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Fractional Semilinear Neumann Problem With Critical Nonlinearity, Zhenfeng Jin, Hongrui Sun
Fractional Semilinear Neumann Problem With Critical Nonlinearity, Zhenfeng Jin, Hongrui Sun
Turkish Journal of Mathematics
In this paper, we consider the following critical fractional semilinear Neumann problem \begin{equation*} \begin{cases} (-\Delta)^{1/2}u+\lambda u=u^{\frac{n+1}{n-1}},~u>0\quad&\, \mathrm{in}\ \Omega,\\ \partial_\nu{u}=0 &\mathrm{on}\ \partial\Omega, \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^n~(n\geq5)$ is a smooth bounded domain, $\lambda>0$ and $\nu$ is the outward unit normal to $\partial\Omega$. We prove that there exists a constant $\lambda_0>0$ such that the above problem admits a minimal energy solution for $\lambda<\lambda_0$. Moreover, if $\Omega$ is convex, we show that this solution is constant for sufficiently small $\lambda$.
Lions-Type Theorem Of The P-Laplacian And Applications, Yu Su, Zhaosheng Feng
Lions-Type Theorem Of The P-Laplacian And Applications, Yu Su, Zhaosheng Feng
School of Mathematical and Statistical Sciences Faculty Publications and Presentations
In this article, our aim is to establish a generalized version of Lions-type theorem for the p-Laplacian. As an application of this theorem, we consider the existence of ground state solution for the quasilinear elliptic equation with the critical growth.
Schrödinger-Poisson Systems With Singular Potential And Critical Exponent, Senli Liu, Haibo Chen, Zhaosheng Feng
Schrödinger-Poisson Systems With Singular Potential And Critical Exponent, Senli Liu, Haibo Chen, Zhaosheng Feng
School of Mathematical and Statistical Sciences Faculty Publications and Presentations
In this article we study the Schrödinger-Poisson system−∆ u+ V (| x|) u+ λφu= f (u), x∈ R3,−∆ φ= u2, x∈ R3, where V is a singular potential with the parameter α and the nonlinearity f satisfies critical growth. By applying a generalized version of Lions-type theorem and the Nehari manifold theory, we establish the existence of the nonnegative ground state solution when λ= 0. By the perturbation method, we obtain a nontrivial solution to above system when λ= 0.
New Multiple Solutions For A Schrödinger-Poisson System Involving Concave-Convex Nonlinearities, Chun-Yu Lei, Gao-Sheng Liu, Chang-Mu Chu, Hong-Min Suo
New Multiple Solutions For A Schrödinger-Poisson System Involving Concave-Convex Nonlinearities, Chun-Yu Lei, Gao-Sheng Liu, Chang-Mu Chu, Hong-Min Suo
Turkish Journal of Mathematics
In this paper, we study the following critical growth Schrödinger-Poisson system with concave-convex nonlinearities term $\left\{\begin{array} -\Delta u + u + \eta\varphi u = \lambda f(x) u^{q-1} + u^5, in R^3, \\ -\Delta \varphi = u^2, in R^3,\end{array}\right. $ where $1 < q < 2, \eta\in \mathbb{R}, \lambda > 0$ is a real parameter and $f \in L^{\frac{6}{6-q}} (\mathbb{R}^3)$ is a nonzero nonnegative function. Using the variational method, we obtain that there exists a positive constant $\lambda_* > 0$ such that for all $\lambda \in (0,\lambda_*)$, the system has at least two positive solutions.
A Nonexistence Result For Blowing Up Sign-Changing Solutions Of The Brezis-Nirenberg-Type Problem, Yessine Dammak
A Nonexistence Result For Blowing Up Sign-Changing Solutions Of The Brezis-Nirenberg-Type Problem, Yessine Dammak
Turkish Journal of Mathematics
We consider the Brezis-Nirenberg problem: $ -\triangle u= u ^{p-1}u\pm\varepsilon u\mbox{ in }\Omega;, \mbox{ with } u=0 \mbox{ on }\partial\Omega,$ where $\Omega$ is a smooth bounded domain in $\mathbb{R}^n$, $n\geq4$, $p+1=2n/(n-2)$ is the critical Sobolev exponent, and $\varepsilon > 0$ is a positive parameter. The main result of this paper shows that if $n\geq4$ there are no sign-changing solutions $u_\varepsilon$ of $(P_{-\varepsilon})$ with two positive and one negative blow up points.
Multiple Solutions Of A P(X)-Laplacian Equation Involving Critical Nonlinearities, Yuan Liang, Xianbin Wu, Qihu Zhang, Chunshan Zhao
Multiple Solutions Of A P(X)-Laplacian Equation Involving Critical Nonlinearities, Yuan Liang, Xianbin Wu, Qihu Zhang, Chunshan Zhao
Department of Mathematical Sciences Faculty Publications
In this paper, we consider the existence of multiple solutions for the following p(x)-Laplacian equations with critical Sobolev growth conditions
−div(|∇u| p(x)−2 ∇u) + |u| p(x)−2 u = f(x, u) in Ω,
u = 0 on ∂Ω.
We show the existence of infinitely many pairs of solutions by applying the Fountain Theorem and the Dual Fountain Theorem respectively. We also present a variant of the concentration-compactness principle, which is of independent interest.
On The Number Of Radially Symmetric Solutions To Dirichlet Problems With Jumping Nonlinearities Of Superlinear Order, Alfonso Castro, Hendrik J. Kuiper
On The Number Of Radially Symmetric Solutions To Dirichlet Problems With Jumping Nonlinearities Of Superlinear Order, Alfonso Castro, Hendrik J. Kuiper
All HMC Faculty Publications and Research
This paper is concerned with the multiplicity of radially symmetric solutions u(x) to the Dirichlet problem
Δu+f(u)=h(x)+cφ(x)
on the unit ball Ω⊂RN with boundary condition u=0 on ∂Ω. Here φ(x) is a positive function and f(u) is a function that is superlinear (but of subcritical growth) for large positive u, while for large negative u we have that f'(u)<μ, where μ is the smallest positive eigenvalue for Δψ+μψ=0 in Ω with ψ=0 on ∂Ω. It is shown that, given any integer k≥0, the value c may be chosen so large that there are 2k+1 solutions with k or less interior nodes. Existence of positive solutions is excluded for large enough values of c.
Positive Solutions For A Semilinear Elliptic Problem With Critical Exponent, Ismail Ali, Alfonso Castro
Positive Solutions For A Semilinear Elliptic Problem With Critical Exponent, Ismail Ali, Alfonso Castro
All HMC Faculty Publications and Research
No abstract provided in article.
Radial Solutions To A Dirichlet Problem Involving Critical Exponents When N=6, Alfonso Castro, Alexandra Kurepa
Radial Solutions To A Dirichlet Problem Involving Critical Exponents When N=6, Alfonso Castro, Alexandra Kurepa
All HMC Faculty Publications and Research
In this paper we show that, for each λ>0, the set of radially symmetric solutions to the boundary value problem
-Δu(x) = λu(x) + u(x)|u(x)|, x ε B := {x ε R6:|x|<1},
u(x) = 0, x ε ∂B
is bounded. Moreover, we establish geometric properties of the branches of solutions bifurcating from zero and from infinity.
Radially Symmetric Solutions To A Dirichlet Problem Involving Critical Exponents, Alfonso Castro, Alexandra Kurepa
Radially Symmetric Solutions To A Dirichlet Problem Involving Critical Exponents, Alfonso Castro, Alexandra Kurepa
All HMC Faculty Publications and Research
In this paper we answer, for N = 3,4, the question raised in [1] on the number of radially symmetric solutions to the boundary value problem -Δu(x) = λu(x) + u(x)|u(x)|^{4/(N-2)}, x ε B: = x ε RN:{|x| < 1}, u(x)=0, x ε ∂B, where Δ is the Laplacean operator and λ>0. Indeed, we prove that if N = 3,4, then for any λ>0 this problem has only finitely many radial solutions. For N = 3,4,5 we show that, for each λ>0, the set of radially symmetric solutions is bounded. Moreover, we establish geometric properties of the branches of solutions bifurcating from zero and from infinity.