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 superlinearelliptic problem $-\unicode[STIX]{x1D6E5}u=a(x)|u|^{p-1}u$ with zero Dirichlet boundary condition in a boundedLipschitz domain. Our result improves the well-known a priori $L^{\infty }$ -estimate and provides information about the boundary decayrate 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.

We consider the nonlinear Dirac equation

The potential function V satisfies the conditions that the essential spectrum of the Dirac operator is and this Dirac operator has infinitely many eigenvalues in (−1, 1) accumulating at 1. This potential function V may change sign in ℝ3 and contains the classical Coulomb potential V (x) = −γ/|x| with γ > 0 as a special case. The nonlinearity F satisfies the resonance-type condition lim. Under some additional conditions on V and F, we prove that this equation has infinitely many solutions.

We prove the existence of positive radial solutions to a class of semipositone p-Laplacian problems on the exterior of a ball subject to Dirichlet and nonlinear boundary conditions. Using variational methods we prove the existence of a solution, and then use a priori estimates to prove the positivity of the solution.

We consider weak solutions to

with p > 1, q ≥ max{p − 1, 1}. We exploit the Moser iteration technique to prove a Harnack comparison inequality for C1 weak solutions. As a consequence we deduce a strong comparison principle.

Using a mixture of classical and probabilistic techniques, we investigate the convexity of solutions to the elliptic partial differential equation associated with a certain generalized Ornstein–Uhlenbeck process.

We study a quasilinear degenerately elliptic system involving the operator curl. The leading-order term of the associated energy functional is a q-power of curl of the unknowns. It is interesting to us that the structure of the lower-order terms will play an important role in both the existence and regularity of the solutions. When the lower-order part of the energy functional is convex we obtain weak solutions by minimizing the functional in some suitable spaces of vector fields. When it is concave we obtain critical points of the truncated functional, which are weak solutions of a nonlinear eigenvalue problem. We also examine the interior Hölder regularity of the weak solutions, which indicates that the weak solution is composed of a ‘good’ divergence-free part and a ‘bad’ part in gradient form. The analysis involves local higher integrability and local Hölder gradient estimates for a translated q′-Laplace equation and a translated quasilinear equation with p-growth.

We consider bifurcation of solutions from a given trivial branch for a class of strongly indefinite elliptic systems via the spectral flow. Our main results establish bifurcation invariants that can be obtained from the coefficients of the systems without using explicit solutions of their linearizations at the given branch. Our constructions are based on a comparison principle for the spectral flow and a generalization of a bifurcation theorem due to Szulkin.

In this paper we analyse possible extensions of the classical Steklov eigenvalue problem to the fractional setting. In particular, we find a non-local eigenvalue problem of fractional type that approximates, when taking a suitable limit, the classical Steklov eigenvalue problem.