The purpose of this paper is to study $T$-periodic solutions to
[(-Δ_{x}+m^{2})^{s}-m^{2s}]u=f(x,u) &\mbox{in} (0,T)^{N} (P)
u(x+Te_{i})=u(x) &\mbox{for all} x \in \R^{N}, i=1, \dots, N
where $s\in (0,1)$, $N>2s$, $T>0$, $m> 0$ and $f(x,u)$ is a continuous function, $T$-periodic in $x$ and satisfying a suitable growth assumption weaker than the Ambrosetti-Rabinowitz condition. The nonlocal operator $(-Δ_{x}+m^{2})^{s}$ can be realized as the Dirichlet to Neumann map for a degenerate elliptic problem posed on the half-cylinder $\mathcal{S}_{T}=(0,T)^{N}\times (0,\infty)$. By using a variant of the Linking Theorem, we show that the extended problem in $\mathcal{S}_{T}$ admits a nontrivial solution $v(x,ξ)$ which is $T$-periodic in $x$. Moreover, by a procedure of limit as $m\rightarrow 0$, we also prove the existence of a nontrivial solution to (P) with $m=0$.
We consider a fractional double phase Robin problem involving variable order and variable exponents. The nonlinearity $f$ is a Carathéodory function satisfying some hypotheses which do not include the Ambrosetti-Rabinowitz type condition. By using Variational methods, we investigate the multiplicity of solutions.
We carry out an investigation of the existence of infinitely many solutions to a fractional $p$-Kirchhoff type problem with a singularity and a superlinear nonlinearity with a homogeneous Dirichlet boundary condition. Further the solution(s) will be proved to be bounded and a weak comparison principle has also been proved. A {\it `$C^1$ versus $W_0^{s,p}$'} analysis has also been discussed.
We consider a non-local Shrödinger problem driven by the fractional Orlicz g-Laplace operator as follows \begin{equation}\label{PP} (-\triangle_{g})^αu+g(u)=K(x)f(x,u),\ \ \text{in}\ \mathbb{R}^{d},\tag{P} \end{equation} where $d\geq 3,\ (-\triangle_{g})^α$ is the fractional Orlicz g-Laplace operator, $f:\mathbb{R}^d\times\mathbb{R}\rightarrow \mathbb{R}$ is a measurable function and $K$ is a positive continuous function. Employing the Nehari manifold method and without assuming the well-known Ambrosetti-Rabinowitz and differentiability conditions on the non-linear term $f$, we prove that the …
In this paper, we consider the following quasilinear elliptic problem with potential $$(P) \begin{cases} -\mbox{div}(ϕ(x,|\nabla u|)\nabla u)+ V(x)|u|^{q(x)-2}u= f(x,u) & \ \ \mbox{ in }Ω, u=0 & \ \ \mbox{ on } \partialΩ, \end{cases}$$ where $Ω$ is a smooth bounded domain in $\mathbb{R}^{N}$ ($N\geq 2$), $V$ is a given function in a generalized Lebesgue space $L^{s(x)}(Ω)$, and $f(x,u)$ is a Carathéodory function satisfying suitable growth conditions. Using variational arguments, we study the existence of weak solutions for $(P)$ in the framework of Musielak-Sobolev spaces. The main difficulty…
In this paper, we prove the existence of infinitely many solutions for a class of quasilinear elliptic $m(x)$-polyharmonic Kirchhoff equations where the nonlinear function has a quasicritical growth at infinity and without assuming the Ambrosetti and Rabinowitz type condition. The new aspect consists in employing the notion of a Schauder basis to verify the geometry of the symmetric mountain pass theorem. Furthermore, for the case $m(x)\equiv Const$, we introduce a positive quantity $λ_M$ similar to the first eigenvalue of the $m$-polyharmonic operator to find a mountain pass solution, and als…
We are interested in the existence of solutions for the following fractional $p(x,\cdot)$-Kirchhoff type problem $$ \left\{\begin{array}{ll} M \, \left(\displaystyle\int_{Ω\times Ω} \ \displaystyle{\frac{|u(x)-u(y)|^{p(x,y)}}{p(x,y) \ |x-y|^{N+p(x,y)s}}} \ dx \, dy\right)(-Δ)^{s}_{p(x,\cdot)}u = f(x,u), \quad x\in Ω, \\ \\ u= 0, \quad x\in \partialΩ, \end{array}\right.$$ where $Ω\subset\mathbb{R}^{N}$, $N\geq 2$ is a bounded smooth domain, $s\in(0,1),$ $p: \overlineΩ\times \overlineΩ \rightarrow (1, \infty)$, $(-Δ)^{s}_{p(x,\cdot)}$ denotes the $p(x,\cdot)$-fractional Laplace operator, $M: [0,…
We consider a nonlinear Robin problem driven by the sum of $p$-Laplacian and $q$-Laplacian (i.e. the $(p,q)$-equation). In the reaction there are competing effects of a singular term and a parametric perturbation $λf(z,x)$, which is Carathéodory and $(p-1)$-superlinear at $x\in\mathbb{R},$ without satisfying the Ambrosetti-Rabinowitz condition. Using variational tools, together with truncation and comparison techniques, we prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter $λ>0$ varies.
We apply the Fountain theorem to a class of nonlinear Dirac-Laplace equation on compact spin manifold. We show the standard Ambrosetti-Rabinowitz condition can be replaced by a more natural super-quadratic condition that is enough to obtain the Cerami condition under certain conditions. Multiple solutions of nonlinear Dirac-Laplace equation are obtained in this note.
This paper deals with some classes of Kirchhoff type problems on a double phase setting and with nonlinear boundary conditions. Under general assumptions, we provide multiplicity results for such problems in the case when the perturbations exhibit a suitable behavior in the origin and at infinity, or when they do not necessarily satisfy the Ambrosetti-Rabinowitz condition. To this aim, we combine variational methods, truncation arguments and topological tools.
We deal with a Dirichlet problem driven by the degenerate fractional p-Laplacian and involving a nonlinear reaction which satisfies, among other hypotheses, a (p-1)-linear growth at infinity with non-resonance above the first eigenvalue. The energy functional governing the problem is thus noncoercive. Thus we focus on the behavior of the reaction near the origin, assuming that it has a (p-1)-sublinear growth at zero, vanishes at three points, and satisfies a reverse Ambrosetti-Rabinowitz condition. Under such assumptions, by means of critical point theory and Morse theory, and using suitably t…
We consider the following problem $ -Δ_{p}u= h(x,u) \mbox{ in }Ω$, $u\in W^{1,p}_{0}(Ω)$, where $Ω$ is a bounded domain in $\mathbb{R}^{N}$, $1<p<N$, with a smooth boundary. In this paper we assume that $h(x,u)=a(x)f(u)+b(x)g(u)$ such that $f$ is regularly varying of index $p-1$ and superlinear at infinity. The function $g$ is a $p$-sublinear function at zero. The coefficients $a$ and $b$ belong to $L^{k}(Ω)$ for some $k>\frac{N}{p}$ and they are without sign condition. Firstly, we show a priori bound on solutions, then by using variational arguments, we prove the existence of at least two non…
In this work, we establish some abstract results on the perspective of the fractional Musielak-Sobolev spaces, such as: uniform convexity, Radon-Riesz property with respect to the modular function, $(S_{+})$-property, Brezis-Lieb type Lemma to the modular function and monotonicity results. Moreover, we apply the theory developed to study the existence of solutions to the following class of nonlocal problems \begin{equation*} \left\{ \begin{array}{ll} (-Δ)_{Φ_{x,y}}^s u = f(x,u),& \mbox{in }Ω, u=0,& \mbox{on }\mathbb{R}^N\setminus Ω, \end{array} \right. \end{equation*} where $N\geq 2$, $Ω\subse…
Multiple solutions for Grushin operator without odd nonlinearity
我们分别讨论了非齐次和齐次方程\begin{eqnarray*} (P_g)\quad - Δ_λ u + V(x) u = f(x,u)+g(x),\;\mbox{ in } \R^N,\; \end{eqnarray*}和\begin{eqnarray*} (P_0)\quad - Δ_λ u + V(x) u = K(x)f(x,u),\;\mbox{ in } \R^N,\; \end{eqnarray*}的解的存在性和多解性，其中$Δ_λ$是强退化算子，$V(x)$允许变号，$K\in C(\R^N,\R)$，$g:\R^N\to\R$是扰动，非线性项$f(x,u)$是不满足Ambrosetti-Rabinowitz超二次条件的连续函数（简称$(AR)$）.首先，利用山路引理和Ekeland变分原理，当$f$满足超线性增长条件时，得到了$(P_g)$两个不同解的存在性.此外，我们证明了存在无穷多个解的$(P_0)$，如果$f$是奇数$u$感谢克拉克定理附近的起源。因此，我们的主要结果大大改善了文献中出现的结果。
