Related papers: Bifurcations for a Coupled Schr\"odinger System wi…

In this paper, we consider the following overdetermined eigenvalue problem on an unbounded domain $\Omega\subset\mathbb{R}^{N+1}$ with $N\geq1$ \begin{equation} \left\{ \begin{array}{ll} -\Delta u=\lambda u\,\, &\text{in}\,\, \Omega,\\ u=0…

In this paper we study the bifurcation of branches of non-symmetric solutions from the symmetric branch of solutions to the Euler-Lagrange equations satisfied by optimal functions in functional inequalities of Caffarelli-Kohn-Nirenberg…

In this paper, we consider the singularly perturbed fractional Schr\"{o}dinger equation \begin{equation*} \epsilon^{2\alpha}(-\Delta)^\alpha u+V(x)u=f(u),\quad x\in \mathbb{R}^N, \end{equation*} where $\epsilon>0$ is a small parameter,…

We consider the following elliptic system with Neumann boundary: \begin{equation} \begin{cases} -\Delta u + \mu u=v^p, &\hbox{in } \Omega, \\-\Delta v + \mu v=u^q, &\hbox{in } \Omega, \\\frac{\partial u}{\partial n} = \frac{\partial…

We study positive solutions of the equation $-\Delta_g u + \lambda u = \lambda u^q$, with $\lambda >0$, $q>1$ on the round sphere $\mathbb{S}^n$ . We reduce the equation to an ordinary differential equation by considering isoparametric…

We study the multiplicity of positive solutions of the critical elliptic equation: $ \Delta_{\mathbb{S}^3} U = -(U^5 +\lambda U) \hspace{0.3cm}\hbox{ on } \Omega$ that vanish on the boundary of $\Omega$, where $\Omega$ is a region of…

We consider a Br\'ezis-Nirenberg type critical growth $p$-Laplacian problem involving a parameter $\mu > 0$ in a smooth bounded domain $\Omega$. We prove the existence of multiple nontrivial solutions if either $\mu$ or the volume of…

In this paper we prove the existence of infinitely many saddle-shaped positive solutions for non-cooperative nonlinear elliptic systems with bistable nonlinearities in the phase-separation regime. As an example, we prove that the system \[…

The paper addresses the doubly elliptic eigenvalue problem $$\begin{cases} -\Delta u=\lambda u \qquad &\text{in $\Omega$,}\\ u=0 &\text{on $\Gamma_0$,}\\ -\Delta_\Gamma u +\partial_\nu u =\lambda u\qquad &\text{on $\Gamma_1$,} \end{cases}…

Let $\Omega$ be a bounded, convex, centrally symmetric in $\mathbb{R}^{2}$ with a connected $C^{2,\epsilon}$ ($\epsilon\in(0,1)$) boundary. We show that, if the following overdetermined elliptic problem \begin{equation} -\Delta u=\alpha…

In this article, we study an elliptic problem of mixed order with both local and nonlocal aspects involving singular nonlinearity in combination with critical Hartree-type nonlinearity. Using variational methods together with the critical…

We study the following coupled Schr\"{o}dinger equations which have appeared as several models from mathematical physics: \begin{displaymath} \begin{cases}-\Delta u_1 +\la_1 u_1 = \mu_1 u_1^3+\beta u_1 u_2^2, \quad x\in \Omega,\\ -\Delta…

Operator splitting methods combined with finite element spatial discretizations are studied for time-dependent nonlinear Schr\"odinger equations. In particular, the Schr\"odinger-Poisson equation under homogeneous Dirichlet boundary…

We build blowing-up solutions to the critical elliptic system with Neumann boundary condition, \begin{equation*} \begin{cases} -\Delta u_1 + \lambda u_1 = u_1^{3} -\beta u_1u_2^2 & \text{in } \Omega, -\Delta u_2 + \lambda u_2 = u_2^{3}…

We address a two-dimensional nonlinear elliptic problem with a finite-amplitude periodic potential. For a class of separable symmetric potentials, we study the bifurcation of the first band gap in the spectrum of the linear Schr\"{o}dinger…

We seek discrete approximations to solutions $u:\Omega \to R$ of semilinear elliptic partial differential equations of the form $\Delta u + f_s(u) = 0$, where $f_s$ is a one-parameter family of nonlinear functions and $\Omega$ is a domain…

We consider an asymptotically linear Schr\"odinger equation $-\Delta u + V(x)u = \lambda u + f(x,u), \ x\in R^N$, and show that if $\lambda_0$ is an isolated eigenvalue for the linearization at infinity, then under some additional…

In this article, we prove the existence of at least one positive solution for the mixed local-nonlocal semipositone problem \begin{equation*} \left\{ \begin{aligned} -\Delta_p u+ (-\Delta)^s_p u &= \lambda f(u) && \text{in } \Omega, u &= 0…

We develop a new approach to the $L^p$ Dirichlet problem via $L^2$ estimates and reverse Holder inequalities. We apply this approach to second order elliptic systems and the polyharmonic equation on a bounded Lipschitz domain $\Omega$ in…

In this paper, we shall study global bifurcation phenomenon for the following Kirchhoff type problem \begin{equation} \left\{ \begin{array}{l} -\left(a+b\int_\Omega \vert \nabla u\vert^2\,dx\right)\Delta u=\lambda…