We prove the existence of weak solutions for the strongly nonlinear parabolic problem

in the anisotropic Sobolev space , where the data f are assumed to be in the dual, and the nonlinear term g(x, t, s) has growth and sign conditions on s.

We study the multiplicity and concentration behaviour of positive solutions for a quasi-linear Choquard equation

where Δp is the p-Laplacian operator, 1 < p < N, V is a continuous real function on ℝN, 0 < μ < N, F(s) is the primitive function of f(s), ε is a positive parameter and * represents the convolution between two functions. The question of the existence of semiclassical solutions for the semilinear case p = 2 has recently been posed by Ambrosetti and Malchiodi. We suppose that the potential satisfies the condition introduced by del Pino and Felmer, i.e.V has a local minimum. We prove the existence, multiplicity and concentration of solutions for the equation by the penalization method and Lyusternik–Schnirelmann theory and even show novel results for the semilinear case p = 2.

In this paper we deal with a singular elliptic problem involving an asymptotically linear nonlinearity and depending on two positive parameters. We investigate the existence, uniqueness and non-existence of the minima of the functional associated with the problem and, by employing a natural and very general definition of a weak solution, we also obtain a bifurcation-type result.

We establish that the intrinsic distance dE associated with an indecomposable plane set E of finite perimeter is infinitesimally Euclidean; namely, in E. By this result, we prove through a standard argument that a conservative vector field in a plane set of finite perimeter has a potential. We also provide some applications to complex analysis. Moreover, we present a collection of results that would seem to suggest the possibility of developing a De Rham cohomology theory for integral currents.

We consider the following Fife–Greenlee problem:

where Ω is a smooth and bounded domain in ℝn, ν is the outer unit normal to ∂Ω and a is a smooth function satisfying a(x) ∈ (–1, 1) in . Let K, Ω– and Ω+ be the zero-level sets of a, {a < 0} and {a < 0}, respectively. We assume ∇a ≠ 0 on K. Fife and Greenlee constructed stable layer solutions, while del Pino et al. proved the existence of one unstable layer solution provided that some gap condition is satisfied. In this paper, for each odd integer m ≥ 3, we prove the existence of a sequence ε = εj → 0, and a solution with m-transition layers near K. The distance of any two layers is O(ε log(1/ε)). Furthermore, converges uniformly to ±1 on the compact sets of Ω± as j → +∞

We develop an algebraic approach to constructing short-time asymptotic expansions of solutions of a class of abstract semilinear evolution equations. The expansions are typically valid for both the solution of the equation and its gradient. We apply a perturbation approach based on the symbolic calculus of pseudo-differential operators and heat kernel methods. The construction is explicit and can be done to arbitrary order. All results are rigorously formulated in terms of Banach algebras. As an application we obtain a novel approach to finding approximate solutions of Markovian backward stochastic differential equations.

There is a huge jump in the theory of conservation laws if the convexity assumption is dropped. We study a scalar conservation law without the convexity assumption by monitoring the dynamics in the fundamental solution. We introduce three shock types in addition to the usual genuine shock: left-, right- and double-sided contacts. There are three kinds of phenomenon for these shocks, called branching, merging and transforming. All of these shocks and phenomena can be observed if the flux function has two inflection points. A comprehensive picture of a global dynamics of a non-convex flux is discussed in terms of characteristic maps and dynamical convex–concave envelopes.

We examine the regularity of the extremal solution of the nonlinear eigenvalue problem

on a general bounded domain Ω in ℝN, with Navier boundary condition u = Δu on ∂Ω. Firstly, we prove the extremal solution is smooth for any p > 1 and N ⩽ 4, which improves the result of Guo and Wei (Discrete Contin. Dynam. Syst. A 34 (2014), 2561–2580). Secondly, if p = 3, N = 3, we prove that any radial weak solution of this nonlinear eigenvalue problem is smooth in the case Ω = 𝔹, which completes the result of Dávila et al. (Math. Annalen348 (2009), 143–193). Finally, we also consider the stability of the entire solution of Δ2u = –l/up in ℝN with u > 0.

Let K ⊂ ℝN be any convex body containing the origin. A measurable set G ⊂ ℝN with finite and positive Lebesgue measure is said to be K-dense if, for any fixed r > 0, the measure of G ⋂ (x + rK) is constant when x varies on the boundary of G (here, x + rK denotes a translation of a dilation of K). In a previous work, we proved for the case N = 2 that if G is K-dense, then both G and K must be homothetic to the same ellipse. Here, we completely characterize K-dense sets in ℝN: if G is K-dense, then both G and K must be homothetic to the same ellipsoid. Our proof, which builds upon results obtained in our previous work, relies on an asymptotic formula for the measure of G ⋂ (x + rK) for large values of the parameter r and a classical characterization of ellipsoids due to Petty.