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The paper deals with existence, localization and multiplicity of radial positive solutions in the annulus or the ball, for the Neumann problem involving a general ϕ -Laplace operator. Our results apply in particular to the classical Laplacian and to the mean curvature operators in the Euclidean and Minkowski spaces. Numerical experiments with the MATLAB object-oriented package Chebfun are performed to obtain numerical solutions for some concrete equations. [ABSTRACT FROM AUTHOR]

This article deals with the study of the following Kirchhoff–Choquard problem: M ∫ R N | ∇ u | p (- Δ p) u + V (x) | u | p - 2 u = ∫ R N F (u) (y) | x - y | μ d y f (u) , in R N , u > 0 , in R N , where M models Kirchhoff-type nonlinear term of the form M (t) = a + b t θ - 1 , where a , b > 0 are given constants; 1 < p < N , Δ p = div (| ∇ u | p - 2 ∇ u) is the p-Laplacian operator; potential V ∈ C 2 (R N) ; f is monotonic function with suitable growth conditions. We obtain the existence of a positive high energy solution for θ ∈ 1 , 2 N - μ N - p via the Pohožaev manifold and linking theorem. Apart from this, we also studied the radial symmetry of solutions of the associated limit problem. [ABSTRACT FROM AUTHOR]

This paper concerns with the existence of radial solutions of the biharmonic elliptic equation ▵ 2 u = f (| x | , u , | ∇ u | , ▵ u) in an annular domain Ω = { x ∈ R N : r 1 < | x | < r 2 } ( N ≥ 2 ) with the boundary conditions u | ∂ Ω = 0 and ▵ u | ∂ Ω = 0 , where f : [ r 1 , r 2 ] × R × R + × R → R is continuous. Under certain inequality conditions on f involving the principal eigenvalue λ 1 of the Laplace operator − ▵ with boundary condition u | ∂ Ω = 0 , an existence result and a uniqueness result are obtained. The inequality conditions allow for f (r , ξ , ζ , η) to be a superlinear growth on ξ , ζ , η as | (ξ , ζ , η) | → ∞ . Our discussion is based on the Leray–Schauder fixed point theorem, spectral theory of linear operators and technique of prior estimates. [ABSTRACT FROM AUTHOR]

Abstract This paper studies the existence of radial solutions of the boundary value problem of p-Laplace equation with gradient term { − Δ p u = K ( | x | ) f ( | x | , u , | ∇ u | ) , x ∈ Ω , ∂ u ∂ n = 0 , x ∈ ∂ Ω , lim | x | → ∞ u ( x ) = 0 , $$ \textstyle\begin{cases} -\Delta_{p} u= K( \vert x \vert ) f( \vert x \vert , u, \vert \nabla u \vert ) ,\quad x\in\Omega , \\ \frac{\partial u}{\partial n}=0 ,\quad x\in\partial\Omega, \\ \lim_{ \vert x \vert \to\infty}u(x)=0 , \end{cases} $$ where Ω = { x ∈ R N : | x | > r 0 } $\Omega=\{x\in\mathbb{R}^{N}: |x|>r_{0}\}$ , N ≥ 3 $N\ge3$ , 1 < p ≤ 2 $1< p\le2$ , K : [ r 0 , ∞ ) → R + $K: [r_{0}, \infty)\to\mathbb{R}^{+}$ , and f : [ r 0 , ∞ ) × R × R + → R $f: [r_{0}, \infty)\times\mathbb{R}\times\mathbb{R}^{+}\to \mathbb{R}$ are continuous. Under certain inequality conditions of f, the existence results of radial solutions are obtained.

In this paper, we study the well-posedness, ill-posedness and uniqueness of the stationary 3-D radial solution to the bipolar isothermal hydrodynamic model for semiconductors. The density of electron is imposed with sonic boundary and interiorly subsonic case and the density of hole is fully subsonic case. It is difficult to estimate the upper and lower bounds of the holes due to the coupling of electrons and holes and the degeneracy of electrons at the boundary. Thus, we use the topological degree method to prove the well-posedness of solution. We prove the ill-posedness of subsonic solution under some conditions by direct mathematical analysis and contradiction method. The ill-posedness property shows significant difference to the unipolar model. Another highlight of this paper is the application of specific energy method to obtain the uniqueness of solution in two cases. One case is the relaxation time τ = ∞ , namely, the pure Euler-Poisson case; the other case is j τ ≪ 1 , which means that, when the current flow is sufficiently small and the relaxation time is sufficiently large both can satisfy. [ABSTRACT FROM AUTHOR]

This paper studies the existence of radial solutions of the boundary value problem of p-Laplace equation with gradient term { − Δ p u = K (| x |) f (| x | , u , | ∇ u |) , x ∈ Ω , ∂ u ∂ n = 0 , x ∈ ∂ Ω , lim | x | → ∞ u (x) = 0 , where Ω = { x ∈ R N : | x | > r 0 } , N ≥ 3 , 1 < p ≤ 2 , K : [ r 0 , ∞) → R + , and f : [ r 0 , ∞) × R × R + → R are continuous. Under certain inequality conditions of f, the existence results of radial solutions are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the existence of radial solutions for a class of biharmonic equation with a convection term, involving two real parameters A and p. We mainly use a combination of the fixed point index theory and the Banach contraction theorem to prove that there are A0 > 0 and p0 > 0 such the equation admits at least one radial solution for all (A, p) e [-Ao, x [0, p0]. [ABSTRACT FROM AUTHOR]

We define rigorously a solution to the fourth-order total variation flow equation in $ \mathbb{R}^n $. If $ n\geq3 $, it can be understood as a gradient flow of the total variation energy in $ D^{-1} $, the dual space of $ D^1_0 $, which is the completion of the space of compactly supported smooth functions in the Dirichlet norm. However, in the low dimensional case $ n\leq2 $, the space $ D^{-1} $ does not contain characteristic functions of sets of positive measure, so we extend the notion of solution to a larger space. We characterize the solution in terms of what is called the Cahn-Hoffman vector field, based on a duality argument. This argument relies on an approximation lemma which itself is interesting. We introduce a notion of calibrability of a set in our fourth-order setting. This notion is related to whether a characteristic function preserves its form throughout the evolution. It turns out that all balls are calibrable. However, unlike in the second-order total variation flow, the outside of a ball is calibrable if and only if $ n\neq2 $. If $ n\neq2 $, all annuli are calibrable, while in the case $ n = 2 $, if an annulus is too thick, it is not calibrable. We compute explicitly the solution emanating from the characteristic function of a ball. We also provide a description of the solution emanating from any piecewise constant, radially symmetric datum in terms of a system of ODEs.

This paper concerns with the existence of radial solutions of the biharmonic elliptic equation ▵2u=f(|x|,u,|∇u|,▵u) in an annular domain Ω={x∈RN:r1<|x|

This article discusses the existence and uniqueness of radial solution for the elliptic equation system $ \left \{ \begin{array}{ll} -\triangle u = f(|x|, \ u, \ v, \ |\nabla u|), \; \; x\in \Omega, \\[10pt] -\triangle v = g(|x|, \ u, \ v, \ |\nabla v|), \; \; x\in \Omega, \\[10pt] u|_{\partial \Omega} = 0, \; v|_{\partial \Omega} = 0, \end{array} \right. $ where $ \Omega = \{x\in \mathbb{R}^{N}:\; r_1 < |x| < r_2\}, \; N\ge 3, \; f, \; g:[r_1, \; r_2]\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}^+\to \mathbb{R} $ are continuous. Due to the appearance of the gradient term in the nonlinearity, the equation system has no variational structure and the variational method cannot be applied to it directly. We will give the correlation conditions of $ f $ and $ g $, that is, $ f $ and $ g $ are superlinear or sublinear, and prove the existence and uniqueness of radial solutions by using Leray-Schauder fixed point theorem.

We investigate an n -dimensional free boundary problem modeling tumor growth in radial form with angiogenesis process. This process is reflected in the Robin boundary condition where time-dependent absorption rate β (t) is considered. The system consists of a reaction-diffusion equation for the nutrient u (r , t) with nonlinear consumption and an integro-differential equation for the tumor radius R (t) with nonlinear cell proliferation. It is shown that the large-time behavior of the tumor is greatly tied to the properties of β (t). We prove that for any sufficiently small c > 0 , the tumor radius R (t) will remain bounded or shrink to zero when β (t) is uniformly bounded or tends to zero, respectively; and if β (t) → β ⁎ , the solution (u , R) will converge to the unique steady state (u ⁎ , R ⁎) associating with β ⁎. We also give examples of R (t) blowing up as t → ∞ even if β (t) → 0 when c is not small. [ABSTRACT FROM AUTHOR]

We define rigorously a solution to the fourth-order total variation flow equation in R n . If n ≥ 3 , it can be understood as a gradient flow of the total variation energy in D − 1 , the dual space of D 0 1 , which is the completion of the space of compactly supported smooth functions in the Dirichlet norm. However, in the low dimensional case n ≤ 2 , the space D − 1 does not contain characteristic functions of sets of positive measure, so we extend the notion of solution to a larger space. We characterize the solution in terms of what is called the Cahn-Hoffman vector field, based on a duality argument. This argument relies on an approximation lemma which itself is interesting. We introduce a notion of calibrability of a set in our fourth-order setting. This notion is related to whether a characteristic function preserves its form throughout the evolution. It turns out that all balls are calibrable. However, unlike in the second-order total variation flow, the outside of a ball is calibrable if and only if n ≠ 2. If n ≠ 2 , all annuli are calibrable, while in the case n = 2 , if an annulus is too thick, it is not calibrable. We compute explicitly the solution emanating from the characteristic function of a ball. We also provide a description of the solution emanating from any piecewise constant, radially symmetric datum in terms of a system of ODEs. We define rigorously a solution to the fourth-order total variation flow equation in . If , it can be understood as a gradient flow of the total variation energy in , the dual space of , which is the completion of the space of compactly supported smooth functions in the Dirichlet norm. However, in the low dimensional case , the space does not contain characteristic functions of sets of positive measure, so we extend the notion of solution to a larger space. We characterize the solution in terms of what is called the Cahn-Hoffman vector field, based on a duality argument. This argument relies on an approximation lemma which itself is interesting. We introduce a notion of calibrability of a set in our fourth-order setting. This notion is related to whether a characteristic function preserves its form throughout the evolution. It turns out that all balls are calibrable. However, unlike in the second-order total variation flow, the outside of a ball is calibrable if and only if . If , all annuli are calibrable, while in the case , if an annulus is too thick, it is not calibrable. We compute explicitly the solution emanating from the characteristic function of a ball. We also provide a description of the solution emanating from any piecewise constant, radially symmetric datum in terms of a system of ODEs. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following two-dimensional chemotaxis system of attraction–repulsion with indirect signal production 𝜕tu=Δu−∇·χ1u∇v1+∇·(χ2u∇v2),x∈R2,t>0,0=Δvj−λjvj+w,x∈R2,t>0,(j=1,2),𝜕tw+δw=u,x∈R2,t>0,u(0,x)=u0(x),w(0,x)=w0(x),x∈R2, where the parameters χi≥0, λi>0(i=1,2) and non-negative initial data (u0(x),w0(x))∈L1(R2)∩L∞(R2). We prove the global bounded solution exists when the attraction is more dominant than the repulsion in the case of χ1≥χ2. At the same time, we propose that when the radial solution satisfies χ1−χ2≤2πδ∥u0∥L1(R2)+∥w0∥L1(R2), the global solution is bounded. During the proof process, we found that adding indirect signals can constrict the blow-up of the global solution.

The paper deals with the existence and localization of positive radial solutions for stationary partial differential equations involving a general $\phi $-Laplace operator in the annulus. Three sets of boundary conditions are considered: Dirichlet–Neumann, Neumann–Dirichlet and Dirichlet–Dirichlet. The results are based on the homotopy version of Krasnosel'skii's fixed point theorem and Harnack type inequalities, first established for each one of the boundary conditions. As a consequence, the problem of multiple solutions is solved in a natural way. Numerical experiments confirming the theory, one for each of the three sets of boundary conditions, are performed by using the MATLAB object-oriented package Chebfun.

Copyright of Journal of Zhejiang University (Science Edition) is the property of Journal of Zhejiang University (Science Edition) Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we first deduce the following Sobolev inequality with logarithmic term: (0.1) sup ⁡ { ∫ B | u | 2 ⁎ | ln ⁡ (τ + | u |) | | x | β d x : u ∈ H 0 , r a d 1 (B) , ‖ ∇ u ‖ L 2 (B) = 1 } < ∞ , where β > 0 , τ ≥ 0 are constants, B is the unit ball in R N , N ≥ 3 , and 2 ⁎ = 2 N / (N − 2) is the critical Sobolev exponent. Then we show that the supremum in (0.1) is attained when 0 < β < min ⁡ { N / 2 , N − 2 } and 1 ≤ τ < ∞. The inequality (0.1) can be used to prove the existence of positive solution for the following supercritical problem: (0.2) { − Δ u = u 2 ⁎ − 1 (ln ⁡ (τ + u)) | x | β + g (| x | , u) , u > 0 in B , u = 0 on ∂ B , where g (r , u) ∈ C ([ 0 , 1) × R) is a subcritical perturbation. As a consequence, we can deduce the existence of positive solution for the supercritical problem with non-power nonlinearity: (0.3) { − Δ u = u 2 ⁎ − 1 (ln ⁡ (τ + u)) | x | β , u > 0 in B , u = 0 on ∂ B. This is somewhat surprising, because the problem (0.3) has no nontrivial solution by Pohozaev's identity if the variable exponent | x | β is replaced by any non-negative constant. [ABSTRACT FROM AUTHOR]

We study the radial solutions of the p‐Laplacian Neumann problem with gradient dependence of the type −Δpu=f(|x|,u,|∇u|)inΩ,∂u∂n=0on∂Ω,$$\begin{equation*} {\hspace*{8pc}\left\lbrace \def\eqcellsep{&}\begin{array}{l}-\Delta _{p}u=f(|x|,u,|\nabla u|)\quad \textrm {in} \nobreakspace \Omega ,\\[3pt] \displaystyle \frac{\partial u}{\partial {\bf n}}=0\quad \textrm {on}\nobreakspace \partial \Omega , \end{array} \right.} \end{equation*}$$where Ω⊂RN(N≥2)$\Omega \subset \mathbb {R}^N(N\ge 2)$ is either a ball or an annulus, and p>1$p>1$. By using the topological transversality method together with the barrier strip technique, we obtain existence results of radial solutions to the above problem, where the nonlinearity f does not need to satisfy any growth restriction. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with boundary value problems for local and nonlocal Laplace operator in 2D with exponential nonlinearities, the so-called Liouville type equations. They include the mean field equation and other equations arising in the statistical mechanics. Existence results into an explicit form for the Dirichlet problem in the unit disc B 1 ⊂ R 2 $B_{1} \subset {\mathbf{R}}^{2} $ and in the participation of positive parameters in the right-hand sides are proved in Theorems 2 and 3. Theorem 2 is illustrated by several examples including an application to the differential geometry. In Theorem 4 global radial solution of the Cauchy problem with constant data at ∂ B 1 $\partial B_{1} $ and under appropriate conditions is constructed. It develops logarithmic singularities for r = 0 $r = 0 $ , r = ∞ $r = \infty $ . An illustrative example to Theorem 4 in the case of two exponents is given at the end of the paper.

Using topological transversality method together with barrier strip technique and cut-off technique, we obtain new existence and uniqueness results of radial solutions to the Neumann problems involving prescribed mean curvature operator div ∇ v 1 + | ∇ v | 2 = f | x | , v , d v d r i n Ω , ∂ v ∂ n = 0 o n ∂ Ω , where Ω = { x ∈ R N : R 1 < | x | < R 2 } (0 ≤ R 1 < R 2 , N ≥ 2) , f : [ R 1 , R 2 ] × R 2 → R is continuous. Meanwhile, we demonstrate the importance of our results through an illustrative example. [ABSTRACT FROM AUTHOR]

Abstract In this paper, using variational methods, we prove the existence of at least one positive radial solution for the generalized p ( x ) $p(x)$ -Laplacian problem − Δ p ( x ) u + R ( x ) u p ( x ) − 2 u = a ( x ) | u | q ( x ) − 2 u − b ( x ) | u | r ( x ) − 2 u $$ -\Delta _{p(x)} u + R(x) u^{p(x)-2}u=a (x) \vert u \vert ^{q(x)-2} u- b(x) \vert u \vert ^{r(x)-2} u $$ with Dirichlet boundary condition in the unit ball in R N $\mathbb{R}^{N}$ (for N ≥ 3 $N \geq 3$ ), where a, b, R are radial functions.

We are concerned with the qualitative and asymptotic analysis of solutions to the nonlocal equation $$ (-\Delta)^su+V(|z|)u=Q(|z|)u^p\quad \text{in} \ \mathbb{R}^{N},$$ where $N\geq 3,\ 0\frac{N+2s}{N+2s+1}$, and $\theta_1,\ \theta_2>0,\ V_0,\ Q_0>0$. Under various hypotheses on $a_1,\, a_2,\, \alpha,\, \beta$, we establish the existence of infinitely many radial solutions. A key role in our arguments is played by the Lyapunov–Schmidt reduction method.

In this article we study the existence of radial solutions of $\Delta u + K(|x|)f(u)= 0$ on the exterior of the ball of radius R>0 centered at the origin in $\mathbb{R}^N$ with u=0 on $\partial B_{R}$, and $\lim_{|x| \to \infty} u(x)=0$ where N>2, $f(u) \sim \frac{-1}{|u|^{q-1}u} $ for u near 0 with 0

In this article we study radial solutions of $\Delta u + K(r)f(u)= 0$ on the exterior of the ball of radius R>0 centered at the origin in ${\mathbb R}^N$ where f is odd with f$ on $(\beta, \delta)$, $f\equiv 0$ for $u> \delta$, and where the function K(r) is assumed to be positive and $K(r)\to 0$ as $r \to \infty$. The primitive $F(u) = \int_0^u f(t) \, dt$ has a ``hilltop'' at $u=\delta$. With mild assumptions on f we prove that if $K(r)\sim r^{-\alpha}$ with $2< \alpha< 2(N-1)$ then there are n solutions of $\Delta u + K(r)f(u)= 0$ on the exterior of the ball of radius R such that $u\to 0$ as $r \to \infty$ if R>0 is sufficiently small. We also show there are no solutions if R>0 is sufficiently large.

The paper deals with the existence and localization of positive radial solutions for stationary partial differential equations involving a general ϕ-Laplace operator in the annulus. Three sets of boundary conditions are considered: Dirichlet-Neumann, Neumann-Dirichlet and Dirichlet-Dirichlet. The results are based on the homotopy version of Krasnosel'ski˘ı's fixed point theorem and Harnack type inequalities, first established for each one of the boundary conditions. As a consequence, the problem of multiple solutions is solved in a natural way. Numerical experiments confirming the theory, one for each of the three sets of boundary conditions, are performed by using the MATLAB object-oriented package Chebfun. [ABSTRACT FROM AUTHOR]

In this paper, we consider the weighted fourth order equation Δ (| x | − α Δ u) + λ div (| x | − α − 2 ∇ u) + μ | x | − α − 4 u = | x | β u p in R n ∖ { 0 } , Δ (| x | − α Δ u) + λ div (| x | − α − 2 ∇ u) + μ | x | − α − 4 u = | x | β u p in R n ∖ { 0 } , where n ≥ 5 n ≥ 5 , − n < α < n − 4 − n < α < n − 4 , p > 1 p > 1 and (p , α , β , n) (p , α , β , n) belongs to the critical hyperbola n + α 2 + n + β p + 1 = n − 2. n + α 2 + n + β p + 1 = n − 2. We prove the existence of radial solutions to the equation for some λ λ and μ μ. On the other hand, let v (t) : = | x | n − 4 − α 2 u (| x |) v (t) := | x | n − 4 − α 2 u (| x |) , t = − ln | x | t = − ln ⁡ | x | , then for the radial solution u u with non-removable singularity at origin, v (t) v (t) is a periodic function if α ∈ (− 2 , n − 4) α ∈ (− 2 , n − 4) and λ λ , μ μ satisfy some conditions; while for α ∈ (− n , − 2 ] α ∈ (− n , − 2 ] , there exists a radial solution with non-removable singularity and the corresponding function v (t) v (t) is not periodic. We also get some results about the best constant and symmetry breaking, which is closely related to the Caffarelli-Kohn-Nirenberg type inequality. [ABSTRACT FROM AUTHOR]

We study the existence of radial solutions of indefinite semilinear elliptic equations in the unit ball in Rn (n>=3) with Dirichlet boundary conditions, whose nonlinear term has the form lamda.m(|x|)f(u) where m(|.|) is radially symmetric, discontinuous and changes sign. This study is realized using variational tecniques and especially the Ambrosetti-Rabinowitz’s “Mountain Pass Lemma”.

In this article we prove the existence of an infinite number of radial solutions to $\Delta u+K(r)f(u)=0$ on $\mathbb{R}^{N}$ such that $\lim_{r \to \infty} u(r)=0$ with prescribed number of zeros on the exterior of the ball of radius R>0 where f is odd with f0 on $(\beta,\infty)$ with f superlinear for large $u$, and $K(r) \sim r^{-\alpha}$ with $ \alpha > 2(N-1)$.

The aim of this paper is to draw attention to a general principle for solving control problems for operator equations with the help of fixed point techniques. Three distinct applications are presented: a control problem related to the Lotka–Volterra system, the nonlinear Stokes system, and radial solutions of the Neumann problem for ϕ-Laplace equations. Only for the first application, the control is an explicit external one, while for the next two applications, the control is dependent on the models and arises from the necessity to conform to the actual modeled process or to a certain boundary condition. From the perspective of those interested in applications, the three examples of problems of such a different nature, we believe have been able to suggest the wide applicability of our method thus paving the way for new applications. From a theoretical perspective, the method leading to fixed point equations with composed operators is suitable to be related to advanced research in fixed point theory for single-valued and multi-valued operators.

In this paper, we study superlinear elliptic equations with mixed boundary value conditions in annular domains. It is assumed that the nonlinearities depend on the derivative terms. Some results about existence of solutions are established by using the Nehari manifold technique, as well as iterative technique. [ABSTRACT FROM AUTHOR]

In this paper, we study the positive solutions of the Schrödinger elliptic system { div(G(|∇y|p−2)∇y) = b1(|x|)ψ(y) + h1(|x|)φ(z), x ∈ Rn (n ≥ 3), div⁡ (G(|∇z|p−2)∇z) = b2(|x|)ψ(z) + h2(|x|)φ(y), x ∈ Rn, where G is a nonlinear operator. By using the monotone iterative technique and Arzela-Ascoli theorem, we prove that the system has the positive entire bounded radial solutions. Then, we establish the results for the existence and nonexistence of the positive entire blow-up radial solutions for the nonlinear Schrödinger elliptic system involving a nonlinear operator. Finally, three examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

We prove the existence of infinitely many sign changing radial solutions for a p-Laplacian Dirichlet problem in a ball. Our problem involves a weight function that is positive at the center of the unit ball and negative in its boundary. Standard initial value problems-phase plane analysis arguments do not apply here because solutions to the corresponding initial value problem may blow up near the boundary due to the fact that our weight function is negative at the boundary. We overcome this difficulty by connecting the solutions to a singular initial value problem with those of a regular initial value problem that vanishes at the boundary. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we derive new closed-form radial solutions for the following elliptic Liouville equation △ϕ+φ(x)eϕ=0inΩ⊂Rn,and its generalized equation △ϕ+φ(x)eϕ=f(x)inΩ⊂Rn,where △ is the Laplace operator, φ and f are smooth positive functions.

This paper deals with boundary value problems for local and nonlocal Laplace operator in 2D with exponential nonlinearities, the so-called Liouville type equations. They include the mean field equation and other equations arising in the statistical mechanics. Existence results into an explicit form for the Dirichlet problem in the unit disc B 1 ⊂ R 2 and in the participation of positive parameters in the right-hand sides are proved in Theorems 2 and 3. Theorem 2 is illustrated by several examples including an application to the differential geometry. In Theorem 4 global radial solution of the Cauchy problem with constant data at ∂ B 1 and under appropriate conditions is constructed. It develops logarithmic singularities for r = 0 , r = ∞ . An illustrative example to Theorem 4 in the case of two exponents is given at the end of the paper. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we prove the uniqueness of negative radial solution to a Dirichlet problem of (m , k) -Hessian equation in a finite ball of R n for 1 < k < n m . Our proof is based on a Pohozaev identity and the monotone separation techniques. [ABSTRACT FROM AUTHOR]

In this article we prove the existence of infinitely many radial solutions of $\Delta u + K(r)f(u)= 0$ on the exterior of the ball of radius R>0, $B_R$, centered at the origin in $\mathbb{R}^N$ with u=0 on $\partial B_R$ and $\lim_{r \to \infty} u(r)=0$ where N>2, f is odd with f0 on $(\beta, \infty)$, f is superlinear for large u, $f(u) \sim-1/(|u|^{q-1}u)$ with 0

In this paper, we investigate a class of nonlinear Schrödinger systems containing a nonlinear operator under Osgood-type conditions. By employing the iterative technique, the existence conditions for entire positive radial solutions of the above problem are given under the cases where components μ and ν are bounded, μ and ν are blow-up, and one of the components is bounded, while the other is blow-up. Finally, we present two examples to verify our results.

In this paper, using variational methods, we prove the existence of at least one positive radial solution for the generalized p (x) -Laplacian problem − Δ p (x) u + R (x) u p (x) − 2 u = a (x) | u | q (x) − 2 u − b (x) | u | r (x) − 2 u with Dirichlet boundary condition in the unit ball in R N (for N ≥ 3 ), where a, b, R are radial functions. [ABSTRACT FROM AUTHOR]

In this article we prove the existence of an infinite number of radial solutions to ΔU + K(x)f(U)= 0 on the exterior of the ball of radius R>0 centered at the origin in ℝN with U=0 on ∂BR, and lim|x|→∞ = 0 where N>2, ... for small U ≠ 0 with 01. Also, K(x) ~ |x|-α with α>2(N-1) for large |x|. [ABSTRACT FROM AUTHOR]

In this article we study the existence of radial solutions of Δu + K(|x|)f(u) = 0 on the exterior of the ball of radius R>0 centered at the origin in ℝN with u=0 on ∂BR, and lim|x|→∞ u(x) where N>2, ... for u near 0 with 0

In this article we consider the radial solutions of $\Delta u + K(r)f(u)= 0$ on the exterior of the ball of radius $R>0$, $B_{R}$, centered at the origin in ${\mathbb R}^N$ with $u=0$ on $\partial B_{R}$ and $\lim_{r \to \infty} u(r)=0$ where $N>2$, $f$ is odd with $f0$ on $(\beta, \infty)$, $f(u)\sim u^p$ with $00$ is sufficiently small. If $R>0$ is sufficiently large then there are no solutions with $\lim_{r \to \infty} u(r)=0$.