This paper is concerned with the following nonlinear fractional Schrödinger equationwhere a small parameter, positive function, and . Under some suitable conditions, we prove that for any integer , one can construct -spike solution near local maximum point of

Abstract This paper deals with the following weakly coupled nonlinear Schrödinger system <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mfenced open="{" close=""> <m:mrow> <m:mtable displaystyle="true"> <m:mtr> <m:mtd columnalign="left"> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:msub> <m:mi>u</m:mi> </m:mrow> <m:mn>1</m:mn> </m:msub> <m:mo>+</m:mo> <m:mi>a</m:mi> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo> <m:mo>=</m:mo> <m:mo>∣</m:mo> <m:msup>...

In this paper, by an approximating argument, we obtain infinitely many solutions for the following Hardy-Sobolev fractional equation with critical growth \begin{equation*}\label{0.1} \left\{% \begin{array}{ll} (-Δ)^{s} u-\ds\frac{μu}{|x|^{2s}}=|u|^{2^*_s-2}u+au, &amp; \hbox{$\text{in}~ Ω$},\vspace{0.1cm} u=0,\,\, &amp;\hbox{$\text{on}~\partial Ω$}, \\ \end{array}% \right. \end{equation*} provided $N&gt;6s$, $μ\geq0$, $0&lt; s&lt;1$, $2^*_s=\frac{2N}{N-2s}$, $a&gt;0$ is a constant and $Ω$...

We consider the following nonlinear fractional Schr\"{o}dinger equation $$ (-\Delta)^su+u=K(|x|)u^p,\ \ u>0 \hbox{in}\ R^N, where $K(|x|)$ is a positive radial function, $N\ge 2$, $0<s<1$, $1<p<\frac{N+2s}{N-2s}$. Under some asymptotic assumptions on $K(x)$ at infinity, we show that this problem has infinitely many non-radial solutions, whose energy can be made arbitrarily large.

We establish several new Opial-type inequalities involving different types of boundary conditions.

We study a linearly coupled Schr\"{o}dinger system in $\R^N(N\leq3).$ Assume that the potentials are continuous functions satisfying suitable decay assumptions, but without any symmetry properties and parameters satisfy some restrictions. Using Liapunov-Schmidt reduction methods two times combing localized energy method, we prove problem has infinitely many positive synchronized solutions, which extends result Theorem 1.2 about nonlinearly equations \cite{aw} to our problem.

<p style='text-indent:20px;'>The present paper deals with a class of Schrödinger-poisson system. Under some suitable assumptions on the decay rate coefficients, we derive existence infinitely many positive solutions to problem by using purely variational methods. Comparing previous works, encounter new challenges because nonlocal term. By doing delicate estimates for term overcome difficulty and find solutions.

We study the semilinear equation urn:x-wiley:mma:media:mma3309:mma3309-math-0001 where 0 &lt; s 1, , V ( x ) is a sufficiently smooth non‐symmetric potential with and ϵ &gt; small number. Letting U be radial ground state of (−Δ) + − p =0 in we build solutions form urn:x-wiley:mma:media:mma3309:mma3309-math-0005 for points ϑ j = 1,⋯, m using Lyapunov–Schmidt variational reduction. Copyright © 2014 John Wiley &amp; Sons, Ltd.

In this paper, the shift-HSS splitting (denoted by SFHSS) iteration method is used to solve nonsingular saddle point system with nonsymmetric positive definite (1, 1)-block. Theoretical analysis illustrates that SFHSS converges unique solution of as parameter satisfies certain condition.

Abstract In this paper, we study the following Schrödinger–Poisson system in $\mathbb{R} ^{3}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>R</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:math> $$ \textstyle\begin{cases} (-\Delta )^{\sigma }u+A(y)u+B(y)\phi (y)u=b(y) \vert u ^{p-1}u, &amp; y \in \mathbb{R} ^{3}, \\ }\phi =B(y)u^{2}, y\in \end{cases}...

Abstract The present paper deals with a class of nonlocal problems. Under some suitable assumptions on the decay rate coefficients, we derive existence infinitely many positive solutions to problem by applying reduction method. Comparing previous work, encounter new challenges because competing potentials. By doing delicate estimates for potentials, overcome difficulties and find solutions.

In this paper, we deal with a class of fractional Hénon equation and by using the Lyapunov-Schmidt reduction method, under some suitable assumptions, derive existence infinitely many solutions, whose energy can be made arbitrarily large. Compared to previous works, encounter new challenges because nonlocal property for Laplacian. But doing delicate estimates term overcome difficulty find nonradial solutions.

We investigate the existence and local uniqueness of normalized $k$-peak solutions for fractional Schrödinger equations with attractive interactions a class degenerated trapping potential non-isolated critical points. Precisely, applying finite dimensional reduction method, we first obtain concentrated especially describe relationship between chemical $μ$ interaction $a$. Second, after precise analysis points Lagrange multiplier, prove prescribed $L^2$-norm, by use Pohozaev identities,...

We revisit the following nonlinear critical elliptic equation \begin{equation*} -\Delta u+Q(y)u=u^{\frac{N+2}{N-2}},\;\;\; u>0\;\;\;\hbox{ in } \mathbb{R}^N, \end{equation*} where $N\geq 5.$ There seems to be no results about periodicity of bubbling solutions. Here we try investigate some related problems. Assuming that $Q(y)$ is periodic $y_1$ with period 1 and has a local minimum at 0 satisfying $Q(0)=0,$ prove existence uniqueness infinitely many solutions problem above. This result...