L. Nirenberg which assumes a local linking. Finally, we apply the result on the Dirichlet problem for semilinear stationary PDEs. The analysis includes, for example, the existence of multiple stationary solutions of bistable (or Allen–Cahn) equation and semipositone problems.

Variational Approaches to Kirchhoff-Type Second-Order Impulsive Differential Equations on the Half-Line

In this work we continue the study of the multiplicity results for a Kirchhoff-type second-order impulsive differential equation on the half-line. In fact, using a consequence of the local minimum theorem due Bonanno we look into the existence one solution under algebraic conditions on the nonlinear term and two solutions for the problem under algebraic conditions with the classical Ambrosetti–Rabinowitz condition on the nonlinear term. Furthermore, by employing two critical point theorems, one due Averna and Bonanno, and another one due Bonanno we guarantee the existence of two and three solutions for the problem in a special case.

Elliptic problems involving the p-Laplacian in ℝN with locally Lipschitz functional

ABSTRACTWe study the existence of infinitely many weak solutions for the elliptic type equation via a general nonsmooth critical point theorem.

Multiple Solutions for Fractional Boundary Value Problems

Variational methods and critical point theorems are used to discuss existence and multiplicity of solutions for fractional boundary value problem where Riemann–Liouville fractional derivatives and Caputo fractional derivatives are used. Some conditions to determinate nonnegative solutions are presented. An example is given to illustrate our results.

Multiplicity of solutions of elliptic system using two critical point theorem

In this paper, we consider the system of three elliptic equations using two critical point theorem. We prove the existence of two solutions for suitable forcing terms, under a condition on the linear part which prevents resonance with eigenvalues of the operator.

Existence Results for a 2nth-Order Differential Equation via Variational Methods

In this paper, we discuss the existence of solutions for a 2nth-order differential equation with Sturm–Liouville operator. Using critical point theorems due to Bonanno and Candito, we prove that there exist at least three classical solutions for the boundary value problem.

Existence and multiplicity of solutions for a nonlocal problem with critical Sobolev exponent

In this work, we are interested in considering the following nonlocal problem −a+b∫Ω|∇u|2dxΔu=μ|u|2∗−2u+λ|u|q−2u,x∈Ω,u=0,x∈∂Ω,where Ω⊂RN(N≥4) is a smooth bounded domain, a≥0,b>0,1<q<2,μ,λ>0 and 2∗=2NN−2 is the critical Sobolev exponent. By using the variational method and the critical point theorem, some existence and multiplicity results are obtained.

Schrödinger-Poisson systems with radial potentials and discontinuous quasilinear nonlinearity

We consider the following Schrödinger-Poisson system: $$ \begin{cases} -\triangle u+V(|x|)u+\phi u= Q(|x|) f(u) &amp;\hbox{in } \mathbb{R}^3,\\ -\triangle \phi=u^{2} &amp; \hbox{in } \mathbb{R}^3, \end{cases} $$ \noindent with more general radial potentials $V,Q$ and discontinuous nonlinearity $f$. The Lagrange functional may be locally {\it Lipschitz}. Using nonsmooth critical point theorem, we obtain the multiplicity results of radial solutions, we also show concentration properties of the solutions. This is in contrast with some recent papers concerning similar problems by using the classical Sobolev embedding theorems.

Existence and multiplicity of solutions for critical Kirchhoff-type p-Laplacian problems

In this paper, we deal with the existence, multiplicity and asymptotic behavior of solutions for the following stationary Kirchhoff problems involving the p-Laplacian:−M(∫Ω|∇u|pdx)Δpu=λf(x,u)+|u|p⁎−2uinΩ,u=0 on∂Ω, where Ω is a bounded smooth domain of RN, M:R0+→R0+ is a continuous function satisfying some extra assumptions. By using the mountain pass theorem, the existence of solutions is obtained. By applying an abstract critical point theorem based on the cohomological index, we prove the existence of multiple solutions for the above problem, when parameter λ belongs to some left neighborhood of the eigenvalue of the nonlinear operator −‖∇u‖(θ−1)pΔp, where p<N<θp<p⁎. We also investigate the asymptotic behavior of solutions when λ converges to infinity. 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, i.e. the problem is degenerate.

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.

Variational approach for a p-Laplacian boundary value problem on time scales

ABSTRACTIn this paper, we study a second-order p-Laplacian dynamic equation on a periodic time scale subject to certain boundary value conditions. The variational structure of the above-mentioned boundary value problem is presented. Some new results on the existence of at least one or three distinct periodic solutions are established by means of the saddle point theorem, the least action principle as well as the three critical point theorem.

Existence of solutions for nonlinear p-Laplacian difference equations

The aim of this paper is the study of existence of solutions for nonlinear $2n^{\mathrm{th}}$-order difference equations involving $p$-Laplacian. In the first part, the existence of a nontrivial homoclinic solution for a discrete $p$-Laplacian problem is proved. The proof is based on the mountain-pass theorem of Brezis and Nirenberg. Then, we study the existence of multiple solutions for a discrete $p$-Laplacian boundary value problem. In this case the proof is based on the three critical points theorem of Averna and Bonanno.

Perturbed nonlocal fourth order equations of Kirchhoff type with Navier boundary conditions

We investigate the existence of multiple solutions for perturbed nonlocal fourth-order equations of Kirchhoff type under Navier boundary conditions. We give some new criteria for guaranteeing that the perturbed fourth-order equations of Kirchhoff type have at least three weak solutions by using a variational method and some critical point theorems due to Ricceri. We extend and improve some recent results. Finally, by presenting two examples, we ensure the applicability of our results.

Variational methods for strongly indefinite problems

This paper reviews some recent development in studying variational methods for strongly indefinite problems under the support of National Natural Science Foundation of China. We firstly describe the ideas for establishing the variational setting, then state a deformation theory in locally convex topological linear spaces and several critical point theorems for dealing with strongly indefinite problems based on the deformation. Some applications of the theory are given including mainly: Existence and multiplicity of solutions to non-autonomous stationary Dirac system, in particular, the existence, concentration phenomena and the exponential decay of semi-classical states; existence and multiplicity of global solutions to the nonlinear (non-autonomous, unbounded Hamilton-type) reaction-diffusion system, especially the existence, concentration phenomena and decay of its ground states of the system with singular perturbation; depth study of homoclinic orbits Hamiltonian system, global Schrodinger equation; other initial work, such as bifurcation of Dirac equation on spin manifolds.

Localization of positive critical points in Banach spaces and applications

Two critical point theorems of M. Schechter in a ball of a Hilbert space are extended to uniformly convex Banach spaces by exploiting the properties of the duality mapping. Moreover, the critical points are sought in the intersection of a ball with a wedge, in particular with a cone, making possible applications to positive solutions of variational problems. The extension from Hilbert to Banach spaces not only requires a major refining of reasoning, but also a different statement by adding a third possibility to the original two alternatives from Schechter's results. The theory is applied to positive solutions for $p$-Laplace equations.