Solutions Of Semilinear Elliptic Equations Research Articles

Complete study of the existence and uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary

The purpose of this paper is to study the weak solutions of the fractional elliptic problem (Section.Display) where , or , with is the fractional Laplacian defined in the principle value sense, is a bounded open set in with , is a bounded Radon measure supported in and is defined in the distribution sense, i.e. here denotes the unit inward normal vector at . In this paper, we prove that (0.1) with admits a unique weak solution when g is a continuous nondecreasing function satisfying Our interest then is to analyse the properties of weak solution when with , including the asymptotic behaviour near and the limit of weak solutions as . Furthermore, we show the optimality of the critical value in a certain sense, by proving the non-existence of weak solutions when . The final part of this article is devoted to the study of existence for positive weak solutions to (0.1) when and is a bounded nonnegative Radon measure supported in . We employ the Schauder’s fixed point theorem to obtain positive solution under the hypothesis that g is a continuous function satisfying -pagination

Mild solutions of semilinear elliptic equations in Hilbert spaces

This paper extends the theory of regular solutions (C1 in a suitable sense) for a class of semilinear elliptic equations in Hilbert spaces. The notion of regularity is based on the concept of G-derivative, which is introduced and discussed. A result of existence and uniqueness of solutions is stated and proved under the assumption that the transition semigroup associated to the linear part of the equation has a smoothing property, that is, it maps continuous functions into G-differentiable ones. The validity of this smoothing assumption is fully discussed for the case of the Ornstein–Uhlenbeck transition semigroup and for the case of invertible diffusion coefficient covering cases not previously addressed by the literature. It is shown that the results apply to Hamilton–Jacobi–Bellman (HJB) equations associated to infinite horizon optimal stochastic control problems in infinite dimension and that, in particular, they cover examples of optimal boundary control of the heat equation that were not treatable with the approaches developed in the literature up to now.

On the behavior of positive solutions of semilinear elliptic equations in asymptotically cylindrical domains

The goal of this note is to study the asymptotic behavior of positive solutions for a class of semilinear elliptic equations which can be realized as minimizers of their energy functionals. This class includes the Fisher-KPP and Allen–Cahn nonlinearities. We consider the asymptotic behavior in domains becoming infinite in some directions. We are in particular able to establish an exponential rate of convergence for this kind of problems.

Propagating terraces in a proof of the Gibbons conjecture and related results

The Gibbons conjecture stating the one-dimensional symmetry of certain solutions of semilinear elliptic equations has been proved by several authors. We show how attractivity properties of minimal propagating terraces of one-dimensional parabolic problems can be used in a proof of a version of this result and related statements.

Multiplicity and concentration of positive solutions for semilinear elliptic equations with steep potential

In this paper, we study the existence, multiplicity and concentration of positive solutions for the following indefinite semilinear elliptic equations involving concave-convex nonlinearities: \begin{eqnarray} \left\{\begin{array}{l} -\Delta u+V_{\lambda }\left( x\right) u=f\left( x\right) \left\vert u\right\vert ^{q-2}u+g\left( x\right) \left\vert u\right\vert ^{p-2}u & \text{in }\mathbb{R}^{N}, \\ u\geq 0, & \text{in }\mathbb{R}^{N}, \end{array}\right. \end{eqnarray} where$ 1 < q < 2 < p < 2^{\ast} (2^{\ast } = \frac{2N}{N-2}$ for $N \geq 3) $ the potential $V_{\lambda }(x)=\lambda V^{+}(x)-V^{-}(x)$ with $ V^{\pm }=\max \left\{ \pm V,0\right\} $ and the parameter $\lambda >0.$ We assume that the functions $f,g$ and $V$ satisfy suitable conditions with the potential $V$ and the weight function $g$ without the assumptions of infinite limits.

Precise exponential decay for solutions of semilinear elliptic equations and its effect on the structure of the solution set for a real analytic nonlinearity

We are concerned with the properties of weak solutions of the stationary Schrödinger equation $-\Delta u + Vu = f(u)$, $u\in H^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$, where $V$ is Hölder continuous and $\inf V>0$. Assuming $f$ to be continuous and bounded near $0$ by a power function with exponent larger than $1,$ we provide precise decay estimates at infinity for solutions in terms of Green's function of the Schrödinger operator. In some cases this improves known theorems on the decay of solutions. If $f$ is also real analytic on $(0,\infty)$, we obtain that the set of positive solutions is locally path connected. For a periodic potential $V$ this implies that the standard variational functional has discrete critical values in the low energy range and that a compact isolated set of positive solutions exists, under additional assumptions.

Radial symmetry of $n$-mode positive solutions for semilinear elliptic equations in a disc and its application to the Hénon equation

Let $f \in C((0,1)\times (0,\infty),\mathbb{R})$ and $n \in \mathbb{N}$ with $n \geq 2$ such that for each $u \in (0,\infty)$, $r\mapsto r^{2-2n}f(r,u)\colon (0,1)\rightarrow \mathbb{R}$ is nonincreasing and let $D=\{x=(x_1,x_2)\in\mathbb{R}^2: |x|< 1\}$. We show that each positive solution of $$ \Delta u + f(|x|,u) =0 \quad\text{in $D$,} \qquad u=0 \quad\text{on $\partial D$} $$ which satisfies $u(r,\theta)= u(r,\theta+2\pi/n)$ by the polar coordinates is radially symmetric and $u_r(|x|)< 0$ for each $r=|x| \in (0,1)$. We apply our result to the Hénon equation.

Existence of three nontrivial solutions for semilinear elliptic equations on ℝ N

We establish the existence theorem of three nontrivial solutions for a class of semilinear elliptic equation on ℝN by using variational theorems of mixed type due to Marino and Saccon and linking theorem.

Nonnegative solutions of semilinear elliptic equations in half-spaces

We consider the semilinear elliptic problem(0.1){−Δu=f(u)in R+Nu=0on ∂R+N where the nonlinearity f is assumed to be C1 and globally Lipschitz with f(0)<0, and R+N={x∈RN:xN>0} stands for the half-space. Denoting by u0 the unique solution of the one-dimensional problem −u″=f(u) with u(0)=u′(0)=0, we show that nonnegative solutions u of (0.1) which verify u(x)≥u0(xN) in R+N either are positive and monotone in the xN direction or coincide with u0. As a particular instance, when f(t)=t−1, we prove that the unique nonnegative (not necessarily bounded) solution of (0.1) is u(x)=1−cos⁡xN. This solves in a strengthened form a conjecture posed by Berestycki, Caffarelli and Nirenberg in 1997.

New a Priori Estimates for Semistable Solutions of Semilinear Elliptic Equations

We consider the semilinear elliptic equation -L u = f(u) in a general smooth bounded domain ${\Omega } \subset R^{n}$ with zero Dirichlet boundary condition, where L is a uniformly elliptic operator and f is a C 2 positive, nondecreasing and convex function in $[0,\infty )$ such that $\frac {f(t)}{t}\rightarrow \infty $ as $t\rightarrow \infty $ . We prove that if u is a positive semistable solution then for every 0 ≤ β < 1 we have $$\left|\left|f(u){{\int}_{0}^{u}}f(t)f^{\prime\prime}(t)~e^{2{\beta{\int}_{0}^{t}}\sqrt{\frac{f^{\prime\prime}(s)}{f(s)}}ds}~dt \right|\right|_{L^{1}({\Omega})}\leq C_{\beta}<\infty, $$ where C β is a constant independent of u. As we shall see, a large number of results in the literature concerning a priori bounds are immediate consequences of this estimate. In particular, among other results, we establish a priori $L^{\infty }$ bound in dimensions n ≤ 9, under the extra assumption that $\limsup _{t\rightarrow \infty }\frac {f(t)f^{\prime \prime }(t)}{f^{\prime }(t)^{2}}< \frac {2}{9-2\sqrt {14}}\cong 1.318$ . Also, we establish a priori $L^{\infty }$ bound when n ≤ 5 under the very weak assumption that, for some e > 0, $\liminf _{t\rightarrow \infty }\frac {(tf(t))^{2-\epsilon }}{f^{\prime }(t)}>0$ or $\liminf _{t\rightarrow \infty }\frac {t^{2}f(t)f^{\prime \prime }(t)}{f^{\prime }(t)^{\frac {3}{2}+\epsilon }}>0$ .

Boundary behavior of large solutions for semilinear elliptic equations with weights

Boundary behavior of large solutions for semilinear elliptic equations with weights

Singular solutions of semilinear elliptic equations with fractional Laplacian in entire space

We consider the existence of positive singular solutions of the problem [Formula: see text] where [Formula: see text] and [Formula: see text]. We obtain the existence of positive singular solutions [Formula: see text] of this problem in space dimensions [Formula: see text] for the case [Formula: see text] under the condition [Formula: see text]. By the well-known Caffarelli–Silvestre extension result, the trace of [Formula: see text] is a positive singular solution for the following equation with a supercritical exponent [Formula: see text] where [Formula: see text] denotes the fractional Laplacian.

On the Stability of Solutions of Semilinear Elliptic Equations with Robin Boundary Conditions on Riemannian Manifolds

We investigate existence and nonexistence of stationary stable nonconstant solutions, i.e., patterns, of semilinear parabolic problems in bounded domains on Riemannian manifolds, satisfying Robin boundary conditions. These problems arise in several models in applications, in particular in mathematical biology. We point out the significance both of the nonlinearity and of geometric objects such as the Ricci curvature of the manifold, the second fundamental form of the boundary of the domain, and its mean curvature. Special attention is given to surfaces of revolution and to spherically symmetric manifolds, where we prove refined results.

Positive solutions for semilinear elliptic equations with critical weighted Hardy-Sobolev exponents

In this paper, we study a class of semilinear elliptic equations with critical weighted Hardy-Sobolev exponents and superlinear nonlinearity. By means of the variational methods and some analysis techniques, positive solution is obtained.

Moderate solutions of semilinear elliptic equations with Hardy potential

Let Ω be a bounded smooth domain in RN. We study positive solutions of equation (E) −Lμu+uq=0 in Ω where Lμ=Δ+μδ2, 0<μ, q>1 and δ(x)=dist(x,∂Ω). A positive solution of (E) is moderate if it is dominated by an Lμ-harmonic function. If μ<CH(Ω) (the Hardy constant for Ω) every positive Lμ-harmonic function can be represented in terms of a finite measure on ∂Ω via the Martin representation theorem. However the classical measure boundary trace of any such solution is zero. We introduce a notion of normalized boundary trace by which we obtain a complete classification of the positive moderate solutions of (E) in the subcritical case, 1<q<qμ,c. (The critical value depends only on N and μ.) For q≥qμ,c there exists no moderate solution with an isolated singularity on the boundary. The normalized boundary trace and associated boundary value problems are also discussed in detail for the linear operator Lμ. These results form the basis for the study of the nonlinear problem.

On the Schrödinger–Poisson system with a general indefinite nonlinearity

We study the existence and multiplicity of positive solutions of a class of Schrödinger–Poisson system: {−Δu+u+l(x)ϕu=k(x)g(u)+μh(x)uinR3,−Δϕ=l(x)u2inR3, where k∈C(R3) changes sign in R3, lim∣x∣→∞k(x)=k∞<0, and the nonlinearity g behaves like a power at zero and at infinity. We mainly prove the existence of at least two positive solutions in the case that μ>μ1 and near μ1, where μ1 is the first eigenvalue of −Δ+id in H1(R3) with weight function h, whose corresponding positive eigenfunction is denoted by e1. An interesting phenomenon here is that we do not need the condition ∫R3k(x)e1pdx<0, which has been shown to be a sufficient condition to the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearity (see e.g. Costa and Tehrani, 2001).

On solutions of semilinear elliptic equation with linear growth nonlinearity in $\mathbb{R}^{N}$

We study nontrivial solutions for a class of semilinear elliptic equation which could be resonant at infinity. We establish the existence of solutions for the equation by considering the modified non-resonant problem associated with the original equation through Morse theory. Moreover, only linear growth assumption is imposed on the nonlinearity and condition on the potential is weaker than the coercive assumption.

A priori estimates for semistable solutions of semilinear elliptic equations

We consider positive semistable solutions $u$ of $Lu+f(u)=0$ with zero Dirichlet boundary condition, where $L$ is a uniformly elliptic operator and $f\in C^2$ is a positive, nondecreasing, and convex nonlinearity which is superlinear at infinity. Under these assumptions, the boundedness of all semistable solutions is expected up to dimension $n\leq 9$, but only established for $n\leq 4$. In this paper we prove the $L^\infty$ bound up to dimension $n=5$ under the following further assumption on $f$: for every $\varepsilon>0$, there exist $T=T(\varepsilon)$ and $C=C(\varepsilon)$ such that $f'(t)\leq Cf(t)^{1+\varepsilon}$ for all $t>T$. This bound will follow from a $L^p$-estimate for $f'(u)$ for every $p<3$ (and for all $n\geq 2$). Under a similar but more restrictive assumption on $f$, we also prove the $L^\infty$ estimate when $n=6$. We remark that our results do not assume any lower bound on $f'$.

Singular set of solutions of semilinear elliptic equations

We study the behavior of the singular set {u=∇u=0} for solutions u to the semilinear elliptic equation Δu=f(x,u,∇u),x∈Ω, where Ω is an open set in Rn and |f(x,u,∇u)|≤A|u|p+B|∇u|q, where A,B≥0 and p,q∈(0,∞). We show that in dimension n=2 the singular set is a discrete set, and if min{p,q}<1 then a solution u satisfies an optimal growth βp,q near every non-isolated point of the singular set. Also the same results are true in dimension n≥3 under an (n−1)-dimensional density condition.

Boundary singularities of solutions of semilinear elliptic equations with critical Hardy potentials

We study the boundary behavior of positive functions u satisfying (E) −Δu−κd2(x)u+g(u)=0 in a bounded domain Ω of RN, where 0<κ≤14, g is a continuous nondecreasing function and d(.) is the distance function to ∂Ω. We first construct the Martin kernel associated to the linear operator Lκ=−Δ−κd2(x) and give a general condition for solving equation (E) with any Radon measure μ for boundary data. When g(u)=|u|q−1u we show the existence of a critical exponent qc=qc(N,κ)>1 with the following properties: when 0<q<qc any measure is eligible for solving (E) with μ for boundary data; if q≥qc, a necessary and sufficient condition is expressed in terms of the absolute continuity of μ with respect to some Besov capacity. The same capacity characterizes the removable compact boundary sets. At end any positive solution (F) −Δu−κd2(x)u+|u|q−1u=0 with q>1 admits a boundary trace which is a positive outer regular Borel measure. When 1<q<qc we prove that to any positive outer regular Borel measure we can associate a positive solutions of (F) with this boundary trace.