Multi-bump Solutions Research Articles

Normalized multi-bump solutions for saturable Schrödinger equations

AbstractIn this paper, we are concerned with the existence of multi-bump solutions for a class of semiclassical saturable Schrödinger equations with an density function:$$\begin{array}{} \displaystyle -{\it\Delta} v +{\it\Gamma} \frac{I(\varepsilon x) + v^2}{1+I(\varepsilon x) +v^2} v =\lambda v,\, x\in{{\mathbb{R}}^{2}}. \end{array}$$We prove that, with the density function being radially symmetric, for given integerk≥ 2 there exist a family of non-radial,k-bump type normalized solutions (i.e., with theL2constraint) which concentrate at the global maximum points of density functions whenε→ 0+. The proof is based on a variational method in particular on a convexity technique and the concentration-compactness method.

Numerical solution of the neural field equation in the presence of random disturbance

This paper aims at presenting an efficient and accurate numerical method for treating both deterministic- and stochastic-type neural field equations (NFEs) in the presence of external stimuli input (or without it). The devised numerical integration means belongs to the class of Galerkin-type spectral approximations. The particular effort is focused on an efficient practical implementation of the solution technique because of the partial integro-differential fashion of the NFEs in use, which are to be integrated, numerically. Our method is implemented in Matlab. Its practical performance and efficiency are investigated on three variants of an NFE model with external stimuli inputs. We study both the deterministic case of the mentioned model and its stochastic counterpart to observe important differences in the solution behavior. First, we observe only stable one-bump solutions in the deterministic neural field scenario, which, in general, will be preserved in our stochastic NFE scenario if the level of random disturbance is sufficiently small. Second, if the area of the external stimuli is large enough and exceeds the size of the bump, considerably, the stochastic neural field solution’s behavior may change dramatically and expose also two- and three-bump patterns. In addition, we show that strong random disturbances, which may occur in neural fields, fully alter the behavior of the deterministic NFE solution and allow for multi-bump (and even periodic-type) solutions to appear in all variants of the stochastic NFE model studied in this paper.

Sign-changing multi-bump solutions for the Chern-Simons-Schrödinger equations in ℝ2

Abstract In this paper we consider the nonlinear Chern-Simons-Schrödinger equations with general nonlinearity $$\begin{array}{} \displaystyle -{\it\Delta} u+\lambda V(|x|)u+\left(\frac{h^2(|x|)}{|x|^2}+\int\limits^{\infty}_{|x|}\frac{h(s)}{s}u^2(s)ds\right)u=f(u),\,\, x\in\mathbb R^2, \end{array}$$ where λ > 0, V is an external potential and $$\begin{array}{} \displaystyle h(s)=\frac{1}{2}\int\limits^s_0ru^2(r)dr=\frac{1}{4\pi}\int\limits_{B_s}u^2(x)dx \end{array}$$ is the so-called Chern-Simons term. Assuming that the external potential V is nonnegative continuous function with a potential well Ω := int V–1(0) consisting of k + 1 disjoint components Ω0, Ω1, Ω2 ⋯, Ωk, and the nonlinearity f has a general subcritical growth condition, we are able to establish the existence of sign-changing multi-bump solutions by using variational methods. Moreover, the concentration behavior of solutions as λ → +∞ are also considered.

A nondegeneracy condition for a semilinear elliptic system and the existence of 1- bump solutions

Combining situations originally considered in [7] - [8], a semilinear elliptic system is treated and a nondegeneracy condition leading to the existence of multibump solutions is considerably weakened.

Unstable normalized standing waves for the space periodic NLS

For the stationary nonlinear Schr\"odinger equation $-\Delta u+ V(x)u- f(u) = \lambda u$ with periodic potential $V$ we study the existence and stability properties of multibump solutions with prescribed $L^2$-norm. To this end we introduce a new nondegeneracy condition and develop new superposition techniques which allow to match the $L^2$-constraint. In this way we obtain the existence of infinitely many geometrically distinct solutions to the stationary problem. We then calculate the Morse index of these solutions with respect to the restriction of the underlying energy functional to the associated $L^2$-sphere, and we show their orbital instability with respect to the Schr\"odinger flow. Our results apply in both, the mass-subcritical and the mass-supercritical regime.

Solutions for conformally invariant fractional Laplacian equations with multi-bumps centered in lattices

In this paper, we consider the following nonlinear elliptic equation involving the fractional Laplacian with critical exponent:(−Δ)su=K(x)uN+2sN−2s,u>0inRN, where s∈(0,1) and N>2+2s, K>0 is periodic in (x1,…,xk) with 1≤k<N−2s2. Under some natural conditions on K near a critical point, we prove the existence of multi-bump solutions where the centers of bumps can be placed in some lattices in Rk, including infinite lattices. On the other hand, to obtain positive solution with infinite bumps such that the bumps locate in lattices in Rk, the restriction on 1≤k<N−2s2 is in some sense optimal, since we can show that for k≥N−2s2, no such solutions exist.

Multi-bump solutions for fractional Nirenberg problem

We consider the multi-bump solutions of the following fractional Nirenberg problem (0.1)(−Δ)su=K(x)un+2sn−2s,u>0inRn,where s∈(0,1) and n>2+2s. If K is a periodic function in some k variables with 1≤k<n−2s2, we proved that (0.1) has multi-bump solutions with bumps clustered on some lattice points in Rk via Lyapunov–Schmidt reduction. It is also established that the (0.1) has an infinite-many-bump solutions with bumps clustered on some lattice points in Rn which is isomorphic to Z+k.

Existence and concentration of positive solutions for Schrödinger-Poisson systems with steep well potential

This paper is devoted to study the following Schrödinger-Poisson system where λ is a positive parameter, a ∈ C(R3,R+) has a bounded potential well Ω = a−1(0), b ∈ C(R3, R) is allowed to be sign-changing, K ∈ C(R3, R+) and f ∈ C(R, R). Without the monotonicity of f(t)=/|t|3 and the Ambrosetti-Rabinowitz type condition, we establish the existence and exponential decay of positive multi-bump solutions of the above system for , and obtain the concentration of a family of solutions as λ →+∞, where is determined by terms of a, b, K and f. Our results improve and generalize the ones obtained by C. O. Alves, M. B. Yang [3] and X. Zhang, S. W. Ma [38].

Sign-changing multi-bump solutions for Kirchhoff-type equations in &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$\mathbb{R}^3$\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;

We are interested in the existence of sign-changing multi-bump solutions for the following Kirchhoff equation $ - (a + b\int_{{\mathbb{R}^3}} {|\nabla u{|^2}dx} )\Delta u + \lambda V(x)u = f(u),\;x \in {\mathbb{R}^3},$ where $λ$&gt;0 is a parameter and the potential $V(x)$ is a nonnegative continuous function with a potential well $Ω: = int(V^{-1}(0))$ which possesses $k$ disjoint bounded components $Ω_1,Ω_2,···,Ω_k$. Under some conditions imposed on $f(u)$, multiple sign-changing multi-bump solutions are obtained. Moreover, the concentration behavior of these solutions as $λ→ +∞$ are also studied.

Multi-bump solutions for logarithmic Schrödinger equations

We study spatially periodic logarithmic Schr\"odinger equations: \begin{equation}\tag{LS} -\Delta u + V(x)u=Q(x)u\log u^2, \quad u>0\quad \text{in}\ \mathbb{R}^N, \end{equation} where $N\geq 1$ and $V(x)$, $Q(x)$ are spatially $1$-periodic functions of class $C^1$. We take an approach using spatially $2L$-periodic problems ($L\gg 1$) and we show the existence of infinitely many multi-bump solutions of $(LS)$ which are distinct under $\mathbb{Z}^N$-action.

On multi-bump solutions of nonlinear Schrödinger equation with electromagnetic fields and critical nonlinearity in $$\mathbb {R}^N$$ R N

In this paper, we study a class of nonlinear Schrodinger equations with electromagnetic fields and critical nonlinearity in \(\mathbb {R}^N: -\Delta _Au + (\lambda V(x) + Z(x))u = \beta f(|u|^2)u + |u|^{2^*-2}u\), where f is a continuous function, \(V, Z: \mathbb {R}^N \rightarrow \mathbb {R}\) are continuous functions verifying suitable hypotheses. We show that if the zero set of V(x) has several isolated connected components \(\Omega _1,\ldots ,\Omega _k\) such that the interior of \(\Omega _i\) is not empty and \(\partial \Omega _i\) is smooth, then for \(\lambda > 0\) large there exists, for any non-empty subset \(\Gamma \subset \{1,\ldots ,k\}\), a bump solution trapped in a neighborhood of \(\cup _{j\in \Gamma }\Omega _j\).

On a two-component Bose–Einstein condensate with steep potential wells

In this paper, we study the following two-component systems of nonlinear Schrodinger equations $$\begin{aligned} \left\{ \begin{array}{ll} \Delta u-(\lambda a(x)+a_0(x))u+\mu _1u^3+\beta v^2u=0&{}\quad \text {in }\mathbb \Delta v-(\lambda b(x)+b_0(x))v+\mu _2v^3+\beta u^2v=0&{}\quad \text {in }\mathbb u,v\in H^1(\mathbb {R}^3), u,v>0&{}\quad \text {in }\mathbb {R}^3, \end{array}\right. \end{aligned}$$ where \(\lambda ,\mu _1,\mu _2>0\) and \(\beta <0\) are parameters; \(a(x), b(x)\ge 0\) are steep potentials and \(a_0(x),b_0(x)\) are sign-changing weight functions; a(x), b(x), \(a_0(x)\) and \(b_0(x)\) are not necessarily to be radial symmetric. By the variational method, we obtain a ground state solution and multi-bump solutions for such systems with \(\lambda \) sufficiently large. The concentration behaviors of solutions as both \(\lambda \rightarrow +\infty \) and \(\beta \rightarrow -\infty \) are also considered. In particular, the phenomenon of phase separations is observed in the whole space \(\mathbb {R}^3\). In the Hartree–Fock theory, this provides a theoretical enlightenment of phase separation in \(\mathbb {R}^3\) for the 2-mixtures of Bose–Einstein condensates.

Existence and multiplicity results for a class of quasilinear Schrödinger equations in involving critical growth

The existence of multi-bump solutions for the following class of quasilinear Schrödinger equations in is established, where and h is a continuous function, are continuous functions verifying some hypotheses. By a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem for suitable assumptions. We show that if the zero set of V(x) has several isolated connected components such that the interior of is not empty and is smooth, then for large there exists, for any non-empty subset , a bump solution which is trapped in a neighbourhood of .

Multi-bump solutions for singularly perturbed Schrödinger equations in $\mathbb{R}^2$ with general nonlinearities

We are concerned with the following equation: $$ -\varepsilon^2\Delta u+V(x)u=f(u),\quad u(x)&gt; 0\quad \mbox{in } \mathbb{R}^2. $$ By a variational approach, we construct a solution $u_\varepsilon$ which concentrates, as $\varepsilon \to 0$, around arbitrarily given isolated local minima of the confining potential $V$: here the nonlinearity $f$ has a quite general Moser's critical growth, as in particular we do not require the {\it monotonicity} of $f(s)/s$ nor the {\it Ambrosetti-Rabinowitz} condition.

Multiplicity of Multi-Bump Type Nodal Solutions for A Class of Elliptic Problems with Exponential Critical Growth in ℝ2

AbstractIn this paper we establish the existence and multiplicity of multi-bump nodal solutions for the class of problemswhereλ ∈(0, ∞),fis a continuous function with exponential critical growth andV: ℝ2→ℝ is a continuous function verifying some hypotheses.

Existence of multi-bump solutions for a class of elliptic problems involving the biharmonic operator

Using variational methods, we establish existence of multi-bump solutions for the following class of problems $$\begin{aligned} \left\{ \begin{array}{l} \Delta ^2 u +(\lambda V(x)+1)u = f(u), \quad \text{ in } \quad \mathbb {R}^{N}, u \in H^{2}(\mathbb {R}^{N}), \end{array} \right. \end{aligned}$$ where \(N \ge 1\), \(\Delta ^2\) is the biharmonic operator, f is a continuous function with subcritical growth, \(V : \mathbb {R}^N \rightarrow \mathbb {R}\) is a continuous function verifying some conditions and \(\lambda >0\) is a real constant large enough.

Existence of multi-bump solutions for the fractional Schrödinger-Poisson system

This paper considers the fractional Schrödinger-Poisson system in ℝ3. We prove that the problem has m-bump solutions under some given conditions which are given in the Introduction. Moreover, the system has more and more multi-bump solutions as ϵ → 0.

Existence of positive multi-bump solutions for a Schrödinger-Poisson system in $\mathbb{R}^{3}$

In this paper we are going to study a class of Schrodinger-Poisson system $$ \left\{ \begin{array}{ll} - \Delta u + (\lambda a(x)+1)u+ \phi u = f(u) \mbox{ in } \,\,\, \mathbb{R}^{3},\\ -\Delta \phi=4\pi u^2 \mbox{ in } \,\,\, \mathbb{R}^{3}.\\ \end{array} \right. $$ Assuming that the nonnegative function $a(x)$ has a potential well $int (a^{-1}(\{0\}))$ consisting of $k$ disjoint bounded components $\Omega_1, \Omega_2, ....., \Omega_k$ and the nonlinearity $f(t)$ has a subcritical growth, we are able to establish the existence of positive multi-bump solutions by variational methods.

Chaotic Orbits for Systems of Nonlocal Equations

We consider a system of nonlocal equations driven by a perturbed periodic potential. We construct multibump solutions that connect one integer point to another one in a prescribed way. In particular, heteroclinc, homoclinic and chaotic trajectories are constructed. This is the first attempt to consider a nonlocal version of this type of dynamical systems in a variational setting and the first result regarding symbolic dynamics in a fractional framework.

Multi-bump solutions for a class of Kirchhoff type problems with critical growth in $\mathbb{R}^N$

Using variational methods, we establish existence of multi-bump solutions for a class of Kirchhoff type problems $$ -\bigg(1+b\int_{\mathbb{R}^N}|\nabla u|^pdx\bigg)\Delta_pu + (\lambda V(x) + Z(x))u^{p-1} = \alpha f(u) + u^{p^\ast-1}, $$ where $f$ is a continuous function, $V, Z\colon \mathbb{R}^N \rightarrow\mathbb{R}$ are continuous functions verifying some hypotheses. We show that if the zero set of $V$ has several isolated connected components $\Omega_1,\ldots,\Omega_k$ such that the interior of $\Omega_i$ is not empty and $\partial\Omega_i$ is smooth, then for $\lambda > 0$ large enough there exists, for any non-empty subset $\Gamma \subset \{1,\ldots,k\}$, a bump solution trapped in a neighbourhood of $\bigcup\limits_{j\in \Gamma}\Omega_j$. The results are also new for the case $p=2$.