In this paper, we consider the planar vortex patch problem in an incompressible steady flow in a bounded domain Ω of R2...

Let Ω be a bounded domain in R², u+=u if u⩾0, u+=0 if u<0, u−=u+−u. In this paper we study the existence of solutions to the following problem arising in the study of a simple model of a confined plasma ... where ν is the outward unit normal of ∂Ω at x, c is a constant which is unprescribed, and I is a given positive constant. The set Ωp = {x ∈ Ω, u(x) < 0} is called plasma set. Existence of solutions whose plasma set consisting of one component and asymptotic behavior of plasma set were studied by Caffarelli and Friedman (1980) [3] for large λ. Under the condition that the homology of Ω is nontrivial we obtain in this paper by a constructive way that for any given integer k⩾1, there is λk>0 such that for λ>λk, (Pλ) has a solution with plasma set consisting of k components.

This paper is concerned with the positive solutions for generalized quasilinear Schrödinger equations in R'N' with critical growth which have appeared from plasma physics, as well as high-power ultrashort laser in matter. By using a change of variables and variational argument, we obtain the existence of positive solutions for the given problem.

We prove the existence of positive solutions concentrating simultaneously on some higher dimensional manifolds near and on the boundary of the domain for a nonlinear singularly perturbed elliptic Neumann problem.

Let B₁(0) 'C RN' be the unit ball centred at the origin, N ≥ 3. In this paper, we analyse the profile of the ground state solution of the Hénon equation - ∆u = │x│'au⁻¹ in B₁ (0), u = 0 on ∂B₁ (0). We prove that for fixed p ε (2,2*), (2* = 2N/(N - 2)), the maximum point xₐ of the ground state solution uₐ satisfies a(1 - │xₐ│) → l ε(0,+ ∞) as a → ∞. We also obtain the asymptotic behaviour of uₐ, which shows that the ground state solution is non-radial. Moreover, we prove the existence of multi-peaked solutions and give their asymptotic behaviour.

By variational methods, for a kind of Webster scalar curvature problems on the CR sphere with cylindrically symmetric curvature, we construct some multi-peak solutions as the parameter is sufficiently small under certain assumptions. We also obtain the asymptotic behaviors of the solutions.

In this paper, simulated by the paper of Wei and Yan [33] (see also [30–32]), we intend to find infinitely many positive solutions to (1.2) for all p ∈ (1, 5) under weaker integrability conditions on K(y) and Q(y). In [33], a single equation, this is, K(y) ≡ 0 in (1.2), was studied and infinitely many non-radial solutions were found in the case that Q(y) is radial. For this, they employed a very novel idea, that is, they use k, the number of the bumps of the solutions, as the parameter to construct Infinitely Many Solutions for Schrödinger-Poisson System 1071 spike solutions for the Schrödinger equation considered.

This paper is concerned with the solitary wave solutions for a generalized quasilinear Schrödinger equation in RN involving critical exponents, which have appeared from plasma physics, as well as high-power ultashort laser in matter. We find the related critical exponents for a generalized quasilinear Schrödinger equation and obtain its solitary wave solutions by using a change of variables and variational argument.

In this paper, we will prove the existence of infinitely many solutions for the following elliptic problem with critical Sobolev growth...