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We consider the nonlinear Schrödinger equation \begin{equation*} Δu = \big( 1 +\varepsilon V_1(|y|)\big)u - |u|^{p-1}u \quad \text{in} \quad \mathbb{R}^N, \quad N\ge 3, \quad p \in \left(1, \frac{N+2}{N-2}\right).\end{equation*} The phenomenon of pattern formation has been a central theme in the study of nonlinear Schrödinger equations. However, the following nonexistence of $O(N)$ symmetry breaking solution is well-known: if the potential function is radial and radially nondecreasing, any positive solution must be radial. Therefore, solutions of interesting patterns, such as those with symmetry in a discrete subgroup of $O(N)$, can only exist after violating the assumptions. For a potential function that is radial but asymptotically decreasing, a solution with symmetry merely in a discrete subgroup of $O(2)$ has been presented. These observations pose the question of whether patterns of higher dimensions can appear. In this study, the existence of nonradial solutions whose symmetry group is a discrete subgroup of $O(3)$, more precisely, the orientation-preserving regular tetrahedral group is shown.

The super- and subsoltuion theory is developed when the nonlinear reaction function is discontinuous at stable steady states. The solution is defined in a weak sense using a notion of set-valued in...

We consider the following system of Schr\"odinger equations \begin{equation*}\left.\begin{cases} -\Delta U + \lambda U = \alpha_0 U^3+ \beta UV^2 -\Delta V + \mu(y) V = \alpha_1 V^3+\beta U^2V \end{cases}\right. \text{in} \quad \mathbb{R}^N, \ N=2, 3,\end{equation*} where $\lambda$, $\alpha_0$, $\alpha_1>0$ are positive constants, $\beta \in \mathbb{R}$ is the coupling constant, and $\mu: \mathbb{R}^N \rightarrow \mathbb{R}$ is a potential function. Continuing the work of Lin and Peng \cite{lin_peng_2014}, we present a solution of the type where one species has a peak at the origin and the other species has many peaks over a circle, but as seen in the above, coupling terms are nonlinear.

We consider competition systems of two species which have different dispersal strategies and interspecific competitive strengths. One of the dispersal strategies is random dispersal and the other is a Fokker-Planck diffusion whose motility is piecewise constant and jumps up when the resource is not enough. In this paper, first we show the Fokker-Planck diffusion allows ideal free distribution. Next we show the linear stability of semitrivial steady states is determined exactly by a threshold on the interspecific competitive strengths. Some conditions for coexistence and global asymptotic stability are also provided.

For $k =1,\cdots,K,$ let $M_k$ be a $q_k$-dimensional smooth compact framed manifold in $R^N$ with $q_k \in \{1,\cdots,N-1\} $. We consider the equation $-\varepsilon^2\Delta u + V(x)u - u^p = 0$ in $R^N$ where for each $k \in \{1,\cdots,K\}$ and some $m_k > 0,$ $V(x)=|\textrm{dist}(x,M_k)|^{m_k}+O(|\textrm{dist}(x,M_k)|^{m_k+1})$ as $\textrm{dist}(x,M_k) \to 0 $. For a sequence of $\varepsilon$ converging to zero, we will find a positive solution $u_{\varepsilon}$ of the equation which concentrates on $M_1\cup \dots \cup M_K$.

We find an interval of some parameters where there exists a sign-changing solution of nonlinear Schrodinger systems having a prescribed number of zeros.

We study the nonexistence of positive solutions for nonlinear elliptic systems with potentials vanishing at infinity, and establish the optimal vanishing order of potentials for the nonexistence of positive supersolutions in exterior domains.

For a singularly perturbed nonlinear elliptic equation e 2 Δ u − V ( x ) u + u p = 0 , x ∈ R N , we prove the existence of bump solutions concentrating around positive critical points of V when nonnegative V is not identically zero for p ∈ ( N N − 2 , N + 2 N − 2 ) or nonnegative V satisfies l i m i n f | x | → ∞ V ( x ) | x | 2 log | x | > 0 for p = N N − 2 .