This paper investigates the large-time behaviors of strong solutions for the free-boundary problem of a one-dimensional drift-flux model, which describes a flow scenario in the wellbore flow systems where the gas-liquid twophase flow is separated by a gas-dominated region that holds a specific pressure p(& lowast;) > 0. Without any smallness assumption upon the initial data, we prove the global existence and large-time behaviors of the solution. In particular, we show the exponential decay rates of the solution and the behaviors of the solution near the boundary. The key to the matter is to prevent the formation of the possible singularity of the pressure function and improve the regularity of the solution near the boundary. The analysis is built on the reformulation of the model via the flow map, and a series of weighted energy estimates established in virtue of the Hardy inequality.

We concern standing wave solutions with frequency λ to a two dimensional Gross-Pitaevskii equation with a trap potential under the unit mass constraint, which is used to describe Bose-Einstein condensates with attractive interaction. First, we investigate the necessary conditions for existence of the solutions with concentration phenomena directed along closed smooth curves. Next, not only imposing stationary and non-degeneracy conditions on the curves with respect to an auxiliary weighted length involving the trap potential, but also adding some other technical assumptions, we select a sequence {λ_j} of the frequency λ with -λ_j→ +∞ and construct solutions with concentration directed along the curves. Our result partially answers the conjecture raised in [A. Ambrosetti, A. Malchiodi, W.-M. Ni, Comm. Math. Phys. 2003] about necessary condition for solution concentrating at submanifolds. The solutions constructed in this paper are concentrating on curves whose length are non-uniformly bounded, and hence the situation is quite different from that in [M. del Pino, M. Kowalczyk, J. Wei, Comm. Pure Appl. Math. 2007].

In this article, we study the elliptic equation with critical Sobolev nonlinearity and Hardy potentials (-triangle)p(u )+ a(x)|u|(p-1)u - mu | u | (p - 1) u / | x | p = | u |( p & lowast; - 2) u + f(x, u), u is an element of W-1,W-p(R-N), where 0 < < min{ ( N - p ) /p(p ) , N (p- 1) ( N - p 2 )/ p p } , p & lowast; = Np/ N-p is the critical Sobolev exponent. Through a compactness analysis of the associated functional operator, we obtain the existence of positive solutions under certain assumptions on a(x) and f(x, u).

We consider the prescribed scalar curvature problem on 𝕊^N Δ _𝕊^N v-N(N-2)/2 v+K̃(y) v^N+2/N-2=0 on 𝕊^N, v >0 in 𝕊^N, under the assumptions that the scalar curvature K̃ is rotationally symmetric, and has a positive local maximum point between the poles. We prove the existence of infinitely many non-radial positive solutions, whose energy can be made arbitrarily large. These solutions are invariant under some non-trivial sub-group of O(3) obtained doubling the equatorial. We use the finite dimensional Lyapunov–Schmidt reduction method.

We consider the nonlinear problem of anisotropic Allen-Cahn equa-tion epsilon 2div(Va(y)u) +P(y)u(1 - u2) = 0 in ohm, Va(y)u center dot nu = 0 on partial differential ohm, where ohm is a bounded domain in R2 with smooth boundary, epsilon is a small positive parameter, nu denotes the unit outward normal of partial differential ohm, and P(y) is a uniformly positive smooth potential on ohm over bar . The operator Va(y)u is defined by Va(y)u = (a1(y)uy1 , a2(y)uy2) with a(y) = (a1(y), a2(y)), where a1(y) and a2(y) are two positive smooth functions on ohm over bar . Let Gamma be an interior curve intersecting orthogonally with partial differential ohm at exactly two points or a closed simple curve in ohm, and dividing ohm into two parts. Moreover, Gamma is a non-degenerate geodesic embedded in the Riemannian manifold R2 associated with metric P(y)(a2(y)dy1 (R) dy1 + a1(y)dy2 (R) dy2). By assuming some additional constraints on the functions a(y), P(y) and the curves Gamma, partial differential ohm, we prove that there exists a solution u epsilon with an interface such that: as epsilon -> 0, u epsilon approaches +1 in one part of ohm, while tends to -1 in the other part, except a small neighborhood of Gamma.

We consider the inhomogeneous Allen–Cahn equation $$\begin{aligned} \epsilon ^2\Delta u\,+\,V(y)(1-u^2)\,u\,=\,0\quad \text{ in }\ \Omega , \qquad \frac{\partial u}{\partial \nu }\,=\,0\quad \text{ on }\ \partial \Omega , \end{aligned}$$ where $$\Omega $$ is a bounded domain in $${\mathbb {R}}^2$$ with smooth boundary $$\partial \Omega $$ and V(x) is a positive smooth function, $$\epsilon >0$$ is a small parameter, $$\nu $$ denotes the unit outward normal of $$\partial \Omega $$ . For any fixed integer $$N\ge 2$$ , we will show the existence of a clustered solution $$u_{\epsilon }$$ with N-transition layers near $$\partial \Omega $$ with mutual distance $$O(\epsilon |\ln \epsilon |)$$ , provided that the generalized mean curvature $$\mathcal {H} $$ of $$\partial \Omega $$ is positive and $$\epsilon $$ stays away from a discrete set of values at which resonance occurs. Our result is an extension of those (with dimension two) by Malchiodi et al. (Pac. J. Math. 229(2):447–468, 2007) and Malchiodi et al. (J. Fixed Point Theory Appl. 1(2):305–336, 2007).

We consider the problem epsilon(2)div(del(a(y))u) - V(y)u + u(p) = 0, u > 0 in Omega, where Omega is a bounded domain in R-2 with smooth boundary, the exponent pis greater than 1, epsilon > 0 is a small parameter, Vis a uniformly positive smooth potential on (Omega) over bar, and nu denotes the outward normal of partial derivative Omega. For two positive smooth functions a(1)(y), a(2)(y) on (Omega) over bar, the operator del(a(y)) is given by del(a(y))u - (a1(y)partial derivative u/partial derivative y(1), alpha(2)(y)partial derivative u/partial derivative y(2)). (1). Let Gamma subset of (Omega) over bar be a smooth curve intersecting orthogonally with partial derivative Omega at exactly two points and dividing Omega into two parts. Moreover, Gamma is a non-degenerate geodesicembedded in the Riemannian manifold R2with metric V-2 sigma(y)[a(2)(y)dy(1)(2)+ a(1)(y)dy(2)(2)], where sigma = p+1/p- 1- 1/2. By assuming some additional constraints on the functions a(y), V(y) and the curves G, partial derivative Omega, we prove that there exists a sequence of epsilon such that the problem has solutions u(epsilon) with clustering concentration layers directed along Gamma, exponentially small in epsilon at any positive distance from it. (2). If (Gamma) over tilde is a simple closed smooth curve in Omega(not touching the boundary partial derivative Omega), which is also a non-degenerate geodesicembedded in the Riemannian manifold R-2 with metric V-2 sigma(y)[a(2)(y)dy(1)(2)]+ a(1)(y)dy(2)(2)], then a similar result of concentrated solutions is still true. (C) 2021 Elsevier Inc. All rights reserved.

This paper deals with the following non-linear equation with a fractional Laplacian operator and almost critical exponents: \[ (-\Delta)^{s} u=K(|y'|,y'')u^{({N+2s})/(N-2s)\pm\epsilon},\quad u > 0,\quad u\in D^{1,s}(\mathbb{R}^{N}), \] where N ⩾ 4, 0 < s < 1, ( y ′, y ″) ∈ ℝ 2 × ℝ N −2 , ε > 0 is a small parameter and K ( y ) is non-negative and bounded. Under some suitable assumptions of the potential function K ( r , y ″), we will use the finite-dimensional reduction method and some local Pohozaev identities to prove that the above problem has a large number of bubble solutions. The concentration points of the bubble solutions include a saddle point of K ( y ). Moreover, the functional energies of these solutions are in the order $\epsilon ^{-(({N-2s-2})/({(N-2s)^2})}$ .

We construct infinite time blow-up solution to the following heat equation with Sobolev critical exponent and drift terms $$\begin{aligned} {\left\{ \begin{array}{ll} u_t \,=\, \Delta u\,+\,\nabla b (x) \cdot \nabla u\,+\, u^{\frac{n+2}{n-2}} ~ \text{ in } ~ \mathbb {R}^n\times (0,+\infty ),\\ u(\cdot ,0)=u_0 ~ \text{ in } ~ \mathbb {R}^n, \end{array}\right. } \end{aligned}$$where b(x) is a smooth bounded function in $$\mathbb {R}^{n}$$ with $$n\ge 5$$ and the initial datum $$u_0$$ is positive and smooth. Let $$q_j \in \mathbb {R}^n,j=1,\ldots ,k$$, be distinct nondegenerate local minimum points of b(x). Assume that an eigenvalue condition (1.6) is satisfied. We prove the existence of a positive smooth solution u(x, t) which blows up at infinite time near those points with the form $$\begin{aligned} u(x,t) \approx \sum _{j=1}^k \alpha _n \left( \frac{ \mu _j(t)}{ \mu _j(t)^2 \,+\, |x-\xi _j(t)|^2 } \right) ^{\frac{n-2}{2}}, \quad \text{ as } t\rightarrow +\infty . \end{aligned}$$Here $$\xi _j(t) \rightarrow q_j$$ and $$0<\mu _j(t)\rightarrow 0$$ exponentially as $$t\rightarrow +\infty $$.

We consider the nonlinear problem of inhomogeneous Allen-Cahn equation epsilon(2)Delta u + V(y) (1 - u(2)) u = 0 in Omega, partial derivative u/partial derivative v = 0 on partial derivative Omega, where Omega is a bounded domain in R-2 with smooth boundary, epsilon is a small positive parameter, nu denotes the unit outward normal of partial derivative Omega, V is a positive smooth function on (Omega) over bar. Let Gamma subset of Omega be a smooth curve dividing Omega into two disjoint regions and intersecting orthogonally with partial derivative Omega at exactly two points P-1 and P-2. Moreover, by considering R-2 as a Riemannian manifold with the metric g = V(y) (dy(1)(2) + dy(2)(2)), we assume that: the curve Gamma is a non-degenerate geodesic in the Riemannian manifold (R-2, g), the Ricci curvature of the Riemannian manifold (R-2, g) along the normal n of is positive at Gamma, the generalized mean curvature of the submanifold partial derivative Omega in (R-2, g) vanishes at P-1 and P-2. Then for any given integer N >= 2, we construct a solution exhibiting N-phase transition layers near Gamma (the zero set of the solution has N components, which are curves connecting partial derivative Omega and directed along the direction of Gamma) with mutual distance O(epsilon vertical bar log epsilon vertical bar), provided that epsilon stays away from a discrete set of values to avoid the resonance of the problem. Asymptotic locations of these layers are governed by a Toda system.

We consider the Fife-Greenlee problem epsilon(2)Delta u + (u - a(y)) (1-u(2)) = 0 in Omega, partial derivative u/partial derivative v = 0 on partial derivative Omega, where Omega is a bounded domain in R-2 with smooth boundary, epsilon > 0 is a small parameter, nu denotes the unit outward normal of partial derivative Omega. Let Gamma ={y epsilon Omega : a(y) = 0} be a simple smooth curve intersecting orthogonally with partial derivative Omega at exactly two points and dividing Omega into two disjoint nonempty components. We assume that -1 < a(y) < 1 on Omega and del a not equal 0 on Gamma, and also some admissibility conditions hold for a, Gamma and partial derivative Omega. For any fixed integer N = 2m + 1 >= 3, we will show the existence of a clustered solution uewith N-transition layers near Gamma with mutual distance O(epsilon vertical bar log epsilon vertical bar), provided that estays away from a discrete set of values at which resonance occurs. (C) 2020 Elsevier Inc. All rights reserved.

This paper deals with the following prescribed scalar curvature problem−Δu=Q(|y′|,y″)uN+2N−2,u>0,y=(y′,y″)∈R2×RN−2, where Q(y) is nonnegative and bounded. By combining a finite reduction argument and local Pohozaev type of identities, we prove that if N≥5 and Q(r,y″) has a stable critical point (r0,y0″) with r0>0 and Q(r0,y0″)>0, then the above problem has infinitely many solutions, whose energy can be made arbitrarily large. Here, instead of estimating directly the derivatives of the reduced functional, we apply some local Pohozaev identities to locate the concentration points of the bump solutions. Moreover, the concentration points of the bump solutions include a saddle point of Q(y).

This paper deals with the following nonlinear perturbed fractional Laplacian equation $$(-\Delta)^s u = K(|y'|,y'')u^{\frac{N+2s}{N-2s}\pm\epsilon},\,\,u>0,\,\,u\in D^{1,s}(\mathbb{R}^N),$$ where $0 0$ is a small parameter and $K(y)$ is nonnegative and bounded. By combining a finite reduction argument and local Pohozaev type of identities, we prove that if $N\geq 4,\max\{\frac{N+1-\sqrt{N^{2}-2N+9}}{4},\frac{3-\sqrt{N^{2}-6N+13}}{2}\} 0$ and $K(r_0, y_0'')>0,$ then the above problem has large number of bubble solutions if $\epsilon>0$ is small enough. Also there exist solutions whose functional energy is in the order $\epsilon^{-\frac{N-2s-2}{(N-2s)^{2}}}$. Here, instead of estimating directly the derivatives of the reduced functional, we apply some local Pohozaev identities to locate the concentration points of the bubble solutions. Moreover, the concentration points of the bubble solutions include a saddle point of $K(y)$.

We consider the Fife-Greenlee problem epsilon(2) Delta u + (u - a(y)) (1 - u(2)) = 0 in Omega, partial derivative u/partial derivative v = 0 on partial derivative Omega, where Omega is a bounded domain in R-2 with smooth boundary, epsilon > 0 is a small parameter, v denotes the unit outward normal of partial derivative Omega. Let Gamma = {y is an element of Omega : a(y) = 0} be a simple smooth curve intersecting orthogonally with partial derivative Omega at exactly two points and dividing Omega into two disjoint nonempty components. We assume that -1 < a(y) < 1 on Omega and del a not equal 0 on Gamma, and also some admissibility conditions between the curves Gamma, partial derivative Omega and the inhomogeneity a hold at the connecting points. We can prove that there exists a solution u, such that: as epsilon -> 0, u(epsilon) approaches +1 in one part, while tends to -1 in the other part, except a small neighborhood of Gamma.

We consider the problem $$ \epsilon^2 \Delta u-V(y)u+u^p\,=\,0,~~u>0~~\quad\mbox{in}\quad\Omega,~~\quad\frac {\partial u}{\partial \nu}\,=\,0\quad\mbox{on}~~~\partial \Omega, $$ where $\Omega$ is a bounded domain in $\mathbb R^2$ with smooth boundary, the exponent $p>1$, $\epsilon>0$ is a small parameter, $V$ is a uniformly positive, smooth potential on $\bar{\Omega}$, and $\nu$ denotes the outward normal of $\partial \Omega$. Let $\Gamma$ be a curve intersecting orthogonally with $\partial \Omega$ at exactly two points and dividing $\Omega$ into two parts. Moreover, $\Gamma$ satisfies stationary and non-degeneracy conditions with respect to the functional $\int_{\Gamma}V^{\sigma}$, where $\sigma=\frac {p+1}{p-1}-\frac 12$. We prove the existence of a solution $u_\epsilon$ concentrating along the whole of $\Gamma$, exponentially small in $\epsilon$ at any positive distance from it, provided that $\epsilon$ is small and away from certain critical numbers. In particular, this establishes the validity of the two dimensional case of a conjecture by A. Ambrosetti, A. Malchiodi and W.-M. Ni(p.327, [4]).