Nonhomogeneous Dirichlet problems for the p-Laplacian

In this work, we are concerned with questions of existence and multiplicity of solutions for a nonlocal and non-homogeneous Neumann boundary value problems involving the p(x)-Laplace operator and critical growth, from the point of view of the trace, via a truncation argument on generalized Lebesgue-Sobolev spaces.

The inverse scattering transform for the defocusing Manakov system with nonzero boundary conditions at infinity is rigorously studied. Several new results are obtained: (i) The analyticity of the Jost eigenfunctions is investigated, and precise conditions on the potential that guarantee such analyticity are provided. (ii) The analyticity of the scattering coefficients is established. (iii) The behavior of the eigenfunctions and scattering coefficients at the branch points is discussed. (iv) New symmetries are derived for the analytic eigenfunctions (which differ from those in the scalar case). (v) These symmetries are used to obtain a rigorous characterization of the discrete spectrum and to rigorously derive the symmetries of the associated norming constants. (vi) The asymptotic behavior of the Jost eigenfunctions is derived systematically. (vii) A general formulation of the inverse scattering problem as a Riemann--Hilbert problem is presented. (viii) Precise results guaranteeing the existence and uniqueness of solutions of the Riemann--Hilbert problem are provided. (ix) Explicit relations among all reflection coefficients are given, and all entries of the scattering matrix are determined in the case of reflectionless solutions. (x) A compact, closed-form expression is presented for general soliton solutions, including any combination of dark-dark and dark-bright solitons. (xi) A consistent framework is formulated for obtaining solutions corresponding to double zeros of the analytic scattering coefficients, leading to double poles in the Riemann--Hilbert problem, and such solutions are constructed explicitly.

In this paper, we are concerned with a semi-discrete complex short-pulse (sdCSP) equation of both focusing and defocusing types, which can be viewed as an analogue to the Ablowitz–Ladik lattice in the ultra-short-pulse regime. By using a generalized Darboux transformation method, various soliton solutions to this newly integrable semi-discrete equation are studied with both zero and non-zero boundary conditions. To be specific, for the focusing sdCSP equation, the multi-bright solution (zero boundary conditions), multi-breather and high-order rogue wave solutions (non-zero boundary conditions) are derived, while for the defocusing sdCSP equation with non-zero boundary conditions, the multi-dark soliton solution is constructed. We further show that, in the continuous limit, all the solutions obtained converge to the ones for its original CSP equation (Linget al. 2016Physica D327, 13–29 (doi:10.1016/j.physd.2016.03.012); Fenget al. 2016Phys. Rev. E93, 052227 (doi:10.1103/PhysRevE.93.052227)).

ABSTRACTIn this paper, we are concerned with questions of the existence of solution for a class of nonlocal and nonhomogeneous Neumann boundary value problems involving the -Laplacian in which the nonlinear terms assume both critical growth. The main tools used are the Lions' Concentration-Compactness Principle [Lions PL. The concentration-compactness principle in the calculus of variations. The limit case I, part 1. Rev Mat Iberoam. 1985;1(1):145–201.] for variable exponent spaces and the Mountain Pass Theorem.

In this paper, we are interested in the study of the existence of solutions of a class of nonlinear fractional differential equations coupled with Dirichlet boundary conditions. We make an exhaustive study of the sign of Green’s function related to the linear problem and obtain the exact values for which it has the constant sign on its whole square of definition. This constant sign is equivalent to the validity of anti-maximum principles. The existence of solutions is deduced from a recent fixed point theorem valid for operators defined on suitable cones of Banach spaces. Moreover, the method of lower and upper solutions is developed for the non-homogeneous Dirichlet problem.

In this article, the Ekeland variational principle, mountain pass theorem and Trudinger–Moser inequality are applied to establish the existence and multiplicity of solutions for the following nonhomogeneous Kirchhoff-type elliptic problem: where Ω is a smooth bounded domain in , is a Kirchhoff function, is a continuous function, , , , and ε is a small positive parameter. The nonlinearity term behaves like when for some .

In this paper, we obtain a multiplicity result for a nonhomogeneous elliptic problem involving critical growth and an improved Hardy potential. The Moser inequality and Hardy inequality due to Adimurthi and Sandeep, are combined with critical point

The discrete modified Korteweg–de Vries equation under nonzero boundary conditions

Inverse scattering transform for the coupled modified Korteweg-de Vries equation with nonzero boundary conditions

The nonlinear Schrödinger equation is a classical nonlinear evolution equation with wide applications. This paper explores the asymptotic behavior of solutions to the nonlinear Schrödinger equation with non-zero boundary conditions in the presence of a pair of second-order discrete spectra. We analyze the Riemann–Hilbert problem in the inverse scattering transform by the Deift–Zhou nonlinear steepest descent method. Then we propose a proper deformation to deal with the growing time term and give the conditions for the series in the process of deformation by the Laurent expansion. Finally, we provide the characterization of the interactions between the solitary waves corresponding to second-order discrete spectra and the coherent oscillations produced by the perturbation. Numerical verifications are also performed.

Under investigation in this work is a newly proposed nonlocal long-wave–short-wave resonance interaction (LSRI) equation with the self-induced parity-time (PT) symmetric potential. This equation offers PT symmetry analogues of the classical integrable LSRI equation and may be important for the occurence of such equations in nonlinear optics as the nonlocal NLS equation. General soliton solutions to the nonlocal LSRI equation with nonzero boundary condition are derived by using the Hirota’s bilinear method combined with the Kadomtsev–Petviashvili (KP) hierarchy reduction method. These solutions are expressed in terms of Gramian determinants and include dark–dark solitons, dark–antidark solitons and antidark–antidark solitons. Three typical cases of the two solitons, namely two dark–dark solitons, two dark–antidark solitons and two antidark–antidark solitons, are demonstrated.

In this work we study the fully nonhomogeneous initial boundary value problem for the two-dimensional time-dependent Navier–Stokes equations in a general open space domain in R2 with low regularity assumptions on the initial and the boundary value data. We show that the perturbed Navier–Stokes operator is a diffeomorphism from a suitable function space onto its own dual and as a corollary we get that the Navier–Stokes equations are uniquely solvable in these spaces and that the solution depends smoothly on all involved data. Our source data space and solution space are in complete natural duality and in this sense, without any smallness assumptions on the data, we solve the equations for data with optimally low regularity in both space and time.

This paper is concerned with the following Klein‐Gordon‐Maxwell system: where ω > 0 is a constant, , and obeys exponential critical growth in the sense of the Trudinger‐Moser inequality. We give some new sufficient conditions on f, specifically related to exponential growth, to obtain the existence of nontrivial solutions. Furthermore, we prove a nonexistence result with a Pohozaev‐type argument which also provides important information to get the above existence results. In particular, some new analytical approaches and estimates are used to overcome the difficulty arising from the critical growth of Trudinger‐Moser type.

We study the variational Dirichlet problem with nonhomogeneous boundary conditions for degenerate elliptic operators in a bounded domain with summable lower coefficients in the case when the corresponding sesquilinear forms may not satisfy the coercivity condition. A case is singled out when nonhomogeneous boundary conditions are specified explicitly and their number depends on the degree of degeneration of the leading coefficients of the operator under study. An inequality is proved in which the norm of the solution of the nonhomogeneous variational Dirichlet problem is estimated from above by the norms of the boundary functions and the right-hand side of the equation.

We prove the existence of multiple positive solutions of fractional Laplace problems with critical growth, we consider the concave power case or the convex power case. We establish the relationship between the number of the local maximum points of the coefficient function of the critical nonlinearity and the number of the positive solutions of the equation
 For more information see https://ejde.math.txstate.edu/Volumes/2021/23/abstr.html