Normalized solutions for a nonlinear Schrödinger equation via a fixed point theorem

This paper is concerned with the following Klein‐Gordon‐Maxwell system: where ω > 0 is a constant, , and obeys exponential critical growth in the sense of the Trudinger‐Moser inequality. We give some new sufficient conditions on f, specifically related to exponential growth, to obtain the existence of nontrivial solutions. Furthermore, we prove a nonexistence result with a Pohozaev‐type argument which also provides important information to get the above existence results. In particular, some new analytical approaches and estimates are used to overcome the difficulty arising from the critical growth of Trudinger‐Moser type.

This study focuses on the existence and concentration of ground state solutions for a class of fractional Kirchhoff–Schrodinger equations. We first study the problem $$\left\{ \begin{array}{ll} M ([u]^{2}_{s} + \int_{\mathbb{R}^{N}} V(x)u^{2}) ((-{\Delta})^{s}u + V (x)u) = \bar{c}u + f(u)\, {\rm in}\,\, \mathbb{R}^N,\\ u > 0, u\, {\in} \, {H}^{s} (\mathbb{R}^N),\end{array} \right.$$ where $$s \in (0,1), N > 2s, [\cdot]_s$$ is the Gagliardo semi-norm, $$\bar{c}$$ is a suitable constant,M is a non-degenerate continuous Kirchhoff function that behaves like $$t^{\alpha}, V(x) = {\lambda}a(x) + 1, {\rm with}\,\, a(x) \geq 0$$ and a is identically zero on the bounded set $${\Omega}_{\Upsilon}$$ , and f denotes a continuous nonlinearity with subcritical growth at infinity. The proof relies on penalization arguments and variational methods to obtain the existence of a solution with minimal energy for a large value of $$\lambda$$ . Moreover, assuming that $$M(t) = m_{0} + b_{0}t^{\alpha}$$ and utilizing the same techniques combined with a concentration-compactness lemma, we can establish the existence and concentration of solutions for the problem $$\left\{\begin{array}{ll} M ([u]^2_s+\int_{\mathbb{R}^N}V(x)u^2) ((-\Delta)^s u + V(x)u)= h(x)u + u^{2^*_s -1} \ {\rm in} \ \mathbb{R}^N,\\ u>0, \quad u\in H^s (\mathbb{R}^N), \end{array}\right.$$ if the value of $$\lambda$$ is large enough and b0 is small or m0 is large.

In this paper, we study the following nonlocal nonautonomous Hamiltonian system on whole \({\mathbb {R}}\)$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^\frac{1}{2}~ u +u=Q(x) g(v)&{}\quad \text{ in } {\mathbb {R}},\\ (-\Delta )^\frac{1}{2}~ v+v = P(x)f(u)&{}\quad \text{ in } {\mathbb {R}}, \end{array}\right. \end{aligned}$$where \((-\Delta )^\frac{1}{2}\) is the square root Laplacian operator. We assume that the nonlinearities f, g have critical growth at \(+\,\infty \) in the sense of Trudinger–Moser inequality and the nonnegative weights P(x) and Q(x) vanish at \(+\infty \). Using suitable variational method combined with the generalized linking theorem, we obtain the existence of at least one positive solution for the above system.

The main purpose of this paper is to investigate the existence of nontrivial solutions to a class of quasilinear non-local problems which do not satisfy the Ambrosetti–Rabinowitz (AR) condition where the nonlinear terms are superlinear at 0 and of subcritical or critical exponential growth (subcritical polynomial growth) at $$\infty $$ . Some existence results for nontrivial solution are obtained using mountain pass theorem combined with the fractional Moser–Trudinger inequality.

Sign-changing solutions for coupled nonlinear Schrödinger equations with critical growth

The nonlinear Schrdinger equation is one of the most important nonlinear models with widely applications in physics. Based on a similarity transformation, the (2+1)-dimensional nonlinear Schrdinger equation with distributed coefficients is transformed into a traceable nonlinear Schrdinger equation, and then two types of rational solutions and several spatial solitons are derived.

The purpose of this paper is three-fold. First, we establish singular Trudinger–Moser inequalities with less restrictive constraint: (0.1) sup u ∈ H 1 ( R 2 ) , ∫ R 2 ( | ∇ u | 2 + V ( x ) u 2 ) d x ≤ 1 ∫ R 2 e 4 π 1 − β 2 u 2 − 1 | x | β d x < + ∞ , $$\underset{u\in {H}^{1}\left({\mathbb{R}}^{2}\right),\underset{{\mathbb{R}}^{2}}{\int }\left(\vert \nabla u{\vert }^{2}+V\left(x\right){u}^{2}\right)\mathrm{d}x\le 1}{\mathrm{sup}}\underset{{\mathbb{R}}^{2}}{\int }\frac{{e}^{4\pi \left(1-\frac{\beta }{2}\right){u}^{2}}-1}{\vert x{\vert }^{\beta }}\mathrm{d}x{< }+\infty ,$$ where 0 < β < 2 $0{< }\beta {< }2$ , V ( x ) ≥ 0 $V\left(x\right)\ge 0$ and may vanish on an open set in R 2 ${\mathbb{R}}^{2}$ . Second, we consider the existence of ground states to the following Schrödinger equations with critical exponential growth in R 2 ${\mathbb{R}}^{2}$ : (0.2) − Δ u + γ u = f ( u ) | x | β , $$-{\Delta}u+\gamma u=\frac{f\left(u\right)}{\vert x{\vert }^{\beta }},$$ where the nonlinearity f $f$ has the critical exponential growth. In order to overcome the lack of compactness, we develop a method which is based on the threshold of the least energy, an embedding theorem introduced in (C. Zhang and L. Chen, “Concentration-compactness principle of singular Trudinger-Moser inequalities in R n ${\mathbb{R}}^{n}$ and n $n$ -Laplace equations,” Adv. Nonlinear Stud., vol. 18, no. 3, pp. 567–585, 2018) and the Nehari manifold to get the existence of ground states. Furthermore, as an application of inequality (0.1), we also prove the existence of ground states to the following equations involving degenerate potentials in R 2 ${\mathbb{R}}^{2}$ : (0.3) − Δ u + V ( x ) u = f ( u ) | x | β . $$-{\Delta}u+V\left(x\right)u=\frac{f\left(u\right)}{\vert x{\vert }^{\beta }}.$$

In this paper, we study the existence of positivesolution for the following p-Laplacain type equations with critical nonlinearity \begin{equation*} \left\{ \renewcommand{\arraystretch}{1.25} \begin{array}{ll} -\Delta_p u + V （x)|u|^{p-2}u = K(x)f(u)+P(x)|u|^{p^*-2}u, \quad x\in\mathbb{R}^N,\\ u \in \mathcal{D}^{1,p}(\mathbb{R}^N), \end{array} \right. \end{equation*}where $\Delta_p u = div(|\nabla u|^{p-2} \nabla u),\ 1 < p < N,\ p^* =\frac{Np}{N-p}$, $V(x)$, $K(x)$ are positive continuous functions which vanish atinfinity, $f$ is a function with a subcritical growth, and $P(x)$ is a bounded,nonnegative continuous function.By working in the weighted Sobolev spaces, and using variational method, weprove that the given problem has at least one positive solution.

Chapter 7 Multiplicity techniques for problems without compactness

In this paper, the general variable coefficient nonlinear Schrdinger equation for the optical fiber is investigated based on the extended cubic nonlinear Klein-Gordon equation. The soliton solutions under non-chirp case are discussed, and some new exact solutions, including the bright soliton, the dark soliton and the soliton-like solution are obtained. The standard nonlinear Schrdinger equation and four kinds of the variable coefficient nonlinear Schrdinger equation are all corresponding special cases. In addition, some interesting results for the optical fiber system with periodic gain or loss are also obtained.

We consider $$\varepsilon $$-perturbed nonlinear Schrodinger equations of the form $$\begin{aligned} - \varepsilon ^2\Delta u + V(x)u = Q(x)f(u) \quad \text {in } \mathbb {R}^2, \end{aligned}$$where V and Q behave like $$(1+|x|)^{-\alpha }$$ with $$\alpha \in (0,2)$$ and $$(1+|x|)^{-\beta }$$ with $$\beta \in (\alpha , + \infty )$$, respectively. When f has subcritical exponential growth—by means of a weighted Trudinger–Moser-type inequality and the mountain pass theorem in weighted Sobolev spaces—we prove the existence of nontrivial mountain pass solutions, for any $$\varepsilon >0$$, and in the semi-classical limit, these solutions concentrate at a global minimum point of $$\mathcal A=V/Q$$. Our existence result holds also when f has critical growth, for any $$\varepsilon >0$$.

We discuss a new kind of nonlinear Schrodinger equation from the viewpoint of prolongation theory. It is shown that the equation possess a Lax pair with a 3 × 3 matrix structure. It is further demonstrated that by a multiple scale perturbation of Zakharovet al. it can be reduced to the usual KdV equation.

Quasilinear equation with critical exponential growth in the zero mass case

This paper focuses on the existence of normalized solutions for the Chern–Simons–Schrödinger system with mixed dispersion and critical exponential growth. These solutions correspond to critical points of the underlying energy functional under the ‐norm constraint, namely, . Under certain mild assumptions, we establish the existence of nontrivial solutions by developing new mathematical strategies and analytical techniques for the given system. These results extend and improve the results in the existing literature.

Multiplicity results for semilinear heterogeneous problems in exterior domains of [formula omitted] involving subcritical or critical nonlinearities à la Trudinger-Moser