Three nontrivial solutions of boundary value problems for semilinear $\Delta_{\gamma}-$Laplace equation

In this paper, we study the existence of multiple solutions for the boundary value problem\begin{equation}\Delta_{\gamma} u+f(x,u)=0 \quad \mbox{ in } \Omega, \quad \quad u=0 \quad \mbox{ on } \partial \Omega, \notag\end{equation}where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^N \ (N \ge 2)$ and $\Delta_{\gamma}$ is the subelliptic operator of the type $$\Delta_\gamma: =\sum\limits_{j=1}^{N}\partial_{x_j} \left(\gamma_j^2 \partial_{x_j} \right), \ \partial_{x_j}=\frac{\partial }{\partial x_{j}}, \gamma = (\gamma_1, \gamma_2, ..., \gamma_N), $$the nonlinearity $f(x , \xi)$ is subcritical growth and may be not satisfy the Ambrosetti-Rabinowitz (AR) condition. We establish the existence of three nontrivial solutions by using Morse theory.

The classical solution of one problem of an absolutely inelastic impact on a long elastic semi-infinite bar

In this article, we study the classical solution of the mixed problem in a quarter of a plane for a one-dimensional wave equation. On the bottom boundary, the Cauchy conditions are specified, meanwhile, the second of them has a discontinuity of the first kind at one point. The smooth boundary condition, which has the first and the second order derivatives, is set at the side boundary. The solution is built using the method of characteristics in an explicit analytical form. The uniqueness is proved and the conditions are established under which a piecewise-smooth solution exists. The problem with matcing conditions is considered.

Differences Between Robin and Neumann Eigenvalues

Let $$\Omega {\subset } {\mathbb {R}}^2$$ Ω ⊂ R 2 be a bounded planar domain, with piecewise smooth boundary $$\partial \Omega $$ ∂ Ω . For $$\sigma >0$$ σ > 0 , we consider the Robin boundary value problem $$\begin{aligned} -\Delta f =\lambda f, \qquad \frac{\partial f}{\partial n} + \sigma f = 0 \text{ on } \partial \Omega \end{aligned}$$ - Δ f = λ f , ∂ f ∂ n + σ f = 0 on ∂ Ω where $$ \frac{\partial f}{\partial n} $$ ∂ f ∂ n is the derivative in the direction of the outward pointing normal to $$\partial \Omega $$ ∂ Ω . Let $$0<\lambda ^\sigma _0\le \lambda ^\sigma _1\le \ldots $$ 0 < λ 0 σ ≤ λ 1 σ ≤ … be the corresponding eigenvalues. The purpose of this paper is to study the Robin–Neumann gaps $$\begin{aligned} d_n(\sigma ):=\lambda _n^\sigma -\lambda _n^0 . \end{aligned}$$ d n ( σ ) : = λ n σ - λ n 0 . For a wide class of planar domains we show that there is a limiting mean value, equal to $$2{\text {length}}(\partial \Omega )/{\text {area}}(\Omega )\cdot \sigma $$ 2 length ( ∂ Ω ) / area ( Ω ) · σ and in the smooth case, give an upper bound of $$d_n(\sigma )\le C(\Omega ) n^{1/3}\sigma $$ d n ( σ ) ≤ C ( Ω ) n 1 / 3 σ and a uniform lower bound. For ergodic billiards we show that along a density-one subsequence, the gaps converge to the mean value. We obtain further properties for rectangles, where we have a uniform upper bound, and for disks, where we improve the general upper bound.

A weak solution for a $(p(x),q(x))$-Laplacian elliptic problem with a singular term

Here, we consider the following elliptic problem with variable components: $$ -a(x)\Delta _{p(x)}u - b(x) \Delta _{q(x)}u+ \frac{u \vert u \vert ^{s-2}}{|x|^{s}}= \lambda f(x,u), $$ − a ( x ) Δ p ( x ) u − b ( x ) Δ q ( x ) u + u | u | s − 2 | x | s = λ f ( x , u ) , with Dirichlet boundary condition in a bounded domain in $\mathbb{R}^{N}$ R N with a smooth boundary. By applying the variational method, we prove the existence of at least one nontrivial weak solution to the problem.

Multiple positive solutions for a p-Laplace Benci–Cerami type problem (1 < p < 2), via Morse theory

Let us consider the quasilinear problem [Formula: see text] where [Formula: see text] is a bounded domain in [Formula: see text] with smooth boundary, [Formula: see text], [Formula: see text], [Formula: see text] is a parameter and [Formula: see text] is a continuous function with [Formula: see text], having a subcritical growth. We prove that there exists [Formula: see text] such that, for every [Formula: see text], [Formula: see text] has at least [Formula: see text] solutions, possibly counted with their multiplicities, where [Formula: see text] is the Poincaré polynomial of [Formula: see text]. Using Morse techniques, we furnish an interpretation of the multiplicity of a solution, in terms of positive distinct solutions of a quasilinear equation on [Formula: see text], approximating [Formula: see text].

On the fractional Lazer-McKenna conjecture with critical growth

This paper deals with the following fractional elliptic equation with critical exponent \[ \begin{cases} \displaystyle (-\Delta )^{s}u=u_{+}^{2_{s}^{*}-1}+\lambda u-\bar{\nu}\varphi_{1}, & \text{in}\ \Omega,\\ \displaystyle u=0, & \text{in}\ {{\mathfrak R}}^{N}\backslash \Omega, \end{cases}\] where $\lambda$ , $\bar {\nu }\in {{\mathfrak R}}$ , $s\in (0,1)$ , $2^{*}_{s}=({2N}/{N-2s})\,(N>2s)$ , $(-\Delta )^{s}$ is the fractional Laplace operator, $\Omega \subset {{\mathfrak R}}^{N}$ is a bounded domain with smooth boundary and $\varphi _{1}$ is the first positive eigenfunction of the fractional Laplace under the condition $u=0$ in ${{\mathfrak R}}^{N}\setminus \Omega$ . Under suitable conditions on $\lambda$ and $\bar {\nu }$ and using a Lyapunov-Schmidt reduction method, we prove the fractional version of the Lazer-McKenna conjecture which says that the equation above has infinitely many solutions as $|\bar \nu | \to \infty$ .

Global Existence of a Chemotactic Movement with Singular Sensitivity by Two Stimuli

In this study, we deal with the chemotaxis system with singular sensitivity by two stimuli under homogeneous Neumann boundary conditions in a bounded domain with smooth boundary. Under appropriate regularity assumptions on the initial data, we show that the system possesses global classical solution. Our results generalize and improve previously known ones.

Boundedness and Asymptotic Behavior to a Chemotaxis System with Indirect Signal Generation and Singular Sensitivity

In this paper, we consider the following indirect signal generation and singular sensitivity n t = Δ n + χ ∇ ⋅ n / φ c ∇ c , x ∈ Ω , t > 0 , c t = Δ c − c + w , x ∈ Ω , t > 0 , w t = Δ w − w + n , x ∈ Ω , t > 0 , in a bounded domain Ω ⊂ R N N = 2 , 3 with smooth boundary ∂ Ω . Under the nonflux boundary conditions for n , c , and w , we first eliminate the singularity of φ c by using the Neumann heat semigroup and then establish the global boundedness and rates of convergence for solution.