Multiplicity and concentration behavior of solutions for the generalized quasilinear Schrödinger equation with critical growth

We consider the nonlinear Schrödinger equation \begin{equation*} -\Delta u + (1+\mu g(x))u = f(u) \quad \text{in } \mathbb{R}^N, \end{equation*} where $N \ge 3$, $\mu \ge 0$; the function $g \ge 0$ has a potential well and $f$ has critical growth. By using variational methods, the existence and concentration behavior of the ground state solution are obtained.

We consider the Kirchhoff type equation with steep potential well and critical growth. By developing some techniques in variational methods, we obtain existence, multiplicity, and concentration behavior of positive solutions under suitable conditions.

Some general foundational issues of quantum mechanics are considered and are related to aspects of quantum computation. The importance of quantum entanglement and quantum information is discussed a...

We study the existence and asymptotic behavior of least energy sign-changing solutions to a gauged nonlinear Schrödinger equation with critical exponential growth where are constants and Under some suitable assumptions on , we apply the constraint minimization argument to establish a least energy sign-changing solution with precisely two nodal domains. Moreover, we show that the energy of is strictly larger than two times of the ground state energy and analyze the asymptotic behavior of as . Our results generalize the existing ones, see Li G. et al. (Sign-changing solutions to a gauged nonlinear Schrödinger equation. J Math Anal Appl. 2017;455:1559–1578) and Liu Z. et al. (Existence and multiplicity of sign-changing standing waves for a gauged nonlinear Schrödinger equation in . Nonlinearity. 2019;32:3082–3111) for example, to the gauged nonlinear Schrödinger equation with critical exponential growth.

In this paper, we study the following Kirchhoff-type equation:<disp-formula> <label/> <tex-math id="FE1"> $ \begin{equation*} M\left(\displaystyle{\int}_{\mathbb{R}^2}(|\nabla u|^2 +u^2)\mathrm{d} x\right)(-\Delta u+u) + \mu V(x)u = K(x) f(u) \ \ \mathrm{in} \ \ \mathbb{R}^2, \end{equation*} $ </tex-math></disp-formula>where $ M \in C(\mathbb{R}^+, \mathbb{R}^+) $ is a general function, $ V \geq 0 $ and its zero set may have several disjoint connected components, $ \mu > 0 $ is a parameter, $ K $ is permitted to be unbounded above, and $ f $ has exponential critical growth. By using the truncation technique and developing some approaches to deal with Kirchhoff-type equations with critical growth in the whole space, we get the existence and concentration behavior of solutions. The results are new even for the case $ M \equiv 1 $.

Asymptotic behaviour of small solutions of quasilinear elliptic equations with critical and supercritical growth

In this paper, we study two classes of quasilinear Schrödinger equations that describe several phenomena in the real world, such as superfluid thin films or self-channeling of a high-power ultrashort laser in matter. Combining analytical skills and the Trudinger–Moser inequality, we establish the existence of nontrivial solutions for such problems when the nonlinear reaction term satisfies subcritical and critical exponential growth, respectively. In particular, we prove the existence of ground state sign-changing solutions via the constrained variational method and quantitative deformation techniques. In this paper, we are also concerned with the asymptotic behavior of these solutions on the vanishing set of the source potential. Our results extend and complement several recent contributions in the literature.

In this paper, we investigate the following Chern–Simons–Schrödinger system where , and the nonlinearity behaves like as . By using the variational methods and the Trudinger–Moser inequality, we obtain the existence of positive solutions for this system. Moreover, we observe the concentrate behavior of mountain-pass type solutions of this system as .

We study the asymptotic behavior of least energy solutions to the equation Δ2u=c0K(x)upε with the Navier boundary condition as ε→+0, where Ω is a smooth bounded domain in RN (N≥5), c0=(N−4)(N−2)N(N+2) and pε=(N+4)/(N−4)−ε, ε>0. Under some assumptions on the coefficient function K, we obtain fairly precise asymptotics of the L∞-norm of least energy solutions.

In this article, we study the following Schrödinger equation \begin{align*} \begin{cases} -\Delta u -\frac{\mu}{|x|^2} u+\lambda u =f(u), &\text{in}~ \mathbb{R}^N\backslash\{0\},\\ \int_{\mathbb{R}^{N}}|u|^{2}\mathrm{d} x=a, & u\in H^1(\mathbb{R}^{N}), \end{cases} \end{align*} where $N\geq 3$ , a > 0, and $\mu \lt \frac{(N-2)^2}{4}$ . Here $\frac{1}{|x|^2} $ represents the Hardy potential (or ‘inverse-square potential’), λ is a Lagrange multiplier, and the nonlinearity function f satisfies the general Sobolev critical growth condition. Our main goal is to demonstrate the existence of normalized ground state solutions for this equation when $0 \lt \mu \lt \frac{(N-2)^2}{4}$ . We also analyse the behaviour of solutions as $\mu\to0^+$ and derive the existence of normalized ground state solutions for the limiting case where µ = 0. Finally, we investigate the existence of normalized solutions when µ < 0 and analyse the asymptotic behaviour of solutions as $\mu\to 0^-$ .

In this paper, we consider the following quasilinear Choquard equation − ϵ 2 Δ u + V ( x ) u − ϵ 2 Δ ( u 2 ) u = ϵ μ − 2 ( 1 | x | μ ∗ F ( u ) ) f ( u ) in R 2 , where ϵ > 0 is a parameter, 0 < μ < 2 , ∗ is the convolution product in R 2 , V ( x ) is a continuous real function in R 2 , F ( u ) is the primitive function of f ( u ) and f has critical exponential growth with respect to the Trudinger-Moser inequality. By employing a change of variables, the quasilinear equation can be reduced to a semilinear equation, whose associated functional is well defined in a nonstandard Orlicz space and exhibits a mountain pass geometry. Under suitable assumptions on V and f, we investigate the existence and concentration behavior of positive ground state solutions for the above problem by variational methods.

Asymptotic behaviour of positive solutions of elliptic equations with critical and supercritical growth. II. The nonradial case

In this paper, we analyze the concentration behavior of a positive solution to an evolution equation with critical exponential growth on a closed Riemann surface, and particularly derive an energy identity for such a solution. This extends a result of Lamm-Robert-Struwe and complements that of Yang.

Nodal solutions for singularly perturbed equations with critical exponential growth

Dynamical boundary problem for Dirichlet-to-Neumann operator with critical Sobolev exponent and Hardy potential