mountain pass
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2022 ◽  
Vol 2022 (1) ◽  
Author(s):  
Qilin Xie ◽  
Huafeng Xiao

AbstractIn the present paper, we consider the following discrete Schrödinger equations $$ - \biggl(a+b\sum_{k\in \mathbf{Z}} \vert \Delta u_{k-1} \vert ^{2} \biggr) \Delta ^{2} u_{k-1}+ V_{k}u_{k}=f_{k}(u_{k}) \quad k\in \mathbf{Z}, $$ − ( a + b ∑ k ∈ Z | Δ u k − 1 | 2 ) Δ 2 u k − 1 + V k u k = f k ( u k ) k ∈ Z , where a, b are two positive constants and $V=\{V_{k}\}$ V = { V k } is a positive potential. $\Delta u_{k-1}=u_{k}-u_{k-1}$ Δ u k − 1 = u k − u k − 1 and $\Delta ^{2}=\Delta (\Delta )$ Δ 2 = Δ ( Δ ) is the one-dimensional discrete Laplacian operator. Infinitely many high-energy solutions are obtained by the Symmetric Mountain Pass Theorem when the nonlinearities $\{f_{k}\}$ { f k } satisfy 4-superlinear growth conditions. Moreover, if the nonlinearities are sublinear at infinity, we obtain infinitely many small solutions by the new version of the Symmetric Mountain Pass Theorem of Kajikiya.


Author(s):  
Meiqi Liu ◽  
Quanqing Li

We study the following coupled fractional Schrödinger system: $$ \bcs (-\De)^s u=\la_1 u+\mu_1|u|^{p-2}u+\beta r_1|u|^{r_1-2}u|v|^{r_2}\quad &\hbox{in}\;\mathbb{R}^N, \\ (-\De)^s v=\la_2 v+\mu_2|v|^{q-2}v+\beta r_2|u|^{r_1}|v|^{r_2-2}v\quad &\hbox{in}\;\mathbb{R}^N, \\ %\int_{\mathbb{R}^N} u^2=a\quad and\quad \int_{\mathbb{R}^N} v^2=b, \ecs $$ with prescribed mass \[ \int_{\mathbb{R}^N} u^2=a\quad \hbox{and}\quad \int_{\mathbb{R}^N} v^2=b. \] Here, $a, b>0$ are prescribed, $N>2s, s>\frac{1}{2}$, $2+\frac{4s}{N}0$ sufficiently large, a mountain pass-type normalized solution exists provided $2\leq N\leq 4s$ and $ 2+\frac{4s}{N}


2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Debajyoti Choudhuri ◽  
Dušan D. Repovš

AbstractIn this paper we establish the existence and multiplicity of nontrivial solutions to the following problem: $$\begin{aligned} \begin{aligned} (-\Delta )^{\frac{1}{2}}u+u+\bigl(\ln \vert \cdot \vert * \vert u \vert ^{2}\bigr)&=f(u)+\mu \vert u \vert ^{- \gamma -1}u,\quad \text{in }\mathbb{R}, \end{aligned} \end{aligned}$$ ( − Δ ) 1 2 u + u + ( ln | ⋅ | ∗ | u | 2 ) = f ( u ) + μ | u | − γ − 1 u , in  R , where $\mu >0$ μ > 0 , $(*)$ ( ∗ ) is the convolution operation between two functions, $0<\gamma <1$ 0 < γ < 1 , f is a function with a certain type of growth. We prove the existence of a nontrivial solution at a certain mountain pass level and another ground state solution when the nonlinearity f is of exponential critical growth.


Entropy ◽  
2021 ◽  
Vol 23 (10) ◽  
pp. 1352
Author(s):  
Jean Mawhin ◽  
Ewa Skrzypek ◽  
Katarzyna Szymańska-Dȩbowska

This paper is devoted to study the existence of solutions and their regularity in the p(t)– Laplacian Dirichlet problem on a bounded time scale. First, we prove a lemma of du Bois–Reymond type in time-scale settings. Then, using direct variational methods and the mountain pass methodology, we present several sufficient conditions for the existence of solutions to the Dirichlet problem.


Author(s):  
Radu Precup

A new notion of linking is introduced to treat minima as minimax points in a unitary way. Critical points are located in conical annuli making possible to obtain multiplicity. For functionals defined on a Cartesian product, the localization of critical points is given on components and the variational properties of the components can differ, part of them being of minimum type, others of mountain pass type.


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