In this paper, we study the existence and multiplicity of solutions for Kirchhoff-type superlinear problems involving non-local integro-differential operators. As a particular case, we consider the following Kirchhoff-type fractional Laplace equation:

$$\matrix{ {\left\{ {\matrix{ {M\left( {\int\!\!\!\int\limits_{{\open R}^{2N}} {\displaystyle{{ \vert u(x)-u(y) \vert ^2} \over { \vert x-y \vert ^{N + 2s}}}} {\rm d}x{\rm d}y} \right){(-\Delta )}^su = f(x,u)\quad } \hfill & {{\rm in }\Omega ,} \hfill \cr {u = 0\quad } \hfill & {{\rm in }{\open R}^N{\rm \setminus }\Omega {\mkern 1mu} ,} \hfill \cr } } \right.} \hfill \cr } $$

$M:{\open R}_0^ + \to {\open R}^ + $

We are concerned with an elliptic problem which describes a mean field equation of the equilibrium turbulence of vortices with variable intensities. In the first part of the paper, we describe the blow-up picture and highlight the differences from the standard mean field equation as we observe non-quantization phenomenon. In the second part, we discuss the Moser–Trudinger inequality in terms of the blow-up masses and get the existence of solutions in a non-coercive regime by means of a variational argument, which is based on some improved Moser–Trudinger inequalities.

By using variational and some new analytic techniques, we prove the existence of ground state solutions for the quasilinear Schrödinger equation with variable potentials and super-linear nonlinearities. Moreover, we establish a minimax characterisation of the ground state energy. Our result improves and extends the existing results in the literature.

In this paper, we study a class of Brezis–Nirenberg problems for nonlocal systems, involving the fractional Laplacian $(-\unicode[STIX]{x1D6E5})^{s}$ operator, for $0<s<1$, posed on settings in which Sobolev trace embedding is noncompact. We prove the existence of infinitely many solutions in large dimension, namely when $N>6s$, by employing critical point theory and concentration estimates.

In this paper, we consider the existence of multiple solutions for the quasilinear Schrödinger equation

$$\left\{ {\matrix{ {-\Delta u-\Delta (\vert u \vert ^\alpha )\vert u \vert ^{\alpha -2}u = g(x,u) + \theta h(x,u),\;\;x\in \Omega } \hfill \cr {u = 0,\;\;x\in \partial \Omega ,} \hfill \cr } } \right.$$

We introduce an Almgren frequency function of the sub-$p$-Laplace equation on the Heisenberg group to establish a doubling estimate under the assumption that the frequency function is locally bounded. From this, we obtain some partial results on unique continuation for the sub-$p$-Laplace equation.

We consider the unique recovery of a non-compactly supported and non-periodic perturbation of a Schrödinger operator in an unbounded cylindrical domain, also called waveguide, from boundary measurements. More precisely, we prove recovery of a general class of electric potentials from the partial Dirichlet-to-Neumann map, where the Dirichlet data is supported on slightly more than half of the boundary and the Neumann data is taken on the other half of the boundary. We apply this result in different contexts including recovery of some general class of non-compactly supported coefficients from measurements on a bounded subset and recovery of an electric potential, supported on an unbounded cylinder, of a Schrödinger operator in a slab.

The present work is part of a series of studies conducted by the authors on analytical models of avascular tumour growth that exhibit both geometrical anisotropy and physical inhomogeneity. In particular, we consider a tumour structure formed in distinct ellipsoidal regions occupied by cell populations at a certain stage of their biological cycle. The cancer cells receive nutrient by diffusion from an inhomogeneous supply and they are subject to also an inhomogeneous pressure field imposed by the tumour microenvironment. It is proved that the lack of symmetry is strongly connected to a special condition that should hold between the data imposed by the tumour’s surrounding medium, in order for the ellipsoidal growth to be realizable, a feature already present in other non-symmetrical yet more degenerate models. The nutrient and the inhibitor concentration, as well as the pressure field, are provided in analytical fashion via closed-form series solutions in terms of ellipsoidal eigenfunctions, while their behaviour is demonstrated by indicative plots. The evolution equation of all the tumour’s ellipsoidal interfaces is postulated in ellipsoidal terms and a numerical implementation is provided in view of its solution. From the mathematical point of view, the ellipsoidal system is the most general coordinate system that the Laplace operator, which dominates the mathematical models of avascular growth, enjoys spectral decomposition. Therefore, we consider the ellipsoidal model presented in this work, as the most general analytic model describing the avascular growth in inhomogeneous environment. Additionally, due to the intrinsic degrees of freedom inherited to the model by the ellipsoidal geometry, the ellipsoidal model presented can be adapted to a very populous class of avascular tumours, varying in figure and in orientation.

In the present paper we deal with a quasilinear problem involving a singular term. By combining truncation techniques with variational methods, we prove the existence of three weak solutions. As far as we know, this is the first contribution in this direction in the high-dimensional case.

We give two-sided estimates for positive solutions of the superlinear elliptic problem $-\unicode[STIX]{x1D6E5}u=a(x)|u|^{p-1}u$ with zero Dirichlet boundary condition in a bounded Lipschitz domain. Our result improves the well-known a priori$L^{\infty }$-estimate and provides information about the boundary decay rate of solutions.

In this paper we consider the existence of least energy nodal solution for the defocusing quasilinear Schrödinger equation

$$-\Delta u - u \Delta u^2 + V(x)u = a(x)[g(u) + \lambda \vert u \vert ^{p-2}u] \hbox{in} {\open R}^N,$$

$L</italic>^{\infty}({\open R}^{N})$

We are concerned with the following Kirchhoff-type equation

$$ - \varepsilon ^2M\left( {\varepsilon ^{2 - N}\int_{{\open R}^N} {\vert \nabla u \vert^2{\rm d}x} } \right)\Delta u + V(x)u = f(u),\quad x \in {{\open R}^N},\quad N{\rm \ges }2,$$

At the ANZIAM conference in Hobart in February 2018, there were several talks on the solution of Laplace problems in multiply connected domains by means of conformal mapping. It appears to be not widely known that such problems can also be solved by the elementary method of series expansions with coefficients determined by least-squares fitting on the boundary. (These are not convergent series; the coefficients depend on the degree of the approximation.) Here we give a tutorial introduction to this method, which converges at an exponential rate if the boundary data are sufficiently well-behaved. The mathematical foundations go back to Runge in 1885 and Walsh in 1929. One of our examples involves an approximate Cantor set with up to 2048 components.

We study space-like hypersurfaces with functionally bounded mean curvature in Lorentzian warped products , where F is a (non-compact) complete Riemannian manifold whose universal covering is parabolic. In particular, we provide several rigidity results under appropriate mathematical and physical assumptions. As an application, several Calabi–Bernstein-type results are obtained which widely extend the previous ones in this setting.

We prove the uniqueness of a solution for a problem whose simplest model is

with k ≥ 1, 0 f ∈ L∞(Ω) and Ω is a bounded domain of ℝN, N ≥ 2. So far, uniqueness results are known for k < 1, while existence holds for any k ≥ 1 and f positive in open sets compactly embedded in a neighbourhood of the boundary. We extend the uniqueness results to the k ≥ 1 case and show, with an example, that existence does not hold if f is zero near the boundary. We even deal with the uniqueness result when f is replaced by a nonlinear term λuq with 0 < q < 1 and λ > 0.

We consider a nonlinear Robin problem driven by a non-homogeneous differential operator plus an indefinite potential term. The reaction function is Carathéodory with arbitrary growth near±∞. We assume that it is odd and exhibits a concave term near zero. Using a variant of the symmetric mountain pass theorem, we establish the existence of a sequence of distinct nodal solutions which converge to zero.

In this paper, we study the existence of a solution for the following class of non-local problems: P

$$\eqalign{\big\{&-\Delta u=\left(\lambda f(x)-\int_{{open R}^N}K(x,y)\vert u(y)\vert ^{\gamma}\hbox{d}y\right)u\quad \mbox{in } \R^{N}, \cr &\lim_{\vert x\vert \to +\infty}u(x)=0,\quad u \gt 0 \quad \text{in } {open R}^{N},}$$

Let L be a one-to-one operator of type ω in L2(ℝn), with ω∈[0, π/2), which has a bounded holomorphic functional calculus and satisfies the Davies–Gaffney estimates. Let p(·): ℝn→(0, 1] be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors introduce the variable Hardy space $H_L^{p(\cdot )} ({\open R}^n)$ associated with L. By means of variable tent spaces, the authors establish the molecular characterization of $H_L^{p(\cdot )} ({\open R}^n)$. Then the authors show that the dual space of $H_L^{p(\cdot )} ({\open R}^n)$ is the bounded mean oscillation (BMO)-type space ${\rm BM}{\rm O}_{p(\cdot ),{\kern 1pt} L^ * }({\open R}^n)$, where L* denotes the adjoint operator of L. In particular, when L is the second-order divergence form elliptic operator with complex bounded measurable coefficients, the authors obtain the non-tangential maximal function characterization of $H_L^{p(\cdot )} ({\open R}^n)$ and show that the fractional integral L−α for α∈(0, (1/2)] is bounded from $H_L^{p(\cdot )} ({\open R}^n)$ to $H_L^{q(\cdot )} ({\open R}^n)$ with (1/p(·))−(1/q(·))=2α/n, and the Riesz transform ∇ L−1/2 is bounded from $H_L^{p(\cdot )} ({\open R}^n)$ to the variable Hardy space Hp(·)(ℝn).

We are concerned with the existence of positive weak solutions, as well as the existence of bound states (i.e. solutions in W1, p (ℝN)), for quasilinear scalar field equations of the form

$$ - \Delta _pu + V(x) \vert u \vert ^{p - 2}u = K(x) \vert u \vert ^{q - 2}u + \vert u \vert ^{p^ * - 2}u,\qquad x \in {\open R}^N,$$

For bounded domains Ω, we prove that the Lp-norm of a regular function with compact support is controlled by weighted Lp-norms of its gradient, where the weight belongs to a class of symmetric non-negative definite matrix-valued functions. The class of weights is defined by regularity assumptions and structural conditions on the degeneracy set, where the determinant vanishes. In particular, the weight A is assumed to have rank at least 1 when restricted to the normal bundle of the degeneracy set S. This generalization of the classical Poincaré inequality is then applied to develop a robust theory of first-order Lp-based Sobolev spaces with matrix-valued weight A. The Poincaré inequality and these Sobolev spaces are then applied to produce various results on existence, uniqueness and qualitative properties of weak solutions to boundary-value problems for degenerate elliptic, degenerate parabolic and degenerate hyperbolic partial differential equations (PDEs) of second order written in divergence form, where A is calibrated to the matrix of coefficients of the second-order spatial derivatives. The notion of weak solution is variational: the spatial states belong to the matrix-weighted Sobolev spaces with p = 2. For the degenerate elliptic PDEs, the Dirichlet problem is treated by the use of the Poincaré inequality and Lax–Milgram theorem, while the treatment of the Cauchy–Dirichlet problem for the degenerate evolution equations relies only on the Poincaré inequality and the parabolic and hyperbolic counterparts of the Lax–Milgram theorem.