One-dimensional periodic fractional Schrödinger equations with exponential critical growth

In the present paper, we study the existence of nontrivial solutions of the following one-dimensional fractional Schr\“{o}dinger equation $$ (-\Delta)^{1/2}u+V(x)u=f(x,u), \ \ x\in \R, $$ where $(-\Delta)^{1/2}$ stands for the $1/2$-Laplacian, $V(x)\in \mathcal{C}(\R, (0,+\infty))$, and $f(x,u):\R\times\R\to \R$ is a continuous function with an exponential critical growth. Comparing with the existing works in the field of exponential-critical-growth fractional Schr\”{o}dinger equations, we encounter some new challenges due to the weaker assumptions on the reaction term $f$. By using some sharp energy estimates, we present a detailed analysis of the energy level, which allows us to establish the existence of nontrivial solutions for a wider class of nonlinear terms. Furthermore, we use the non-Nehari manifold method to establish the existence of Nehari-type ground state solutions of the one-dimensional fractional Schr\”{o}dinger equations.

Existence result for critical Klein-Gordon-Maxwell system involving steep potential well

This paper deals with the following system \begin{equation*} \left\{\begin{aligned} &{-\Delta u+ (\lambda A(x)+1)u-(2\omega+\phi) \phi u=\mu f(u)+u^{5}}, & & {\quad x \in \mathbb{R}^{3}}, \\ &{\Delta \phi=(\omega+\phi) u^{2}}, & & {\quad x \in \mathbb{R}^{3}}, \end{aligned}\right. \end{equation*} where $\lambda, \mu>0$ are positive parameters. Under some suitable conditions on $A$ and $f$, we show the boundedness of Cerami sequence for the above system by adopting Poho\v{z}aev identity and then prove the existence of ground state solution for the above system on Nehari manifold by using Br\’{e}zis-Nirenberg technique, which improve the existing result in the literature.

Ground state solutions and infinitely many solutions for a nonlinear Choquard equation

In this paper we study the existence and multiplicity of solutions for the following nonlinear Choquard equation: $$\begin{aligned} -\Delta u+V(x)u=\bigl[ \vert x \vert ^{-\mu }\ast \vert u \vert ^{p}\bigr] \vert u \vert ^{p-2}u,\quad x \in \mathbb{R}^{N}, \end{aligned}$$ − Δ u + V ( x ) u = [ | x | − μ ∗ | u | p ] | u | p − 2 u , x ∈ R N , where $N\geq 3$ N ≥ 3 , $0<\mu <N$ 0 < μ < N , $\frac{2N-\mu }{N}\leq p<\frac{2N-\mu }{N-2}$ 2 N − μ N ≤ p < 2 N − μ N − 2 , ∗ represents the convolution between two functions. We assume that the potential function $V(x)$ V ( x ) satisfies general periodic condition. Moreover, by using variational tools from the Nehari manifold method developed by Szulkin and Weth, we obtain the existence results of ground state solutions and infinitely many pairs of geometrically distinct solutions for the above problem.

Ground state solution for the Schrödinger equation with Hardy potential and critical Sobolev exponent

In this paper, we consider the following Schrödinger equation (0.1) − Δ u − μ u | x | 2 + V ( x ) u = K ( x ) | u | 2 ∗ − 2 u + f ( x , u ) , x ∈ R N , u ∈ H 1 ( R N ) , where N ⩾ 4, 0 ⩽ μ < μ ‾, μ ‾ = ( N − 2 ) 2 4 , V is periodic in x, K and f are asymptotically periodic in x, we take advantage of the generalized Nehari manifold approach developed by Szulkin and Weth to look for the ground state solution of (0.1).

Multiple positive solutions for a class of Kirchhoff type equations with indefinite nonlinearities

We study the following Kirchhoff type equation: − a + b ∫ R N | ∇ u | 2 d x Δ u + u = k ( x ) | u | p − 2 u + m ( x ) | u | q − 2 u in R N , $$\begin{equation*}\begin{array}{ll} -\left(a+b\int\limits_{\mathbb{R}^{N}}|\nabla u|^{2}\mathrm{d}x\right)\Delta u+u =k(x)|u|^{p-2}u+m(x)|u|^{q-2}u~~\text{in}~~\mathbb{R}^{N}, \end{array} \end{equation*}$$ where N=3, a , b > 0 $ a,b \gt 0 $ , 1 < q < 2 < p < min { 4 , 2 ∗ } $ 1 \lt q \lt 2 \lt p \lt \min\{4, 2^{*}\} $ , 2≤=2N/(N − 2), k ∈ C (ℝ N ) is bounded and m ∈ L p/(p−q)(ℝ N ). By imposing some suitable conditions on functions k(x) and m(x), we firstly introduce some novel techniques to recover the compactness of the Sobolev embedding H 1 ( R N ) ↪ L r ( R N ) ( 2 ≤ r < 2 ∗ ) $ H^{1}(\mathbb{R}^{N})\hookrightarrow L^{r}(\mathbb{R}^{N}) (2\leq r \lt 2^{*}) $ ; then the Ekeland variational principle and an innovative constraint method of the Nehari manifold are adopted to get three positive solutions for the above problem.

Multiple solutions for superlinear double phase Neumann problems

We study a double phase Neumann problem with a superlinear reaction which need not satisfy the Ambrosetti-Rabinowitz condition. Using the Nehari manifold method, we show that the problem has at least three nontrivial bounded ground state solutions, all with sign information (positive, negative and nodal).

Existence and nonexistence results for a class of Hamiltonian Choquard-type elliptic systems with lower critical growth on ℝ2

In this paper, we investigate the existence and nonexistence of results for a class of Hamiltonian-Choquard-type elliptic systems. We show the nonexistence of classical nontrivial solutions for the problem \[ \begin{cases} -\Delta u + u= ( I_{\alpha} \ast |v|^{p} )v^{p-1} \text{ in } \mathbb{R}^{N},\\ -\Delta v + v= ( I_{\beta} \ast |u|^{q} )u^{q-1} \text{ in } \mathbb{R}^{N}, \\ u(x),v(x) \rightarrow 0 \text{ when } |x|\rightarrow \infty, \end{cases} \] when $(N+\alpha )/p + (N+\beta )/q \leq 2(N-2)$ (if $N\geq 3$ ) and $(N+\alpha )/p + (N+\beta )/q \geq 2N$ (if $N=2$ ), where $I_{\alpha }$ and $I_{\beta }$ denote the Riesz potential. Second, via variational methods and the generalized Nehari manifold, we show the existence of a nontrivial non-negative solution or a Nehari-type ground state solution for the problem \[ \begin{cases} -\Delta u + u= (I_{\alpha} \ast |v|^{\frac{\alpha}{2}+1})|v|^{\frac{\alpha}{2}-1}v + g(v) \hbox{ in } \mathbb{R}^{2},\\ - \Delta v + v= (I_{\beta} \ast |u|^{\frac{\beta}{2}+1})|u|^{\frac{\beta}{2}-1}u + f(u), \hbox{ in } \mathbb{R}^{2},\\ u,v \in H^{1}(\mathbb{R}^{2}), \end{cases} \] where $\alpha ,\,\beta \in (0,\,2)$ and $f,\,g$ have exponential critical growth in the Trudinger–Moser sense.

Multiple solutions for a quasilinear Schrödinger–Poisson system

In this article, we consider the following quasilinear Schrödinger–Poisson system $$ \textstyle\begin{cases} -\Delta u+V(x)u-u\Delta (u^{2})+K(x)\phi (x)u=g(x,u), \quad x\in \mathbb{R}^{3}, \\ -\Delta \phi =K(x)u^{2}, \quad x\in \mathbb{R}^{3}, \end{cases} $$ { − Δ u + V ( x ) u − u Δ ( u 2 ) + K ( x ) ϕ ( x ) u = g ( x , u ) , x ∈ R 3 , − Δ ϕ = K ( x ) u 2 , x ∈ R 3 , where $V,K:\mathbb{R}^{3}\rightarrow \mathbb{R}$ V , K : R 3 → R and $g:\mathbb{R}^{3}\times \mathbb{R}\rightarrow \mathbb{R}$ g : R 3 × R → R are continuous functions; g is of subcritical growth and has some monotonicity properties. The purpose of this paper is to find the ground state solution of (0.1), i.e., a nontrivial solution with the least possible energy by taking advantage of the generalized Nehari manifold approach, which was proposed by Szulkin and Weth. Furthermore, infinitely many geometrically distinct solutions are gained while g is odd in u.