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This paper deals with the following non-linear equation with a fractional Laplacian operator and almost critical exponents:
where N ⩾ 4, 0 < s < 1, (y′, y″) ∈ ℝ2 × ℝN−2, ε > 0 is a small parameter and K(y) is non-negative and bounded. Under some suitable assumptions of the potential function K(r, y″), we will use the finite-dimensional reduction method and some local Pohozaev identities to prove that the above problem has a large number of bubble solutions. The concentration points of the bubble solutions include a saddle point of K(y). Moreover, the functional energies of these solutions are in the order 
\[ (-\Delta)^{s} u=K(|y'|,y'')u^{({N+2s})/(N-2s)\pm\epsilon},\quad u > 0,\quad u\in D^{1,s}(\mathbb{R}^{N}), \]
$\epsilon ^{-(({N-2s-2})/({(N-2s)^2})}$.
This paper is mainly concerned with the global asymptotic behaviour of the unique solution to a class of singular Dirichlet problems − Δu = b(x)g(u), u > 0, x ∈ Ω, u|∂Ω = 0, where Ω is a bounded smooth domain in ℝn, g ∈ C1(0, ∞) is positive and decreasing in (0, ∞) with 
$\lim _{s\rightarrow 0^+}g(s)=\infty$, b ∈ Cα(Ω) for some α ∈ (0, 1), which is positive in Ω, but may vanish or blow up on the boundary properly. Moreover, we reveal the asymptotic behaviour of such a solution when the parameters on b tend to the corresponding critical values.
We study the quasilinear elliptic inequality
where 
\[ -\Delta_m u - \frac{\mu}{|x|^m}u^{m-1} \geq (I_\alpha*u^p)u^q \quad\mbox{in }\mathbb{R}^N\setminus \overline B_1, N\geq 1, \]
$p>0$, 
$q, \mu \in \mathbb {R}$, 
$m>1$ and 
$I_\alpha$ is the Riesz potential of order 
$\alpha \in (0,N)$. We obtain necessary and sufficient conditions for the existence of positive solutions.
This paper is concerned with the static Choquard equation where 
$$\begin{eqnarray}-\unicode[STIX]{x1D6E5}u=\bigg(\frac{1}{|x|^{N-\unicode[STIX]{x1D6FC}}}\ast |u|^{p}\bigg)|u|^{p-2}u\quad \text{in }\mathbb{R}^{N},\end{eqnarray}$$
$N,p>2$ and 
$\max \{0,N-4\}<\unicode[STIX]{x1D6FC}<N$. We prove that if 
$u\in C^{1}(\mathbb{R}^{N})$ is a stable weak solution of the equation, then 
$u\equiv 0$. This phenomenon is quite different from that of the local Lane–Emden equation, where such a result only holds for low exponents in high dimensions. Our result is the first Liouville theorem for Choquard-type equations with supercritical exponents and 
$\unicode[STIX]{x1D6FC}\neq 2$.
In the present paper, we consider the following Kirchhoff type problem
where b > 0, p ∈ (4, 6), the potential 
$$ -\Big(\varepsilon^2+\varepsilon b \int_{\mathbb R^3} | \nabla v|^2\Big) \Delta v+V(x)v=|v|^{p-2}v \quad {\rm in}\ \mathbb{R}^3, $$
$V\in C(\mathbb R^3,\mathbb R)$ and ɛ is a positive parameter. The existence and multiplicity of semi-classical state solutions are obtained by variational method for this problem with several classes of critical frequency potentials, i.e., 
$\inf _{\mathbb R^N} V=0$. As to Kirchhoff type problem, little has been done for the critical frequency cases in the literature, especially the potential may vanish at infinity.
In this paper, we consider the following fractional Schrödinger–Poisson system with singularity 
\begin{equation*}
\left \{\begin{array}{lcl}
({-}\Delta)^s u+V(x)u+\lambda \phi u = f(x)u^{-\gamma}, &&\quad
x\in\mathbb{R}^3,\\
({-}\Delta)^t \phi = u^2, &&\quad x\in\mathbb{R}^3,\\
u>0,&&\quad x\in\mathbb{R}^3,
\end{array}\right.
\end{equation*}
where 0 < γ < 1, λ > 0 and 0 < s ≤ t < 1 with 4s + 2t > 3. Under certain assumptions on V and f, we show the existence, uniqueness, and monotonicity of positive solution uλ using the variational method. We also give a convergence property of uλ as λ → 0, when λ is regarded as a positive parameter.
We consider the fractional critical problem 
$A_{s}u=K(x)u^{(n+2s)/(n-2s)},u>0$ in 
$\unicode[STIX]{x1D6FA},u=0$ on 
$\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$, where 
$A_{s},s\in (0,1)$, is the fractional Laplace operator and 
$K$ is a given function on a bounded domain 
$\unicode[STIX]{x1D6FA}$ of 
$\mathbb{R}^{n},n\geq 2$. This is based on A. Bahri’s theory of critical points at infinity in Bahri [Critical Points at Infinity in Some Variational Problems, Pitman Research Notes in Mathematics Series, 182 (Longman Scientific & Technical, Harlow, 1989)]. We prove Bahri’s estimates in the fractional setting and we provide existence theorems for the problem when 
$K$ is close to 1.
We are concerned with the semi-classical states for the Choquard equation
where N ⩾ 2, Iα is the Riesz potential with order α ∈ (0, N − 1) and 2 ⩽ p < (N + α)/(N − 2). When the potential V is assumed to be bounded and bounded away from zero, we construct a family of localized bound states of higher topological type that concentrate around the local minimum points of the potential V as ε → 0. These solutions are obtained by combining the Byeon–Wang's penalization approach and the classical symmetric mountain pass theorem.
$$-{\epsilon }^2\Delta v + Vv = {\epsilon }^{-\alpha }(I_\alpha *|v|^p)|v|^{p-2}v,\quad v\in H^1({\mathbb R}^N),$$
We prove that if two 
$C^{1,1}(\unicode[STIX]{x1D6FA})$ solutions of the second boundary value problem for the generated Jacobian equation intersect in 
$\unicode[STIX]{x1D6FA}$ then they are the same solution. In addition, we extend this result to 
$C^{2}(\overline{\unicode[STIX]{x1D6FA}})$ solutions intersecting on the boundary, via an additional convexity condition on the target domain.
We prove the non-degeneracy of the extremals of the Sobolev inequality
when 1 < p < N, as solutions of a critical quasilinear equation involving the p-Laplacian.
\[ \int_{\mathbb R^N}|\nabla u|^p\,\rd x\ge \mathcal S_p\int_{\open R^N}|u|^\frac{Np}{N-p}\,\rd x,\quad u\in \mathcal D^{1,p}(\open R^N) \]
