Let $1<q<p$ and $a\in C(\overlineΩ)$ be sign-changing, where $Ω$ is a bounded and smooth domain of $\mathbb{R}^{N}$. We show that the functional \[ I_{q}(u):=\int_Ω\left( \frac{1}{p}|\nabla u|^{p}-\frac{1}{q}a(x)|u|^{q}\right) , \] has exactly one nonnegative minimizer $U_{q}$ (in $W_{0}^{1,p}(Ω)$ or $W^{1,p}(Ω)$). In addition, we prove that $U_{q}$ is the only possible \textit{positive} solution of the associated Euler-Lagrange equation, which shows that this equation has at most one positive solution. Furthermore, we show that if $q$ is close enough to $p$ then $U_{q}$ is positive, which also guarantees that minimizers of $I_{q}$ do not change sign. Several of these results are new even for $p=2$.
The aim of this paper is analyzing existence, multiplicity, and regularity issues for the positive solutions of a Neumann boundary value problem of superlinear indefinite type related to the mean curvature operator with a sublinear potential at infinity.
We consider quasilinear Schrödinger equations in $\mathbb{R}^{N}$ of the form% \[ -Δu+V(x)u-uΔ(u^{2})=g(u)\text{,}% \] where $g(u)$ is $4$-superlinear. Unlike all known results in the literature, the Schrödinger operator $-Δ+V$ is allowed to be indefinite, hence the variational functional does not satisfy the mountain pass geometry. By a local linking argument and Morse theory, we obtain a nontrivial solution for the problem. In case that $g$ is odd, we get an unbounded sequence of solutions.
It is established existence of bound and ground state solutions for quasilinear elliptic systems driven by (ϕ1, ϕ2)-Laplacian operator. The main feature here is to consider quasilinear elliptic systems involving both nonsingular nonlinearities combined with indefinite potentials and singular cases perturbed by superlinear and subcritical couple terms. These prevent us to use arguments based on Ambrosetti-Rabinowitz condition and variational methods for differentiable functionals. By exploring the Nehari method and doing a fine analysis on the fibering map associated, we get estimates that allo…
In this paper, we study the existence and multiplicity results of nontrivial positive solutions to the following quasilinear elliptic equation on $\RN$, when $N\geq2$, \begin{equation} \Lp u=λ\hspace{0.2mm}K(x)u^{r-1}-V(x)u^{q-1}.\nonumber \end{equation} Here, $K(x),V(x)>0$ are suitable potentials, $1<p<r<q<\infty$, and $λ>0$ is a parameter. To study this problem, some compact embedding results regarding $\MVR\hookrightarrow\LKR$ are proved that unify and extend some recent results of the author \cite{Ha1,Ha2,Ha3}.
In this article we consider a variational problem related to a quasilinear singular problem and obtain a nonexistence result in a metric measure space with a doubling measure and a Poincaré inequality. Our method is purely variational and to the best of our knowledge, this is the first work concerning singular problems in a general metric setting.
Orlicz-Sobolev versus Holder local minimizer and multiplicity results for quasilinear elliptic equations
本文研究如下边值问题（P）- \mathrm{div}（|\nabla使用|）\nabla u）=f（x，u），\ & in $Ω$，u=0，& on $\partialΩ$ }具有非齐次主部.通过假设非线性项$f(x, t)$是次临界增长的，得到了问题（P）的一些抽象结果：（1）正则性;（2）Orlicz-Sobolev与Hölder局部极小;（3）强比较原理.应用这些抽象结果和临界点理论，我们证明了问题（P）在Orlicz-索伯列夫空间中多解的存在性.
We consider a system of quasilinear elliptic equations, with indefinite super-linear nonlinearity, depending on two real parameters $λ,μ$. By using the Nehari manifold and the notion of extremal parameter, we extend some results concerning existence of positive solutions.
In this paper we study the geometry of certain functionals associated to quasilinear elliptic boundary value problems with a degenerate nonlocal term of Kirchhoff type. Due to the degeneration of the nonlocal term it is not possible to directly use classical results such as uniform a-priori estimates and "Sobolev versus Hölder local minimizers" type of results. We prove that results similar to these hold true or not, depending on how degenerate the problem is. We apply our findings in order to show existence and multiplicity of solutions for the associated quasilinear equations, considering se…
We consider the problem $$ (P_λ)\quad -Δ_{p}u=λu^{p-1}+a(x)u^{q-1},\quad u\geq0\quad\mbox{ in }Ω$$ under Dirichlet or Neumann boundary conditions. Here $Ω$ is a smooth bounded domain of $\mathbb{R}^{N}$ ($N\geq1$), $λ\in\mathbb{R}$, $1<q<p$, and $a\in C(\overlineΩ)$ changes sign. These conditions enable the existence of dead core solutions for this problem, which may admit multiple nontrivial solutions. We show that for $λ<0$ the functional \[ I_λ(u):=\int_Ω\left( \frac{1}{p}|\nabla u|^{p}-\frac{λ}{p}|u|^{p}-\frac{1}{q}a(x)|u|^{q}\right) , \] defined in $X=W_{0}^{1,p}(Ω)$ or $X=W^{1,p}(Ω)$, ha…
This paper is devoted to the existence and non-existence of positive solutions for a $(p, q)$-Laplacian system with indefinite nonlinearity depending on two parameters $(λ,μ)$. By using the sub-supersolution method together with adaptations of ideas found in \cite{AAS,CRMZ}, we extend some previous result.
Let $Ω\subset\mathbb{R}^N$, $N\geq 2$, be an open bounded connected set. We consider the fractional weighted eigenvalue problem $(-Δ)^s u =λρu$ in $Ω$ with homogeneous Dirichlet boundary condition, where $(-Δ)^s$, $s\in (0,1)$, is the fractional Laplacian operator, $λ\in \mathbb{R}$ and $ ρ\in L^\infty(Ω)$. We study weak* continuity, convexity and Gâteaux differentiability of the map $ρ\mapsto1/λ_1(ρ)$, where $λ_1(ρ)$ is the first positive eigenvalue. Moreover, denoting by $\mathcal{G}(ρ_0)$ the class of rearrangements of $ρ_0$, we prove the existence of a minimizer of $λ_1(ρ)$ when $ρ$ varies…
Let $p \in (1,\infty)$ and $Ω\subset \mathbb{R}^N$ be a domain. Let $ A: =(a_{ij}) \in L^{\infty}_{\text{loc}}(Ω; \mathbb{R}^{N\times N})$ be a symmetric and locally uniformly positive definite matrix. Set $|ξ|_A^2:= \displaystyle \sum_{i,j=1}^N a_{ij}(x) ξ_i ξ_j$, $ξ\in \mathbb{R}^N$, and let $V$ be a given potential in a certain local Morrey space. We assume that the energy functional $$Q_{p,A,V}(ϕ):=\displaystyle \int_Ω [|\nabla ϕ|_A^p + V|ϕ|^p] {\rm dx} $$ is nonnegative in $W^{1,p}(Ω)\cap C_c(Ω)$. We introduce a generalized notion of $Q_{p,A,V}$-capacity and characterize the space of all …
We establish uniqueness results for quasilinear elliptic problems through the criterion recently provided in \cite{DFMST}. We apply it to generalized $p$-Laplacian subhomogeneous problems that may admit multiple nontrivial nonnegative solutions. Based on a generalized hidden convexity result, we show that uniqueness holds among strongly positive solutions and nonnegative global minimizers. Problems involving nonhomogeneous operators as the so-called $(p,r)$-Laplacian are also treated.
We study the solvability of second boundary value problems of fourth order equations of Abreu type arising from approximation of convex functionals whose Lagrangians depend on the gradient variable, subject to a convexity constraint. These functionals arise in different scientific disciplines such as Newton's problem of minimal resistance in physics and monopolist's problem in economics. The right hand sides of our Abreu type equations are quasilinear expressions of second order; they are highly singular and a priori just measures. However, our analysis in particular shows that minimizers of t…
We consider the $(1+1)$-dimensional quasilinear wave equation $g(x)w_{tt}-w_{xx}+h(x) (w_t^3)_t=0$ on $\mathbb{R}\times\mathbb{R}$ which arises in the study of localized electromagnetic waves modeled by Kerr-nonlinear Maxwell equations. We are interested in time-periodic, spatially localized solutions. Here $g\in L^{\infty}(\mathbb{R})$ is even with $g\not\equiv 0$ and $h(x)=γ\,δ_0(x)$ with $γ\in{\mathbb{R}}\backslash\{0\}$ and $δ_0$ the delta-distribution supported in $0$. We assume that $0$ lies in a spectral gap of the operators $L_k=-\frac{d^2}{dx^2}-k^2ω^2g$ on $L^2(\mathbb{R})$ for all $…
Quasilinear Problems with the Competition Between Convex and Concave Nonlinearities and Variable Potentials
本文证明了一类非线性椭圆问题-Δ p}u=λk（x）u^{q} \pmh（x）u ^{σ}，如果x为\in Ω，满足Dirichlet条件u_{1} = u_{2} =0，在\partial Ω上的解的存在性和不存在性.在我们的结果的证明中，我们使用了次上解方法和变分参数。本文得到的有关结果已在[Z. Guo和Z. Zhang，W^{1，p} versus C^{1} local minimizers and multiplicity results for quasilinear椭圆型椭圆型方程，Journal of Mathematical Analysis and Applications，Volume 286，Issue 1，Pages 32-50，1 October 2003.]对于k（x）=h（x）=1的情况。我们的结果揭示了一些有趣的行为，由于凹凸非线性和可变的潜力之间的相互作用的解决方案。
Periodic waves in the modified Korteweg-de Vries (mKdV) equation are revisited in the setting of the fractional Laplacian. Two families of solutions in the local case are given by the sign-definite dnoidal and sign-indefinite cnoidal solutions. Both solutions can be characterized in the general fractional case as global minimizers of the quadratic part of the energy functional subject to the fixed $L^4$ norm: the sign-definite (sign-indefinite) solutions are obtained in the subspace of even (odd) functions. Morse index is computed for both solutions and the spectral stability criterion is deri…
On singular problems in nonreflexive fractional Orlicz-Sobolev spaces
Instituto de Matemática e Estatística, Universidade Federal de Goiás（戈亚斯联邦大学数学与统计研究所）；Departamento de Matemática, Universidade Federal de Roraima（罗赖马联邦大学数学系）；Departamento de Matemática, Universidade de Brasília（巴西利亚大学数学系）
Maximization and minimization of the principal eigenvalue of the Laplacian with indefinite weight under Dirichlet and Robin boundary conditions on classes of rearrangements
本文研究带不定权重的拉普拉斯算子主特征值在重排类上的最小化与最大化问题，统一处理 Dirichlet 和 Robin 边界条件，并给出 Dirichlet 情形下唯一极大元的完整刻画。
