Singularly perturbed quasilinear Choquard equations with nonlinearity satisfying Berestycki–Lions assumptions

In the present paper, we consider the following singularly perturbed problem: $$ \textstyle\begin{cases} -\varepsilon ^{2}\Delta u+V(x)u-\varepsilon ^{2}\Delta (u^{2})u= \varepsilon ^{-\alpha }(I_{\alpha }*G(u))g(u), \quad x\in \mathbb{R}^{N}; \\ u\in H^{1}(\mathbb{R}^{N}), \end{cases} $$ { − ε 2 Δ u + V ( x ) u − ε 2 Δ ( u 2 ) u = ε − α ( I α ∗ G ( u ) ) g ( u ) , x ∈ R N ; u ∈ H 1 ( R N ) , where $\varepsilon >0$ ε > 0 is a parameter, $N\ge 3$ N ≥ 3 , $\alpha \in (0, N)$ α ∈ ( 0 , N ) , $G(t)=\int _{0}^{t}g(s)\,\mathrm{d}s$ G ( t ) = ∫ 0 t g ( s ) d s , $I_{\alpha }: \mathbb{R}^{N}\rightarrow \mathbb{R}$ I α : R N → R is the Riesz potential, and $V\in \mathcal{C}(\mathbb{R}^{N}, \mathbb{R})$ V ∈ C ( R N , R ) with $0<\min_{x\in \mathbb{R}^{N}}V(x)< \lim_{|y|\to \infty }V(y)$ 0 < min x ∈ R N V ( x ) < lim | y | → ∞ V ( y ) . Under the general Berestycki–Lions assumptions on g, we prove that there exists a constant $\varepsilon _{0}>0$ ε 0 > 0 determined by V and g such that for $\varepsilon \in (0,\varepsilon _{0}]$ ε ∈ ( 0 , ε 0 ] the above problem admits a semiclassical ground state solution $\hat{u}_{\varepsilon }$ u ˆ ε with exponential decay at infinity. We also study the asymptotic behavior of $\{\hat{u}_{\varepsilon }\}$ { u ˆ ε } as $\varepsilon \to 0$ ε → 0 .

Characterization of Minimizers of Aviles–Giga Functionals in Special Domains

We consider the singularly perturbed problem $$F_\varepsilon (u,\Omega ):=\int _\Omega \varepsilon |\nabla ^2u|^2 + \varepsilon ^{-1}|1-|\nabla u|^2|^2$$ F ε ( u , Ω ) : = ∫ Ω ε | ∇ 2 u | 2 + ε - 1 | 1 - | ∇ u | 2 | 2 on bounded domains $$\Omega \subset {\mathbb {R}}^2$$ Ω ⊂ R 2 . Under appropriate boundary conditions, we prove that if $$\Omega $$ Ω is an ellipse, then the minimizers of $$F_\varepsilon (\cdot ,\Omega )$$ F ε ( · , Ω ) converge to the viscosity solution of the eikonal equation $$|\nabla u|=1$$ | ∇ u | = 1 as $$\varepsilon \rightarrow 0$$ ε → 0 .