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In the present paper, a class of Schrödinger equations is investigated, which can be stated as −Δu+V(x)u=f(u), x∈ℝN.- \Delta u + V(x)u = f(u),\;\;\;\;x \in {{\rm{\mathbb R}}^N}. If the external potential V is radial and coercive, then we give the local Ambrosetti-Rabinowitz super-linear condition on the nonlinearity term f ∈ C(ℝ, ℝ) which assures the problem has not only infinitely many radial sign-changing solutions, but also infinitely many non-radial sign-changing solutions.

In this paper, we investigate the non-autonomous Choquard equation

In this paper, we research the following Kirchhoff type problem with critical growth (1) $$\begin{equation} \begin{cases} -\left(a+b\int_{\mathbb{R}^{3}}{|\nabla u|^{2}} \,{\rm d}x\right) \Delta u+ u \\ = \lambda f(x)|u|^{q-2}u+|u|^{4}u, & x\in \mathbb{R}^3,\\ u>0, & u\in H^1(\mathbb{R}^{3}), \end{cases} \end{equation}$$ { − (a + b ∫ R 3 | ∇ u | 2 d x) Δ u + u = λ f (x) | u | q − 2 u + | u | 4 u , x ∈ R 3 , u > 0 , u ∈ H 1 (R 3) , where 10 is a positive constant, $ b,\lambda >0 $ b , λ > 0 are real parameters and $ f\in L^{\frac {6}{6-q}}(\mathbb {R}^3) $ f ∈ L 6 6 − q (R 3) is a nonzero nonnegative function. We prove the problem has at least two positive solutions by using concentration compactness principle and variational method when b is sufficiently small. Moreover, we also explore the asymptotic behaviour of the positive solutions as $ b_n\stackrel {n}{\to } 0^{+} $ b n → n 0 + . [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of Kirchhoff type problems in ℝ3{\mathbb{R}^{3}} involving a critical nonlinearity, namely,

In this article, we establish the existence of bound state solutions for a class of quasilinear Schrödinger equations whose nonlinear term is asymptotically linear in ℝN{\mathbb{R}^{N}}. After changing the variables, the quasilinear equation becomes a semilinear equation, whose respective associated functional is well defined in H1⁢(ℝN){H^{1}(\mathbb{R}^{N})}. The proofs are based on the Pohozaev manifold and a linking theorem.

In this paper, we investigate the following autonomous Choquard equation − Δ u + u = (I α ∗ F (u)) F ′ (u) in R N , where N ≥ 3 , I α denotes the Riesz potential of order α ∈ (0 , N) and F satisfies general critical growth conditions. By using the variational methods and the Pohožaev manifold techniques, we prove the existence of radially symmetric positive ground state solution. [ABSTRACT FROM AUTHOR]

In this paper, we study the nonlinear Kirchhoff type equation { − (a + b ∫ R 3 | ∇ u | 2 d x) △ u + λ V (x) u = | u | 4 u + f (u) , x ∈ R 3 , u ∈ H 1 (R 3) , where a, b are positive constants and λ > 0. Suppose that the nonnegative continuous potential V represents a potential well with the bottom V − 1 (0) and f ∈ C (R , R) satisfies certain assumptions. We obtain the existence of ground state solutions by using the variational methods. Moreover, the concentration behavior of the ground state solutions as λ → ∞ is also worth considering. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the following Chern–Simons–Schrödinger system − Δ u + u + λ ∫ | x | ∞ h (s) s u 2 (s) d s + h 2 (| x |) | x | 2 u = f (x , u) i n R 2 , where λ > 0 , h (s) = ∫ 0 s t 2 u 2 (t) d t and the nonlinearity f (x , s) ∈ C (R 2 × R , R) behaves like e 4 π s 2 as | s | → + ∞. 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 λ → 0. [ABSTRACT FROM AUTHOR]