Class Of Semilinear Elliptic Equations Research Articles

Superharmonic functions related to convexity estimates for a class of semilinear elliptic problems

In this paper, we consider homogeneous Dirichlet problems for a class of semilinear elliptic equations involving the Laplacian with power-type nonlinearities in bounded convex domains of Rn, and find some superharmonic functions which can be used to give convexity estimates for the solutions directly.

A class of semilinear elliptic equations on groups of polynomial growth

In this paper, we study the semilinear elliptic equation−Δu+a(x)|u|p−2u−b(x)|u|q−2u=0 on a Cayley graph of a discrete group of polynomial growth with the homogeneous dimension N≥1, where 2≤p<q<+∞. We first prove the existence of positive solutions to the above equation with constant coefficients a¯,b¯. Then we establish a decomposition of Palais-Smale sequences for the functional with variable coefficients a(x),b(x), which tend to the constants a¯,b¯ at infinity.

A Constant Sign Theorem for a Class of Semi-linear Elliptic Equations

Many problems arising from physics and analytic and geometric properties of solutions to partial differential equations have strong connections with convexity. In the present manuscript, we use the level curve curvature formula to derive a differential inequality and prove a invariant sign property for a class of nonlinear elliptic problems.

Gradient estimates for bounded solutions of semilinear elliptic equations and the Allen-Cahn equation on manifolds

This paper aims at deriving apriori bounds on the gradient of positve solutions to a class of semilinear elliptic equation, with applications focusing on establishing a liouville type property for the bounded solutions of the Allen-Cahn equation on a complete noncompact Riemannian manifold with nonnegative Ricci curvature.

Infinitely many positive solutions for a class of semilinear elliptic equations

&lt;p style='text-indent:20px;'&gt;For &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ N\geq 3 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ 1&amp;lt;p&amp;lt;\frac{N+2}{N-2} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, we consider the following semilinear elliptic equation&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt;&lt;label/&gt;&lt;tex-math id="FE1"&gt; \begin{document}$ \begin{equation*}\label{0-1} \left\{ \begin{aligned} -\Delta u&amp;amp; = V(|x|)(|u-1|^p-1)~~\mathrm{in}~\mathbb{R}^N, \\ u&amp;amp;&amp;gt;0~~\mathrm{in}~\mathbb{R}^N, \quad u\in H^1(\mathbb{R}^N), \end{aligned} \right. ~~~{(1)} \end{equation*}$ \end{document}&lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;where &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ V(r) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is a positive bounded function satisfying&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE2"&gt; \begin{document}$ V(r) = 1+\frac{a_1}{r^\alpha}+\frac{a_2}{r^{\alpha+1}}+O\left(\frac{1}{r^{\alpha+1+\theta}}\right)\quad\mathrm{as}\ r\to+\infty. $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;Here &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ a_{1}, \theta&amp;gt;0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ a_{2}\in \mathbb{R} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ \alpha&amp;gt;2(\min\left\{1, (p-1)\right\})^{-1} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; are some constants. By the finite dimensional Lyapunov-Schmidt reduction method, we show that &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ (1) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; has infinitely many non-radial positive solutions.&lt;/p&gt;

Sign-changing solutions for semilinear elliptic equation with variable exponent

This paper is devoted to study a class of semilinear elliptic equation with variable exponent. By means of perturbation technique, the method of invariant sets for the descending flow and necessary estimates, the existence of infinitely many sign-changing solutions to this problem is obtained.

On strong solvability of the Dirichlet problem for a class of semilinear elliptic equations with discontinuous coefficients

On strong solvability of the Dirichlet problem for a class of semilinear elliptic equations with discontinuous coefficients

Positive radial symmetric solutions for a class of elliptic problems with critical exponent and -1 growth

Abstract In this paper, we consider a class of semilinear elliptic equation with critical exponent and -1 growth. By using the critical point theory for nonsmooth functionals, two positive solutions are obtained. Moreover, the symmetry and monotonicity properties of the solutions are proved by the moving plane method. Our results improve the corresponding results in the literature.

On the existence of positive solutions to a certain class of semilinear elliptic equations

In this paper, we study the following semilinear elliptic equation $$\begin{aligned} \Delta u = \varphi \left( V(x) u - f(x,u(x)) \right) \quad \text {in} \,\, \mathbf {R}^N, \quad u \in H^1( \mathbf {R}^N ) \end{aligned}$$ where $$N \ge 1$$ and $$\varphi (s)$$ , V(x), f(x, s) are given functions. Under some conditions on $$\varphi (s), V(x), f(x,s)$$ , we show the existence of positive solution. In particular, we extend the result of Felmer and Ikoma (J Funct Anal 275(8):2162–2196, 2018). In Felmer and Ikoma (J Funct Anal 275(8):2162–2196, 2018), the existence of positive solution was proved by topological degree theoretic argument. In this paper, we employ the variational method.

On a weighted trace embedding and applications to critical boundary problems†

AbstractWe prove a weighted Sobolev trace embedding in the upper half‐space and give its best constant. This embedding can be employed to study a number of critical boundary problems. In this direction, we obtain existence and nonexistence results for a class of semilinear elliptic equations with nonlinear boundary conditions involving critical growth. These equations are closely related to the study of self‐similar solutions for nonlinear reaction‐diffusion equations.

Infinitely many sign-changing solutions for a semilinear elliptic equation with variable exponent

<abstract> This paper is devoted to study a class of semilinear elliptic equations with variable exponent. By means of perturbation technique, variational methods and a priori estimation, the existence of infinitely many sign-changing solutions to this class of problem is obtained. </abstract>

Positive solutions for a class of elliptic equations

This paper is concerned with a class of semilinear elliptic equations with a potential function−Δu=λ|x|−αu−|x|σupinΩ∖{0}, where λ,σ∈R, α>0, p>1, and Ω⊂RN(N≥3) is a bounded smooth domain with 0∈Ω. We establish the existence, nonexistence and asymptotic behavior of positive solutions when the potential function |x|−α has strong singularity at the origin. When the potential function |x|−α has weak singularity at the origin, we obtain some qualitative properties of positive solutions.

Existence and multiplicity results for a class of semilinear elliptic equations

We study the existence and multiplicity of nonnegative solutions, as well as the behavior of corresponding parameter-dependent branches, to the equation −Δu=(1−u)um−λun in a bounded domain Ω⊂RN endowed with the zero Dirichlet boundary data, where 0<m≤1 and n>0. When λ>0, the obtained solutions can be seen as steady states of the corresponding reaction–diffusion equation describing a model of isothermal autocatalytic chemical reaction with termination. In addition to the main new results, we formulate a few relevant conjectures.

Positive Solutions for Elliptic Problems with the Nonlinearity Containing Singularity and Hardy-Sobolev Exponents

In this paper, we study multiplicity of positive solutions for a class of semilinear elliptic equations with the nonlinearity containing singularity and Hardy-Sobolev exponents. Using variational methods, we establish the existence and multiplicity of positive solutions for the problem.

A remark on partial data inverse problems for semilinear elliptic equations

We show that the knowledge of the Dirichlet-to-Neumann map on an arbitrary open portion of the boundary of a domain in R n \mathbb {R}^n , n ≥ 2 n\ge 2 , for a class of semilinear elliptic equations uniquely determines the nonlinearity.

Nonexistence of positive supersolutions to a class of semilinear elliptic equations and systems in an exterior domain

In this paper, we consider the following semilinear elliptic equation:where $\Omega$ is an exterior domain in $\mathbb{R}^N$ with $N\ge~3$, $h:~\Omega\times~\mathbb{R}^+\to~\mathbb{R}$ is a measurable function, andderive optimal nonexistence results of positive supersolutions. Our argument isbased on a nonexistence result of positive supersolutions of a linear ellipticproblem with Hardy potential. We also establish sharp nonexistence results of positive supersolutionsto an elliptic system.

A note on the uniqueness and the non-degeneracy of positive radial solutions for semilinear elliptic problems and its application

In this paper, we are concerned with the uniqueness and the non-degeneracy of positive radial solutions for a class of semilinear elliptic equations. Using detailed ODE analysis, we extend previous results to cases where nonlinear terms may have sublinear growth. As an application, we obtain the uniqueness and the non-degeneracy of ground states for modified Schrödinger equations.

The Unique Continuation Property of Sublinear Equations

We derive the unique continuation property of a class of semilinear elliptic equations with non-Lipschitz nonlinearities. The simplest type of equations to which our results apply is given as $-\Delta u = |u|^{\sigma-1} u$ in a domain $\Omega \subset \mathbb{R}^{N}$ with $0 \le \sigma <1$. Despite the sublinear character of the nonlinear term, we prove that if a solution vanishes in an open subset of $\Omega$, then it vanishes necessarily in the whole $\Omega$. We then extend the result to equations with variable coefficients operators and inhomogeneous right-hand side.

A discontinuous interpolated finite volume approximation of semilinear elliptic optimal control problems

In this article, we describe a discontinuous finite volume method with interpolated coefficients for the numerical approximation of the distributed optimal control problem governed by a class of semilinear elliptic equations with control constraints. The proposed distributed control problem involves three unknown variable: control, state and costate. For the approximation of control, we have adopted three different methodologies: variational discretization, piecewise constant and piecewise linear discretization, while the approximation of state and costate variables is based on discontinuous piecewise linear polynomials. As the resulted scheme is non‐symmetric, optimize‐then‐discretize approach is used to approximate the control problem. Optimal a priori error estimates in suitable natural norms for state, costate and control variables are derived. Moreover, numerical experiments are presented to support the derived theoretical results. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2090–2113, 2017

Global asymptotic behavior of large solutions for a class of semilinear elliptic problems

By using Karamata regular variation theory and upper and lower solution method, we investigate the existence and the global asymptotic behavior of large solutions to a class of semilinear elliptic equations with nonlinear convection terms. In our study, the weight and nonlinearity are controlled by some regularly varying functions or rapid functions, which is very different from the conditions of previous contexts. Our results largely extend the previous works, and prove that the nonlinear convection terms do not affect the global asymptotic behavior of classical solutions when the index of the convection terms change in a certain range.