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This article concerns the multiplicity of solutions for a class of nonlocal and nonhomogeneous Neumann boundary value problems involving the p ( x )-Laplacian, in which both nonlinear terms assume critical growth. We use variational method, exploring an important truncation argument and properties of the genus.

In present paper, several conditions ensuring existence of three distinct solutions of a system of over-determined Fredholm fractional integro-differential equations on time scales are derived. Variational methods are utilized in the proofs.

Based on the basic theory and critical point theory of variable exponential Lebesgue Sobolev space, this paper investigates the existence and multiplicity of solutions for a class of nonlocal elliptic equations with Navier boundary value conditions when (AR) condition does not hold and improves or generalizes the original conclusions.

In the present work we are concerned with the existence and multiplicity of solutions for the following new Kirchhoff problem involving the p-Laplace operator: $$ \textstyle\begin{cases} - (a-b\int _{\Omega } \vert \nabla u \vert ^{p}\,dx ) \Delta _{p}u = \lambda \vert u \vert ^{q-2}u + g(x, u), & x \in \Omega , \\ u = 0, & x \in \partial \Omega , \end{cases} $$ { − ( a − b ∫ Ω | ∇ u | p d x ) Δ p u = λ | u | q − 2 u + g ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where $a, b > 0$ a , b > 0 , $\Delta _{p} u := \operatorname{div}(|\nabla u|^{p-2}\nabla u)$ Δ p u : = div ( | ∇ u | p − 2 ∇ u ) is the p-Laplace operator, $1 < p < N$ 1 < p < N , $p < q < p^{\ast }:=(Np)/(N-p)$ p < q < p ∗ : = ( N p ) / ( N − p ) , $\Omega \subset \mathbb{R}^{N}$ Ω ⊂ R N ($N \geq 3$ N ≥ 3 ) is a bounded smooth domain. Under suitable conditions on g, we show the existence and multiplicity of solutions in the case of high perturbations (λ large enough). The novelty of our work is the appearance of new nonlocal terms which present interesting difficulties.

In this paper the center problem for cubic differential systems with the line at infinity of multiplicity four is solved.

In this paper, we consider a quasilinear double phase problem on the whole space. Some new criteria to guarantee the existence of ground states for the considered problem is established using the m...

We study the Stokes eigenvalue problem under Navier boundary conditions in $$C^{1,1}$$ C 1 , 1 -domains $$\Omega \subset \mathbb {R}^3$$ Ω ⊂ R 3 . Differently from the Dirichlet boundary conditions, zero may be the least eigenvalue. We fully characterize the domains where this happens and we show that the ball is the unique domain where the zero eigenvalue is not simple, it has multiplicity three. We apply these results to show the validity/failure of a suitable Poincaré-type inequality. The proofs are obtained by combining analytic and geometric arguments.

In a previous paper by Pazsit and Pal [“Multiplicity Theory Beyond the Point Model,” Ann. Nucl. Energy, Vol. 154 (2021)], a general transport theory calculation of the factorial moments of the number of neutrons emitted spontaneously from a sample was elaborated. In contrast to the original derivations by Hage and Cifarelli [“On the Factorial Moments of the Neutron Multiplicity Distribution of Fission Cascades,” Nucl. Instrum. Meth. Phys. Res. A, Vol. 236 (1985)] and Bohnel [“The Effect of Multiplication on the Quantitative Determination of Spontaneously Fissioning Isotopes by Neutron Correlation Analysis,” Nucl. Sci. Eng., Vol. 90 (1985)], also referred to as the point model, in the transport model the spatial and angular dependence of the internal fission chain is taken into account with a one-speed transport theory treatment. Quantitative results were given for a spherical item, and the bias of the point model regarding the estimation of the fission rate as compared to the more exact space-dependent model was estimated as a function of the size of the sphere and the α factor. In the present paper the formalism and the quantitative work are extended to the treatment of items with cylindrical shapes, which are more relevant in many practical applications. Results are presented for both square cylinders (D=H) and for tall (H/D>1) and flat (H/D

A partial-limit method is developed to understand solutions related to real eigenvalues for the Fokas–Lenells equation. By applying a partial-limit procedure to soliton solutions of the Fokas–Lenells equation, new multiple-pole solutions related to real repeated eigenvalues are obtained. For the envelop $$|u|^2$$ , the simplest solution corresponds to a real double eigenvalue, showing a solitary wave with algebraic decay. Two such solitons allow elastic scattering but asymptotically with zero phase shift. Single eigenvalue with higher multiplicity gives rise to rational solutions which contain an intrinsic parameter, live on a zero background, and have slowly changing amplitudes. The partial-limit procedure may be extended to other integrable systems and generate new solutions.

In this paper, we consider the existence and multiplicity of solutions for a discrete Dirichlet boundary value problem involving the p , q -Laplacian. By using the critical point theory, we obtain the existence of infinitely many solutions under some suitable assumptions on the nonlinear term. Also, by our strong maximum principle, we can obtain the existence of infinitely many positive solutions.

In this paper, by using the concentration-compactness principle of Lions for variable exponents found in [Bonder JF, Silva A. Concentration-compactness principal for variable exponent space and app...

In this paper, we study a class of (p, q)-Schrödinger–Kirchhoff type equations involving a continuous positive potential satisfying del Pino–Felmer type conditions and a continuous nonlinearity with subcritical growth at infinity. By applying variational methods, penalization techniques and Lusternik–Schnirelman category theory, we relate the number of positive solutions with the topology of the set where the potential attains its minimum values.

In this paper, we study the number of classical positive radial solutions for Dirichlet problems of type (P) − d i v ∇ u 1 − | ∇ u | 2 = λ f ( u ) in B 1 , u = 0 on ∂ B 1 , $$\left\{\begin{aligned}\hfill & -\mathrm{d}\mathrm{i}\mathrm{v}\left(\frac{\nabla u}{\sqrt{1-\vert \nabla u{\vert }^{2}}}\right)=\lambda f\left(u\right)\quad \text{in} {B}_{1},\hfill \\ \hfill & u=0\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \text{on} \partial {B}_{1}, \hfill \end{aligned}\right.$$ where λ is a positive parameter, B 1 = { x ∈ R N : | x | < 1 } ${B}_{1}=\left\{x\in {\mathbb{R}}^{N}:\vert x\vert {< }1\right\}$ , f : [0, ∞) → [0, ∞) is a continuous function. Using the fixed point index in a cone, we prove the results on both uniqueness and multiplicity of positive radial solutions of (P).

In this paper, using the variational principle, the existence and multiplicity of solutions for p x , q x -Kirchhoff type problem with Navier boundary conditions are proved. At the same time, the sufficient conditions for the multiplicity of solutions are obtained.

This paper deals with the existence, uniqueness and the multiplicity for the boundary value problem of a class of fractional differential equations involving the Riemann-Liouville fractional derivative with nonlocal integral boundary conditions. By using the properties of the Green's function, the Banach contraction principle, the xed point index theory and the Leggett-Williams fixed point theorem, one shows the existence and the uniqueness of positive solutions and the existence and multiplicity of positive solutions. As applications, some examples are presented at the end to illustrate the main results.

In this paper, we show that if the mean curvature of a closed smooth embedded mean curvature flow in R^3 is of type-I, then the rescaled flow at the first finite singular time converges smoothly to a self-shrinker flow with multiplicity one. This result confirms Ilmanen's multiplicity-one conjecture under the assumption that the mean curvature is of type-I. As a corollary, we show that the mean curvature at the first singular time of a closed smooth embedded mean curvature flow in R^3 is at least of type-I.

In this paper, we study the following nonlinear magnetic Kirchhoff equation: { - ( a ⁢ ϵ 2 + b ⁢ ϵ ⁢ [ u ] A / ϵ 2 ) ⁢ Δ A / ϵ ⁢ u + V ⁢ ( x ) ⁢ u = f ⁢ ( | u | 2 ) ⁢ u in ⁢ ℝ 3 , u ∈ H 1 ⁢ ( ℝ 3 , ℂ ) , \left\{\begin{aligned} &\displaystyle{-}(a\epsilon^{2}+b\epsilon[u]_{A/% \epsilon}^{2})\Delta_{A/\epsilon}u+V(x)u=f(\lvert u\rvert^{2})u&&\displaystyle% \phantom{}\text{in }\mathbb{R}^{3},\\ &\displaystyle u\in H^{1}(\mathbb{R}^{3},\mathbb{C}),\end{aligned}\right. where ϵ > 0 {\epsilon>0} , a , b > 0 {a,b>0} are constants, V : ℝ 3 → ℝ {V:\mathbb{R}^{3}\rightarrow\mathbb{R}} and A : ℝ 3 → ℝ 3 {A:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}} are continuous potentials, and Δ A ⁢ u {\Delta_{A}u} is the magnetic Laplace operator. Under a local assumption on the potential V, by combining variational methods, a penalization technique and the Ljusternik–Schnirelmann theory, we prove multiplicity properties of solutions and concentration phenomena for ϵ small. In this problem, the function f is only continuous, which allows to consider larger classes of nonlinearities in the reaction.

This paper concerns the critical points and the level sets of solutions of the Grushin equation in the plane. After exactly establishing descriptions about the critical points of the homogeneous Gruhin-harmonic polynomials and investigating the local geometric properties of the level sets near these critical points, we prove that the critical points of solutions of the Grushin equation are isolated and each critical point has finite multiplicity. We further estimate the numbers of interior critical points of solutions of the Dirichlet boundary value problem.

In this paper, we discuss a class of Kirchhof-type elliptic boundary value problem with Sobolev–Hardy critical exponent and apply the variational method to obtain one positive solution and two nontrivial solutions to the problem under certain conditions.

In this paper, we study the following nonlocal problem: $$ \textstyle\begin{cases} - (a-b \int _{\Omega } \vert \nabla u \vert ^{2}\,dx ) \Delta u= \lambda \vert u \vert ^{q-2}u, & x\in \Omega , \\ u=0, & x\in \partial \Omega , \end{cases} $$ { − ( a − b ∫ Ω | ∇ u | 2 d x ) Δ u = λ | u | q − 2 u , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where Ω is a smooth bounded domain in $\mathbb{R}^{N}$ R N with $N\ge 3$ N ≥ 3 , $a,b>0$ a , b > 0 , $1< q 1 < q < 2 and $\lambda >0$ λ > 0 is a parameter. By virtue of the variational method and Nehari manifold, we prove the existence of multiple positive solutions for the nonlocal problem. As a co-product of our arguments, we also obtain the blow-up and the asymptotic behavior of these solutions as $b\searrow 0$ b ↘ 0 .

We investigate the multiplicity of solutions for the one-dimensional elliptic Dirichlet boundary value problem with jumping nonlinearities. We obtain three theorems depending on the source terms wh...

The aim of this paper is to study the following fractional critical problem with Hardy potential and concave–convex nonlinearities (−Δ)su−μu|x|2s=λuq+u2s∗−1,u>0 in Ω,u=0in RN∖Ω, where (−Δ)s is the ...

We propose a form of writing solutions to linear difference equations which eliminates the need for information on the multiplicity of the roots of the corresponding characteristic equations.

In this paper we study the existence and the multiplicity of nontrivial weak solutions for a fourth order variable exponent Kirchhoff type problem involving p(x)-biharmonic operator with changing sign weight and with no flux boundary condition. By using variational approach and the theory of variable exponent Sobolev spaces, we determine an interval of parameters for which this problem admits at least two nontrivial weak solutions.

We present a general formulation of the averaging method in the setting of a semilinear equation L x = e N ( x , e ) , being L a linear Fredholm mapping of index zero. Our general approach provides new results even in the classical periodic framework. Among the applications we obtained there are: a partial answer to an open problem related to the Liebau phenomenon, the multiplicity of periodic solutions for a planar system with delay and the existence of solution for a nonlocal chemical reactor.

In this paper, we study some results on the existence and multiplicity of solutions for a class of nonlocal quasilinear elliptic systems. In fact, we prove the existence of precise intervals of positive parameters such that the problem admits multiple solutions. Our approach is based on variational methods.

The multiplicity of periodic solutions for a class of second order Hamiltonian system with superquadratic plus subquadratic nonlinearity is studied in this paper. Obtained via the Symmetric Mountain Pass Lemma, two results about infinitely many periodic solutions of the systems extend some previously known results.

We provide sufficient conditions for the existence of one positive solution for a fourth--order beam equation with a discontinuous nonlinear term. Also a multiplicity result is established. They are based on a recent generalization of the Krasnosel’skiĭ fixed point theorem in cones.

In this paper we consider a class of second order Hamiltonian system with the nonlinearity of linear growth. Compared with the existing results, we do not assume an asymptotic of the nonlinearity at infinity to exist. Moreover, we allow the system to be resonant at zero. Under some general conditions, we will establish the existence and multiplicity of nontrivial periodic solutions by using the Morse theory and two critical point theorems.

By using the Krasnoselskii fixed point theorem, sufficient conditions are obtained for the existence and multiplicity of positive periodic solutions for a class of second order damped functional differential equations with multiple delays. Our results are a further expansion of the previous research results.

Using the recent geometric Parter-Wiener, etc. theorem and related results, it is shown that much of the multiplicity theory developed for real symmetric matrices associated with paths and generalized stars remains valid for combinatorially symmetric matrices over a field. A characterization of generalized stars in the case of combinatorially symmetric matrices is given.

We prove the existence and multiplicity of positive solutions to the singular $\phi$-Laplacian BVP \begin{align*} \begin{cases} -(r(t)\phi(u′))′=\lambda g(t)(f(u)-(a/(u^{\alpha}))),& t\in(0,1),\\ u(0)=0, u′(1)+H(u(1))=0 \end{cases} \end{align*} for a certain range of the parameter $\lambda>0$, where $a>0$, $\alpha\in(0,1)$, $\phi$ is an odd, increasing and convex homeomorphism on $\mathbb{R}$, and $f$ is $\phi$-superlinear at $\infty$.