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.

In this paper, we look at a linear system of ordinary differential equations as derived from the two-dimensional Ginzburg–Landau equation. In two cases, it is known that this system admits bounded solutions coming from the invariance of the Ginzburg–Landau equation by translations and rotations. The specific contribution of our work is to prove that in the other cases, the system does not admit any bounded solutions. We show that this bounded solution problem is related to an eigenvalue problem.

We study the existence of entropy solutions by assuming the right-hand side function f to be an integrable function for some elliptic nonlocal p-Laplacian type problems. Moreover, the existence of weak solutions for the corresponding parabolic cases is also established. The main aim of this paper is to provide some positive answers for the two questions proposed by Chipot and de Oliveira (Math. Ann., 2019, 375, 283-306).

We study the quasilinear elliptic inequality

\[ -\Delta_m u - \frac{\mu}{|x|^m}u^{m-1} \geq (I_\alpha*u^p)u^q \quad\mbox{in }\mathbb{R}^N\setminus \overline B_1, N\geq 1, \]

$p>0$

$q, \mu \in \mathbb {R}$

$m>1$

$I_\alpha$

$\alpha \in (0,N)$

This paper is concerned with the static Choquard equation

$$\begin{eqnarray}-\unicode[STIX]{x1D6E5}u=\bigg(\frac{1}{|x|^{N-\unicode[STIX]{x1D6FC}}}\ast |u|^{p}\bigg)|u|^{p-2}u\quad \text{in }\mathbb{R}^{N},\end{eqnarray}$$

$N,p>2$

$\max \{0,N-4\}<\unicode[STIX]{x1D6FC}<N$

$u\in C^{1}(\mathbb{R}^{N})$

$u\equiv 0$

$\unicode[STIX]{x1D6FC}\neq 2$

For Laplacians defined by measures on a bounded domain in ℝn, we prove analogues of the classical eigenvalue estimates for the standard Laplacian: lower bound of sums of eigenvalues by Li and Yau, and gaps of consecutive eigenvalues by Payne, Pólya and Weinberger. This work is motivated by the study of spectral gaps for Laplacians on fractals.

In the present paper, we consider the following Kirchhoff type problem

$$ -\Big(\varepsilon^2+\varepsilon b \int_{\mathbb R^3} | \nabla v|^2\Big) \Delta v+V(x)v=|v|^{p-2}v \quad {\rm in}\ \mathbb{R}^3, $$

$V\in C(\mathbb R^3,\mathbb R)$

$\inf _{\mathbb R^N} V=0$

We show that the spectrum of the relativistic mean curvature operator on a bounded domain Ω ⊂ ℝN (N ⩾ 1) having smooth boundary, subject to the homogeneous Dirichlet boundary condition, is exactly the interval (λ1(2), ∞), where λ1(2) stands for the principal frequency of the Laplace operator in Ω.

Within the framework of the generalised Landau-de Gennes theory, we identify a Q-tensor-based energy that reduces to the four-constant Oseen–Frank energy when it is considered over orientable uniaxial nematic states. Although the commonly considered version of the Landau-de Gennes theory has an elastic contribution that is at most cubic in components of the Q-tensor and their derivatives, the alternative offered here is quartic in these variables. One clear advantage of our approach over the cubic theory is that the associated minimisation problem is well-posed for a significantly wider choice of elastic constants. In particular, this quartic energy can be used to model nematic-to-isotropic phase transitions for highly disparate elastic constants. In addition to proving well-posedness of the proposed version of the Landau-de Gennes theory, we establish a rigorous connection between this theory and its Oseen–Frank counterpart via a Г-convergence argument in the limit of vanishing nematic correlation length. We also prove strong convergence of the associated minimisers.

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 revisit the problem of approximating minimizers of certain convex functionals subject to a convexity constraint by solutions of fourth order equations of Abreu type. This approximation problem was studied in previous articles of Carlier–Radice (Approximation of variational problems with a convexity constraint by PDEs of Abreu type. Calc. Var. Partial Differential Equations 58 (2019), no. 5, Art. 170) and the author (Singular Abreu equations and minimizers of convex functionals with a convexity constraint, arXiv:1811.02355v3, Comm. Pure Appl. Math., to appear), under the uniform convexity of both the Lagrangian and constraint barrier. By introducing a new approximating scheme, we completely remove the uniform convexity of both the Lagrangian and constraint barrier. Our analysis is applicable to variational problems motivated by the original 2D Rochet–Choné model in the monopolist's problem in Economics, and variational problems arising in the analysis of wrinkling patterns in floating elastic shells in Elasticity.

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

$$-{\epsilon }^2\Delta v + Vv = {\epsilon }^{-\alpha }(I_\alpha *|v|^p)|v|^{p-2}v,\quad v\in H^1({\mathbb R}^N),$$

We discuss the existence, nonexistence and multiplicity of solutions for a class of elliptic equations in the unit ball with zero Dirichlet boundary conditions involving nonlinearities with supercritical growth. By using Pohozaev type identity we prove a nonexistence result for a class of supercritical problems with variable exponent which allow us to complement the analysis developed in (Calc. Var. (2016) 55:83). Moreover, we establish existence results of positive solutions for semilinear elliptic equations involving nonlinearities which are subcritical at infinity just in a part of the domain, and can be supercritical in a suitable sense.

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

\[ \int_{\mathbb R^N}|\nabla u|^p\,\rd x\ge \mathcal S_p\int_{\open R^N}|u|^\frac{Np}{N-p}\,\rd x,\quad u\in \mathcal D^{1,p}(\open R^N) \]

We consider the problem of minimizing the weighted Dirichlet energy between homeomorphisms of planar annuli. A known challenge lies in the case when the weight λ depends on the independent variable z. We prove that for an increasing radial weight λ(z) the infimal energy within the class of all Sobolev homeomorphisms is the same as in the class of radially symmetric maps. For a general radial weight λ(z) we establish the same result in the case when the target is conformally thin compared to the domain. Fixing the admissible homeomorphisms on the outer boundary we establish the radial symmetry for every such weight.

For the solution of the Poisson problem with an L∞ right hand side

\begin{cases} -\Delta u(x) = f (x) & {\rm in}\ D, \\ u=0 & {\rm on}\ \partial D \end{cases}

\|u\|_\infty\leq \|f\|_\infty \sigma_D(\|f\|_1/\|f\|_\infty),

$\|f\|_1$

$\|f\|_\infty .$

\sigma_D(t)\leq\sigma_B(t),\text{ for }t\in[0,|D|],

Let$\Omega \subset \mathbb{R}^{N} $, N ≽ 2, be a smooth bounded domain. For s ∈ (1/2, 1), we consider a problem of the form

$$\left\{\begin{array}{@{}ll} (-\Delta)^s u= \mu(x)\, \mathbb{D}_s^{2}(u) + \lambda f(x), & {\rm in}\,\Omega, \\ u= 0, & {\rm in}\,\mathbb{R}^{N} \setminus \Omega,\end{array}\right.$$

$\mu \in L^{\infty}$

$\mathbb {D}_s^2$

$$\mathbb{D}_s^2 (u) = \frac{a_{N,s}}{2} \int_{\mathbb{R}^{N}} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\,{\rm d}y.$$