Solutions Of Nonlinear Elliptic Equations Research Articles

Positive solutions of nonlinear elliptic equations involving unbounded variable exponents and convection term

Positive solutions of nonlinear elliptic equations involving unbounded variable exponents and convection term

On the critical points of solutions of PDE in non-convex settings: The case of concentrating solutions

In this paper we are concerned with the number of critical points of solutions of nonlinear elliptic equations. We will deal with the case of non-convex, contractile and non-contractile planar domains. We will prove results on the estimate of their number as well as their index. In some cases we will provide the exact calculation. The toy problem concerns the multi-peak solutions of the Gel'fand problem, namely{−Δu=λeu in Ωu=0 on ∂Ω, where Ω⊂R2 is a bounded smooth domain and λ>0 is a small parameter.

GIDAS-NI-NIRENBERG TYPE RESULTS FOR THE GELFAND PROBLEM

Abstract. In this paper we are concerned with the number of critical points of solutions of nonlinear elliptic equations. We will deal with the case of non-convex, contractile and non-contractile planar domains. We will prove results on the estimate of their number as well as their index. In some cases we will provide the exact calculation. The toy problem concerns the multi-peak solutions of the Gel'fand problem

Existence of positive radial solutions for nonlinear elliptic equations with gradient terms in an annulus

Existence of positive radial solutions for nonlinear elliptic equations with gradient terms in an annulus

Fucik spectrum with weights and existence of solutions for nonlinear elliptic equations with nonlinear boundary conditions

We consider the boundary value problem $$\displaylines{ - \Delta u + c(x) u = \alpha m(x) u^+ - \beta m(x) u^- +f(x,u), \quad x \in \Omega, \cr \frac{\partial u}{\partial \eta} + \sigma (x) u =\alpha \rho (x) u^+- \beta \rho (x) u^- +g(x,u), \quad x \in \partial \Omega, }$$ where \((\alpha, \beta) \in \mathbb{R}^2\), \(c, m \in L^\infty (\Omega)\), \(\sigma, \rho \in L^\infty (\partial\Omega)\), and the nonlinearities f and g are bounded continuous functions. We study the asymmetric (Fucik) spectrum with weights, and prove existence theorems for nonlinear perturbations of this spectrum for both the resonance and non-resonance cases. For the resonance case, we provide a sufficient condition, the so-called generalized Landesman-Lazer condition, for the solvability. The proofs are based on variational methods and rely strongly on the variational characterization of the spectrum. See also https://ejde.math.txstate.edu/special/02/m2/abstr.html

Regular solutions for nonlinear elliptic equations, with convective terms, in Orlicz spaces

AbstractWe establish some existence and regularity results to the Dirichlet problem, for a class of quasilinear elliptic equations involving a partial differential operator, depending on the gradient of the solution. Our results are formulated in the Orlicz–Sobolev spaces and under general growth conditions on the convection term. The sub‐ and supersolutions method is a key tool in the proof of the existence results.

Radial solutions for nonlinear elliptic equation with nonlinear nonlocal boundary conditions

In this article, we prove existence of radial solutions for a nonlinear elliptic equation with nonlinear nonlocal boundary conditions. Our method is based on some fixed point theorem in a cone.

Critical points of positive solutions of nonlinear elliptic equations: multiplicity, location, and non-degeneracy

Critical points of positive solutions of nonlinear elliptic equations: multiplicity, location, and non-degeneracy

Existence of Classical Solutions for Nonlinear Elliptic Equations with Gradient Terms.

This paper deals with the existence of solutions of the elliptic equation with nonlinear gradient term -Δu=f(x,u,∇u) on Ω restricted by the boundary condition u|∂Ω=0, where Ω is a bounded domain in RN with sufficiently smooth boundary ∂Ω, N≥2, and f:Ω¯×R×RN→R is continuous. The existence results of classical solutions and positive solutions are obtained under some inequality conditions on the nonlinearity f(x,ξ,η) when |(ξ,η)| is small or large enough.

Generalized Pohozhaev’s identity for radial solutions of p-Laplace equations

Abstract We derive a generalized Pohozhaev’s identity for radial solutions of p-Laplace equations, by using the approach in [P. Korman, Global Solution Curves for Semilinear Elliptic Equations, World Scientific Publishing, Hackensack, 2012], thus extending the work [H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 1983, 4, 437–477], where this identity was implicitly used for the Laplace equation.

A variational approach for mixed elliptic problems involving the p-Laplacian with two parameters

By exploiting an abstract critical-point result for differentiable and parametric functionals, we show the existence of infinitely many weak solutions for nonlinear elliptic equations with nonhomogeneous boundary conditions. More accurately, we determine some intervals of parameters such that the treated problem admits either an unbounded sequence of solutions or a pairwise distinct sequence of solutions that strongly converges to zero. No symmetric condition on the nonlinear term is considered.

An inequality of Bessel functions and applications to transcritical bifurcation problems of nonlinear elliptic equations

We prove a weighted integral inequality of the Bessel function of zero order, which implies the positivity of the integral of the cube of nodal radial eigenfunctions of the Laplacian operator in the planar disk. This solves an open problem in Bonheure et al. (2016) and Corsato et al. (2020). The positivity plays a crucial role in transcritical bifurcation problems of nodal radial solutions of nonlinear elliptic equations.

Existence of solutions for nonlinear elliptic equations modeling the steady flow of the antarctic circumpolar current

In this paper, we study the steady and purely azimuthal flow model representing the Antarctic Circumpolar Current, which is described as a semi-linear elliptic equation. By means of the Mercator projection, the mathematical model is transferred into a second order elliptic equation. We give sufficient conditions to guarantee the existence of solutions for this second elliptic equation with two forms of nonlinear ocean vorticity term.

Schrödinger equations in [formula omitted] with critical exponential growth and concave nonlinearities

In this paper, we consider the existence of solutions for nonlinear elliptic equations of the form(0.1)−Δu+V(|x|)u=Q(|x|)f(u)+λg(|x|,u),x∈R2, where the nonlinear term f(s) has critical exponential growth which behaves like eαs2, g(r,s) is a concave term on s, the radial potentials V,Q:R+→R are unbounded, singular at the origin or decaying to zero at infinity and λ>0 is a parameter. Based on the known Trudinger-Moser inequality in H0,rad1(B1), we establish a new version of Trudinger-Moser inequality in the working space of the associated with the energy functional related to the above problem. By combining the variational methods, Trudinger-Moser inequality and some new approaches to estimate precisely the minimax level of the energy functional, we prove the existence of a nontrivial solution for the above problem under some weak assumptions. Our results show that the presence of the concave term (i.e. λ>0) is very helpful to the existence of nontrivial solutions for Eq. (0.1) in one sense.

Existence of solutions of nonlinear elliptic equations with measure data in Musielak-Orlicz spaces

Abstract A second-order quasilinear elliptic equation with a measure of special form on the right-hand side is considered. Restrictions on the structure of the equation are imposed in terms of a generalized -function such that the conjugate function obeys the -condition and the corresponding Musielak-Orlicz space is not necessarily reflexive. In an arbitrary domain satisfying the segment property, the existence of an entropy solution of the Dirichlet problem is proved. It is established that this solution is renormalized. Bibliography: 29 titles.

Multi-bump solutions for nonlinear elliptic equations involving critical Sobolev exponents

We study the existence of multi-bump solutions of −Δu−λu=up,u>0inΩt,u=0on∂Ωt,where λ is a real constant, N≥3, p=N+2N−2 , and Ωt is a domain that expands as t→∞. It is known that the topology of the domain Ωt affects the existence and non-existence of solutions. In this article, we prove the existence of multi-bump solutions for more general domains; for example, Ωtball≡{tq+v|q∈M,v∈(TqM)⊥,|v|<1} or Ωtann={tq+v|q∈M,v∈(TqM)⊥,a<|v|<a+1}, where a>0 and M is a k-dimensional compact smooth submanifold of RN without boundary (k=1,⋯,N−1).

The existence of solutions for nonlinear elliptic equations: Simple proofs and extensions of a paper by Y. Shi

The paper [39] uses the Craig-Wayne-Bourgain method to construct solutions of an elliptic problem involving parameters. The results of [39] include regularity assumptions on the perturbation and involve excluding parameters. The paper [39] also constructs response solutions to a quasi-periodically perturbed (ill-posed evolution) problem.In this paper, we use several classical methods (freezing of coefficients, alternative methods for nonlinear elliptic equations) to extend the results of [39]. We weaken the regularity assumptions on the perturbation and we describe the phenomena that happens for all parameters. In the ill-posed problem, we use a recently developed time-dependent center manifold theorem which allows to reduce the problem to a finite-dimensional ODE with quasi-periodic dependence on time. The bounded and sufficiently small solutions of these ODE give solutions of the ill-posed PDE.

Existence and asymptotic properties of singular solutions of nonlinear elliptic equations in R^{n}backslash{0}

We consider the following singular semilinear problem {Δu(x)+p(x)uγ=0,x∈D(in the distributional sense),u>0,in D,lim|x|→0|x|n−2u(x)=0,lim|x|→∞u(x)=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} \\Delta u(x)+p(x)u^{\\gamma }=0,\\quad x\\in D ~(\\text{in the distributional sense}), \\\\ u>0,\\quad \\text{in }D, \\\\ \\lim_{ \\vert x \\vert \\rightarrow 0} \\vert x \\vert ^{n-2}u(x)=0, \\\\ \\lim_{ \\vert x \\vert \\rightarrow \\infty }u(x)=0,\\end{cases} $$\\end{document} where gamma <1, D=mathbb{R}^{n}backslash {0} (ngeq 3) and p is a positive continuous function in D, which may be singular at x=0. Under sufficient conditions for the weighted function p(x), we prove the existence of a positive continuous solution on D, which could blow-up at the origin. The global asymptotic behavior of this solution is also obtained.

Singular solutions for nonlinear elliptic equations on bounded domains

Singular solutions for nonlinear elliptic equations on bounded domains

On the numerical solution for nonlinear elliptic equations with variable weight coefficients in an integral boundary conditions

In the paper the two-dimensional elliptic equation with integral boundary conditions is solved by finite difference method. The main aim of the paper is to investigate the conditions for the convergence of the iterative methods for the solution of system of nonlinear difference equations. With this purpose, we investigated the structure of the spectrum of the difference eigenvalue problem. Some sufficient conditions are proposed such that the real parts of all eigenvalues of the corresponding difference eigenvalue problem are positive. The proof of convergence of iterative method is based on the properties of the M-matrices not requiring the symmetry or diagonal dominance of the matrices. The theoretical statements are supported by the results of the numerical experiment.