On the existence and multiplicity of solutions for Dirichlet's problem for fractional differential equations

In this paper, by using variational methods and critical point theorems, we prove the existence and multiplicity of solutions for boundary value problem for fractional order differential equations where Riemann-Liouville fractional derivatives and Caputo fractional derivatives are used. Our results extend the second order boundary value problem to the non integer case. Moreover, some conditions to determinate nonnegative solutions are presented and examples are given to illustrate our results.

Three solutions for a mixed boundary value problem involving the one-dimensional p-Laplacian

AbstractThis paper deals with two mixed nonlinear boundary value problems depending on a parameter λ. For each of them we prove the existence of at least three generalized solutions when λ lies in an exactly determined open interval. Usefulness of this information on the interval is then emphasized by means of some consequences. Our main tool is a very recent three critical points theorem stated in [Topol. Methods Nonlinear Anal. 22 (2003) 93–104].

On a mixed boundary value problem involving the p-Laplacian

In this paper we prove the existence of infinitely many solutions for a mixed boundary value problem involving the one dimensional p-Laplacian. A result on the existence of three solutions is also established. The approach is based on multiple critical points theorems.

Multiple solutions for nonlinear nonhomogeneous resonant coercive problems

We consider a nonlinear, nonhomogeneous Dirichlet problem driven by the sum of a \begin{document}$p$\end{document} -Laplacian ( \begin{document}$2 ) and a Laplacian. The reaction term is a Caratheodory function \begin{document}$f(z,x)$\end{document} which is resonant with respect to the principal eigenvalue of ( \begin{document}$-\Delta_p,\, W^{1,p}_0(\Omega)$\end{document} ). Using variational methods combined with truncation and comparison techniques and Morse theory (critical groups) we prove the existence of three nontrivial smooth solutions all with sign information and under three different conditions concerning the behavior of \begin{document}$f(z,\cdot)$\end{document} near zero. By …

EXISTENCE OF THREE SOLUTIONS FOR A MIXED BOUNDARY VALUE PROBLEM WITH THE STURM-LIOUVILLE EQUATION

Abstract. The aim of this paper is to establish the existence of threesolutions for a Sturm-Liouville mixed boundary value problem. The ap-proach is based on multiple critical points theorems. 1. IntroductionThe aim of this paper is to establish, under a suitable set of assumptions, theexistence of at least three solutions for the following Sturm-Liouville problemwith mixed boundary conditions(RS λ )ˆ−(pu ′ ) ′ +qu = λf(t,u) in I =]a,b[u(a) = u ′ (b) = 0,where λ is a positive parameter and p, q, f are regular functions. To be precise,if f : [a,b] × R→ Ris a L 2 -Carath´eodory function and p,q ∈ L ∞ ([a,b]) suchthatp 0 := essinf t∈[a,b] p(t) > 0, q 0 := essinf t∈[a,b] q(t) ≥ 0,then we prove …

Existence theorems for inclusions of the type

For a family of operator inclusions with convex closed-valued right-hand sides in Banach spaces, the existence of solutions is obtained by chiefly using Ky Fan's fixed point principle. The main result of the paper improves Theorem 1 in [16] as well as Theorem 2.2 of [3]. Some meaningful concrete cases are also presented and discussed.

Esistenza e molteplicità di soluzioni per problemi differenziali non lineari con condizioni miste

A Mountain Pass Theorem for a Suitable Class of Functions

Ordinary (p_1,...,p_m)-Laplacian system with mixed boundary value

In this paper we prove the existence of multiple weak solutions for an ordinary mixed boundary value system with (p_1,...,p_m)-Laplacian by using recent results of critical points.

Three solutions for a quasilinear two point boundary value problem involving the one-dimensional p-Laplacian

In this paper we prove the existence of at least three classical solutions for the problem \begin{equation*} \left\{ \begin{array}{l} - \left( |u'|^{p-2} u' \right)' = \lambda f(t,u) h(u') \\ u(a)=u(b)=0, \end{array} \right. \end{equation*} \noindent when $\lambda$ lies in an explicitly determined open interval. Our main tool is a very recent three critical points theorem stated in D.Averna, G.Bonanno, {\em A three critical point theorem and its applications to the ordinary Dirichlet problem}, Topol. Methods Nonlinear Anal., 22 (2003), p.93-104.

Existence and asymptotic properties for quasilinear elliptic equations with gradient dependence

Abstract The paper focuses on a Dirichlet problem driven by the ( p , q ) -Laplacian containing a parameter μ > 0 in the principal part of the elliptic equation and a (convection) term fully depending on the solution and its gradient. Existence of solutions, uniqueness, a priori estimates, and asymptotic properties as μ → 0 and μ → ∞ are established under suitable conditions.

Multiple solutions for a Sturm-Liouville problem with periodic boundary conditions

The main purpose of this paper is to establish the existence of multiple solutions for a Sturm-Liouville problem with periodic boundary conditions. The approach is based on variational methods and multiple critical points theorems

Three solutions for a Neumann boundary value problem involving the p-Laplacian

In this note we prove the existence of an open interval ]λ', λ"[ for each λ of which a Neumann boundary value problem involving the p-Laplacian and depending on λ admits at least three solutions. The result is based on a recent three critical points theorem.

Positive solutions for the Neumann p-Laplacian

We examine parametric nonlinear Neumann problems driven by the p-Laplacian with asymptotically ( $$p-1$$ )-linear reaction term f(z, x) (as $$x\rightarrow +\infty $$ ). We determine the existence, nonexistence and minimality of positive solutions as the parameter $$\lambda >0$$ varies.

Infinitely many solutions to boundary value problem for fractional differential equations

Variational methods and critical point theorems are used to discuss existence of infinitely many solutions to boundary value problem for fractional order differential equations where Riemann-Liouville fractional derivatives and Caputo fractional derivatives are used. An example is given to illustrate our result.

Ordinary (p1,…,pm)-Laplacian systems with mixed boundary value conditions

Abstract In this paper we prove the existence of multiple weak solutions for an ordinary mixed boundary value system with ( p 1 , … , p m )-Laplacian by using recent results of critical points.

Existence of three solutions for a mixed boundary value system with (p_1,...,p_m)-Laplacian

In this paper we prove the existence of at least three weak solutions for a mixed boundary value system with (p_1,,...,p_m)-Laplacian. The approach is based on variational methods.

Multiple solutions for a Dirichlet problem with p-Laplacian and set-valued nonlinearity

AbstractThe existence of a negative solution, of a positive solution, and of a sign-changing solution to a Dirichlet eigenvalue problem with p-Laplacian and multi-valued nonlinearity is investigated via sub- and supersolution methods as well as variational techniques for nonsmooth functions.

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.

Infinitely many weak solutions for a mixed boundary value system with (p_1,…,p_m)-Laplacian

The aim of this paper is to prove the existence of infinitely many weak solu- tions for a mixed boundary value system with (p1, . . . , pm)-Laplacian. The approach is based on variational methods.

Positive solutions for nonlinear Robin problems with convection

We consider a nonlinear Robin problem driven by the p-Laplacian and with a convection term f(z,x,y). Without imposing any global growth condition on f(z,·,·) and using topological methods (the Leray-Schauder alternative principle), we show the existence of a positive smooth solution.