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 here is that the nonlinearity $f(x,u)$ considered does not satisfy the well-known Ambrosetti-Rabinowitz condition.
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 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,…
In this paper, we study the existence and multiplicity of weak solutions to a general type of Kirchhoff equations in magnetic fractional Orlicz-Sobolev spaces. Specifically, we appeal to Critical Point Theory to prove the existence of non-trivial solutions under the so-called Ambrosetti-Rabinowitz condition. We also state the existence of ground-state solutions. Moreover, multiplicity results which yield the existence of an unbounded sequence of solutions are also provided. Finally, we show existence under a weak-type Ambrosetti-Rabinowitz condition formulated in the framework of Orlicz spaces…
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…
We consider a double phase Robin problem with a Carathéodory nonlinearity. When the reaction is superlinear but without satisfying the Ambrosetti-Rabinowitz condition, we prove an existence theorem. When the reaction is resonant, we prove a multiplicity theorem. Our approach is Morse theoretic, using the notion of homological local linking.
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.
In this paper, we study the existence of non-negative non-trivial solutions for a class of double-phase problems where the source term is a Caratheodory function that satisfies the Ambrosetti-Rabinowitz type condition in the framework of Sobolev-Orlicz spaces with variable exponents in complete compact Riemannian n-manifolds. Our approach is based on the Nehari manifold and some variational techniques. Furthermore, the Hölder inequality, continuous and compact embedding results are proved.
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 are concerned with the following nonlinear Schrödinger equation \begin{eqnarray*} \begin{aligned} \begin{cases} -Δu+λu=f(u) \ \ {\rm in}\ \mathbb{R}^{2},\\ u\in H^{1}(\mathbb{R}^{2}),~~~ \int_{\mathbb{R}^2}u^2dx=ρ, \end{cases} \end{aligned} \end{eqnarray*} where $ρ>0$ is given, $λ\in\mathbb{R}$ arises as a Lagrange multiplier and $f$ satisfies an exponential critical growth. Without assuming the Ambrosetti-Rabinowitz condition, we show the existence of normalized ground state solutions for any $ρ>0$. The proof is based on a constrained minimization method and the Trudinger-Moser inequality …
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。
In this paper, we consider the following Kirchhoff type equation $$ -\left(a+ b\int_{\R^3}|\nabla u|^2\right)\triangle {u}+V(x)u=f(u),\,\,x\in\R^3, $$ where $a,b>0$ and $f\in C(\R,\R)$, and the potential $V\in C^1(\R^3,\R)$ is positive, bounded and satisfies suitable decay assumptions. By using a new perturbation approach together with a new version of global compactness lemma of Kirchhoff type, we prove the existence and multiplicity of bound state solutions for the above problem with a general nonlinearity. We especially point out that neither the corresponding Ambrosetti-Rabinowitz conditio…
In this paper we establish a new existence result for the quasilinear elliptic problem \[ -{\rm div}(A(x,u)|\nabla u|^{p-2}\nabla u) +\frac1p A_t(x,u)|\nabla u|^p + V(x)|u|^{p-2} u = g(x,u)\quad\mbox{ in } \mathbb{R}^N, \] with $N\ge 2$, $p>1$ and $V:\mathbb{R}^N\to\mathbb{R}$ suitable measurable positive function, which generalizes the modified Schrödinger equation. Here, we suppose that $A:\mathbb{R}^N\times\mathbb{R}\rightarrow\mathbb{R}$ is a $\mathcal{C}^{1}$-Carathéodory function such that $A_t(x,t) = \frac{\partial A}{\partial t} (x,t)$ and a given Carathéodory function $g:\mathbb{R}^N\…
