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According to a very recent paper [5], the positive solution of the two-dimensional Hénon equation will concentrate on finite points when $ p\to \infty $. In this short note, we prove the uniform boundedness of the solution and then obtain a formula to specify the location of the concentration points by a Kirchoff-Routh type function. [ABSTRACT FROM AUTHOR]

Let N ≥ 3 , R > ρ > 0 and A ρ : = { x ∈ R N ; ρ < | x | < R } . Let U n , ρ ± , n ≥ 1 , be a radial solution with n nodal domains of Δ U + | x | α | U | p - 1 U = 0 in A ρ , U = 0 on ∂ A ρ. We show that if p = N + 2 + 2 α N - 2 , α > - 2 and N ≥ 3 , then U n , ρ ± is nondegenerate for small ρ > 0 and the Morse index m (U n , ρ ±) satisfies m (U n , ρ ±) = n (N + 2 ℓ - 1) (N + ℓ - 1) ! (N - 1) ! ℓ ! for small ρ > 0 , where ℓ = [ α 2 ] + 1 . Using Jacobi elliptic functions, we show that if (p , α) = (3 , N - 4) and N ≥ 3 , then the Morse index of a positive and negative solutions m (U 1 , ρ ±) is completely determined by the ratio ρ / R ∈ (0 , 1) . Upper and lower bounds for m (U n , ρ ±) , n ≥ 1 , are also obtained when (p , α) = (3 , N - 4) and N ≥ 3 . [ABSTRACT FROM AUTHOR]

This paper is concerned with the following supercritical Hénon equation with variable exponent ..., where B ⊂ RN (N ≥ 3) is the unit ball, α > 0, 0 < β < min {(N+α)/2, N-2} and 2*α = (2N + 2α)/(N-2). We obtain the existence of positive ground state solution by applying the mountain pass theorem, concentration-compactness principle and approximation techniques. [ABSTRACT FROM AUTHOR]

We consider the following Hénon equation with critical growth: - Δ u = K (| y |) u N + 2 N - 2 , u > 0 , in B 1 (0) u = 0 , on ∂ B 1 (0) , where B 1 (0) is the unit ball in R N , K : [ 0 , 1 ] → R + is a bounded function and K ′ ′ (1) exists. We prove a non-degeneracy result of the bubble solutions constructed in [24] via the local Pohozaev identities for N ≥ 5 . Then, as applications, by using reduction arguments combined with delicate estimates for the modified Green function and the error, we prove the new existence of infinitely many non-radial solutions, whose energy can be arbitrarily large. [ABSTRACT FROM AUTHOR]

In this paper, we study the positive radial solutions for the Hénon equations with weighted critical exponents on the unit ball B$$ B $$ in ℝN$$ {\mathbb{R}}^N $$ with N⩾3$$ N\geqslant 3 $$. We first confirm that 2∗(α)=2(N+α)N−2$$ {2}^{\ast}\left(\alpha \right)=\frac{2\left(N+\alpha \right)}{N-2} $$ with α>0$$ \alpha >0 $$ is the critical exponent for the embedding from H0,r1(B)$$ {H}_{0,r}^1(B) $$ into Lp(B;|x|α)$$ {L}^p\left(B;{\left|x\right|}^{\alpha}\right) $$ and name 2∗(α)$$ {2}^{\ast}\left(\alpha \right) $$ as the Hénon‐Sobolev critical exponent. Then, following the great ideas of Brezis and Nirenberg (Comm Pure Appl Math. 1983;36:437‐477), we establish the existence and nonexistence of positive radial solutions of the problems with single Hénon‐Sobolev critical exponent and linear or nonlinear but subcritical perturbations. We further study the problems with multiple critical exponents, which may be Hénon‐Sobolev critical exponents, Hardy‐Sobolev critical exponents, or Sobolev critical exponents. The methods and arguments involved with are the mountain pass theorem and the strong maximum principle and the Pohozaev identity. [ABSTRACT FROM AUTHOR]

Here, we consider positive singular solutions of { − Δ u = | x | α | ∇ u | p u = 0 in on Ω ∖ { 0 } , ∂ Ω , { − Δ u = | x | α | ∇ u | p in Ω ∖ { 0 } , u = 0 on ∂ Ω , where Ω Ω is a small smooth perturbation of the unit ball in R N R N and α α and p p are parameters in a certain range. Using an explicit solution on B 1 B 1 and a linearization argument, we obtain positive singular solutions on perturbations of the unit ball. [ABSTRACT FROM AUTHOR]

Abstract This paper is devoted to studying the semilinear elliptic system of Hénon type {−ΔBNu=K(d(x))Qu(u,v),−ΔBNv=K(d(x))Qv(u,v),u,v∈Hr1(BN),N≥3, $$ \textstyle\begin{cases} -\Delta _{\mathbb{B}^{N}}u= K(d(x))Q_{u}(u,v), \\ -\Delta _{\mathbb{B}^{N}}v= K(d(x))Q_{v}(u,v), \\ \quad u, v\in H_{r}^{1}(\mathbb{B}^{N}),\quad N\geq 3, \end{cases} $$ in the hyperbolic space BN $\mathbb{B}^{N}$, where Hr1(BN)={u∈H1(BN):u is radial} $H_{r}^{1}(\mathbb{B} ^{N})=\{u\in H^{1}(\mathbb{B}^{N}): u \text{ is radial}\}$ and −ΔBN $-\Delta _{\mathbb{B}^{N}}$ denotes the Laplace–Beltrami operator on BN $\mathbb{B}^{N}$, d(x)=dBN(0,x) $d(x)=d_{\mathbb{B}^{N}}(0,x)$, Q∈C1(R×R,R) $Q \in C^{1}( \mathbb{R}\times \mathbb{R},\mathbb{R})$ is p-homogeneous, and K≥0 $K\geq 0 $ is a continuous function. We prove a compactness result and, together with Clark’s theorem, we establish the existence of infinitely many solutions.

We consider the equation − Δ u = | x | α | u | p − 1 u for any α ≥ 0 , either in R 2 or in the unit ball B of R 2 centered at the origin with Dirichlet or Neumann boundary conditions. We give a sharp description of the asymptotic behavior as p → + ∞ of all the radial solutions to these problems and we show that there is no uniform a priori bound for nodal solutions under Neumann or Dirichlet boundary conditions. This contrasts with the existence of uniform bounds for positive solutions, as shown in [32] for α = 0 and Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR]

In the present paper we study the existence of ground state solutions for Hénon equations involving with single or multiple upper weighted critical exponents with general perturbational term, where the upper weighted critical exponents include upper Hardy-Sobolev, Sobolev or Hénon-Sobolev critical exponents for the embedding from H r 1 (R N) into L q (R N ; | x | α) with α > − 2. The Nehari manifold method and the mountain pass theorem are applied to obtain the ground state solutions with different assumptions on general perturbational term. The existence of ground state solutions is closely related to the sign of the parameters in the equations. [ABSTRACT FROM AUTHOR]

We consider the Hénon equation (P α) − Δ u = | x | α | u | p − 1 u in B N , u = 0 on ∂ B N , where B N ⊂ R N is the open unit ball centered at the origin, N ≥ 3 , p > 1 and α > 0 is a parameter. We show that, after a suitable rescaling, the two-dimensional Lane-Emden equation − Δ w = | w | p − 1 w in B 2 , w = 0 on ∂ B 2 , where B 2 ⊂ R 2 is the open unit ball, is the limit problem of (P α) , as α → ∞ , in the framework of radial solutions. We exploit this fact to prove several qualitative results on the radial solutions of (P α) with any fixed number of nodal sets: asymptotic estimates on the Morse indices along with their monotonicity with respect to α ; asymptotic convergence of their zeros; blow up of the local extrema and on compact sets of B N. All these results are proved for both positive and nodal solutions. [ABSTRACT FROM AUTHOR]

This article establishes existence, non-existence and Liouville-type theorems for nonlinear equations of the form −div(|x|aDu) = ƒ(x,u), u > 0, in Ω, where N ≥ 3, Ω is an open domain in RN containing the origin, N − 2 + a > 0 and ƒ satisfies structural conditions, including certain growth properties. The first main result is a non-existence theorem for boundary-value problems in bounded domains star-shaped with respect to the origin, provided ƒexhibits supercritical growth. A consequence of this is the existence of positive entire solutions to the equation for ƒ exhibiting the same growth. A Liouville-type theorem is then established, which asserts no positive solution of the equation in Ω = RN exists provided the growth ofƒ is subcritical. The results are then extended to systems of the form −div(|x|aDu1) = ƒ1(x,u1,u2),−div(|x|aDu2) = ƒ2(x,u1,u2),u1,u2 > 0, in Ω, but after overcoming additional obstacles not present in the single equation. Specific cases of our results recover classical ones for a renowned problem connected with finding best constants in Hardy-Sobolev and Caffarelli-Kohn-Nirenberg inequalities as well as existence results for well-known elliptic systems. [ABSTRACT FROM AUTHOR]

We investigate the Henon type equation involving the critical Sobolev exponent with Dirichret boundary condition $$ - \Delta u = \lambda \Psi u + | x |^\alpha u^{2^*-1} $$ in $\Omega$ included in a unit ball, under several conditions. Here, $\Psi$ is a non-trivial given function with $0 \leq \Psi \leq 1$ which may vanish on $\partial \Omega$. Let $\lambda_1$ be the first eigenvalue of the Dirichret eigenvalue problem $-\Delta \phi = \lambda \Psi \phi$ in $\Omega$. We show that if the dimension $N \geq 4$ and $0 < \lambda < \lambda_1$, there exists a positive solution for small $\alpha > 0$. Our methods include the mountain pass theorem and the Talenti function.

We study the spectral asymptotics of nodal (i.e., sign-changing) solutions of the problemin the unit ball , in the limit. More precisely, for a given positive integer , we derive asymptotic -expansions for the negative eigenvalues of the linearization of the unique radial solution of with precisely nodal domains and. As an application, we derive the existence of an unbounded sequence of bifurcation points on the radial solution branch which all give rise to bifurcation of nonradial solutions whose nodal sets remain homeomorphic to a disjoint union of concentric spheres. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the semilinear elliptic system of Hénon type { − Δ B N u = K (d (x)) Q u (u , v) , − Δ B N v = K (d (x)) Q v (u , v) , u , v ∈ H r 1 (B N) , N ≥ 3 , in the hyperbolic space B N , where H r 1 (B N) = { u ∈ H 1 (B N) : u is radial } and − Δ B N denotes the Laplace–Beltrami operator on B N , d (x) = d B N (0 , x) , Q ∈ C 1 (R × R , R) is p-homogeneous, and K ≥ 0 is a continuous function. We prove a compactness result and, together with Clark's theorem, we establish the existence of infinitely many solutions. [ABSTRACT FROM AUTHOR]

Abstract We consider the equation − Δ u = | x | α | u | p − 1 u , x ∈ B , u = 0 on ∂ B , where B ⊂ R 2 is the unit ball centered at the origin, α ≥ 0 , p > 1 , and we prove some results on the Morse index of radial solutions. The contribution of this paper is twofold. Firstly, fixed the number of nodal sets n ≥ 1 of the solution u α , n , we prove that the Morse index m (u α , n) is monotone non-decreasing with respect to α. Secondly, we provide a lower bound for the Morse indices m (u α , n) , which shows that m (u α , n) → + ∞ as α → + ∞. [ABSTRACT FROM AUTHOR]

We show the existence of a symmetry-breaking bifurcation point for the one-dimensional Hénon equation u ″ + | x | l u p = 0 , x ∈ (− 1 , 1) , u (− 1) = u (1) = 0 , where l > 0 and p > 1. Moreover, employing a variant of Rabinowitz's global bifurcation, we obtain the unbounded connected set (the first of the alternatives about Rabinowitz's global bifurcation), which emanates from the symmetry-breaking bifurcation point. Finally, we give an example of a bounded branch connecting two symmetry-breaking bifurcation points (the second of the alternatives about Rabinowitz's global bifurcation) for the problem u ″ + | x | l (λ) u p = 0 , x ∈ (− 1 , 1) , u (− 1) = u (1) = 0 , where l is a specified continuous parametrization function. [ABSTRACT FROM AUTHOR]

We consider the following Hénon equation with supercritical exponent: − Δ u = K (| y |) u p + ɛ , u > 0 , in B 1 (0) , u = 0 , on ∂ B 1 (0) , where B 1 (0) is the unit ball in R N (N ≥ 4) , p = 2 ∗ − 1 , 2 ∗ = 2 N N − 2 , K (r) ∈ C 1 [ 0 , 1 ]. We prove the asymptotic non-degeneracy of multipeak solution constructed in Liu and Peng (2016) via the local Pohozaev identities. [ABSTRACT FROM AUTHOR]

For n≥3 and p=(n+2)/(n-2), we consider the Hénon equation with the homogeneous Neumann boundary condition -Δu+u=|x|αup,u>0inΩ, ∂u/∂ν=0 on ∂Ω, where Ω⊂B(0,1)⊂Rn,n≥3, α≥0 and ∂∗Ω≡∂Ω∩∂B(0,1)≠∅. It is well known that for α=0, there exists a least energy solution of the problem. We are concerned on the existence of a least energy solution for α>0 and its asymptotic behavior as the parameter α approaches from below to a threshold α0∈(0,∞] for existence of a least energy solution. [ABSTRACT FROM AUTHOR]

Consider the Hénon equation with the homogeneous Neumann boundary condition − Δ u + u = | x | α u p , u > 0 in Ω , ∂ u ∂ ν = 0 on ∂ Ω , where Ω ⊂ B ( 0 , 1 ) ⊂ R N , N ≥ 2 and ∂ Ω ∩ ∂ B ( 0 , 1 ) ≠ ∅ . We are concerned on the asymptotic behavior of ground state solutions as the parameter α → ∞ . As α → ∞ , the non-autonomous term | x | α is getting singular near | x | = 1 . The singular behavior of | x | α for large α > 0 forces the solution to blow up. Depending subtly on the ( N − 1 ) − dimensional measure | ∂ Ω ∩ ∂ B ( 0 , 1 ) | N − 1 and the nonlinear growth rate p , there are many different types of limiting profiles. To catch the asymptotic profiles, we take different types of renormalization depending on p and | ∂ Ω ∩ ∂ B ( 0 , 1 ) | N − 1 . In particular, the critical exponent 2 ⁎ = 2 ( N − 1 ) N − 2 for the Sobolev trace embedding plays a crucial role in the renormalization process. This is quite contrasted with the case of Dirichlet problems, where there is only one type of limiting profile for any p ∈ ( 1 , 2 ⁎ − 1 ) and a smooth domain Ω. [ABSTRACT FROM AUTHOR]

In this work we study the following Hénon-type equation − div ∇ u p − 2 ∇ u x a p = x β f ( u ) , in B ; u > 0 , in B ; u = 0 , on ∂ B ; where B ≔ x ∈ R N ; x < 1 is a ball centered at the origin, the parameters verify the inequalities 0 ≤ a < N − p p , N ≥ 4 , β > 0 , 2 ≤ p < N p + p β N − p ( a + 1 ) , and the nonlinearity f is nonhomogeneous. By minimization on the Nehari manifold, we prove that for large values of the parameter β there is a symmetry breaking and non radial solutions appear. [ABSTRACT FROM AUTHOR]

We consider non-autonomous semilinear elliptic equations of the type where is either a ball or an annulus centered at the origin, and is on bounded sets of . We address the question of estimating the Morse index of a sign changing radial solution . We prove that for every and that if is even. If is superlinear the previous estimates become and , respectively, where denotes the number of nodal sets of , i.e. of connected components of . Consequently, every least energy nodal solution is not radially symmetric and as along the sequence of even exponents . [ABSTRACT FROM AUTHOR]

In this article, we consider the existence and asymptotic behavior of solutions for the Henon equation $$\displaylines{ -\Delta_{\mathbb{B}^N}u=(d(x))^{\alpha}|u|^{p-2}u, \quad x\in \Omega\cr u=0 \quad x\in \partial \Omega, }$$ where $\Delta_{\mathbb{B}^N}$ denotes the Laplace Beltrami operator on the disc model of the Hyperbolic space $\mathbb{B}^N$, $d(x)=d_{\mathbb{B}^N}(0,x)$, $\Omega \subset \mathbb{B}^N$ is geodesic ball with radius $1$, $\alpha>0, N\geq 3$. We study the existence of hyperbolic symmetric solutions when $2

This paper is concerned with ground state solutions for the Hénon type equation \(-\Delta u(x)=|y|^{\alpha} u^{p-1}(x)\) in \(\Omega\), where \(\Omega=B^k(0,1)\times B^{n-k}(0,1)\subset \mathbb{R}^n\) and \(x=(y,z) \in \mathbb{R}^k \times \mathbb{R}^{n-k}\). We study the existence of cylindrically symmetric and non-cylindrically symmetric ground state solutions for the problem. We also investigate asymptotic behavior of the ground state solution when \(p\) tends to the critical exponent \(2^*=\frac {2n}{n-2}\) if \(n\geq 3\).

Following the constructive method of Wei and Yan [22] , with new ingredients to take care of n = 3 , we prove the existence of infinitely many solutions of the Hénon equation − Δ u = | x | α u n + 2 n − 2 in the unit ball of R n ( n ⩾ 3 , α > 0 ) with the Dirichlet boundary condition. [ABSTRACT FROM AUTHOR]

In this paper, we establish some Liouville type theorems for positive solutions of fractional Hénon equation and system in ℝn. First, under some regularity conditions, we show that the above equation and system are equivalent to the some integral equation and system, respectively. Then, we prove Liouville type theorems via the method of moving planes in integral forms. [ABSTRACT FROM AUTHOR]

In this paper we study a Hénon-like equation (see Eq. (1)), where the nonlinearity f (t) is not homogeneous (i.e., it is not a power). By minimization on the Nehari manifold, we prove that for large values of the parameter α there is a breaking of symmetry and non radial solutions appear. This holds for sub- and super-critical growth of the nonlinearity f . [ABSTRACT FROM AUTHOR]

We study the one-dimensional generalized Hénon equation under the Dirichlet boundary condition. It is known that there exist at least three positive solutions if the coefficient function is even. In this paper, without the assumption of evenness, we prove the existence of at least three positive solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem (1){-Δu = |x|α up-1, x ∈ Ω, {u > 0, x ∈Ω, {∂u/∂v + βu = 0 x ∈Ω, where Ω is the unit ball in ℝN centered at the origin with N ≥ 3, α > 0, β > N - 2/2, p ≥ 2 and v is the unit outward vector normal to ∂Ω. We investigate the asymptotic behavior of the ground state solutions up of (1) as p → 2N/N - 2. We show that the ground state solutions up has a unique maximum point xp ∈ Ω̄. In addition, the ground state solutions is non-radial provided that p → 2N/N - 2. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following Robin problem: $$\begin{eqnarray*}\displaystyle \left\{ \begin{array}{ @{}ll@{}} \displaystyle - \Delta u= \mid x{\mathop{\mid }\nolimits }^{\alpha } {u}^{p} , \quad & \displaystyle x\in \Omega , \\ \displaystyle u\gt 0, \quad & \displaystyle x\in \Omega , \\ \displaystyle \displaystyle \frac{\partial u}{\partial \nu } + \beta u= 0, \quad & \displaystyle x\in \partial \Omega , \end{array} \right.&&\displaystyle\end{eqnarray*}$$ where $\Omega $ is the unit ball in ${ \mathbb{R} }^{N} $ centred at the origin, with $N\geq 3$, $p\gt 1$, $\alpha \gt 0$, $\beta \gt 0$, and $\nu $ is the unit outward vector normal to $\partial \Omega $. We prove that the above problem has no solution when $\beta $ is small enough. We also obtain existence results and we analyse the symmetry breaking of the ground state solutions. [ABSTRACT FROM AUTHOR]

In this note, we prove that least energy solutions of the two-dimensional Hénon equation -Δu = |χ|2 α up χ ∈ Ω u > 0 χ ∈ Ω, u = 0 χ ∈ ∂ Ω, where Ω is a smooth bounded domain in ℝ² with 0 ∈ Ω, α ≥ 0 is a constant R and p > 1, have only one global maximum point when α > e - 1 and the nonlinear exponent p is sufficiently large. This answers positively to a recent conjecture by C. Zhao (preprint, 2011). [ABSTRACT FROM AUTHOR]

The Hénon equation on a bounded domain in ℝ² with large exponent in the nonlinear term is studied in this paper. We investigate positive solution obtained by the variational method and give its asymptotic behavior as the nonlinear exponent gets large. [ABSTRACT FROM AUTHOR]

In this paper, we use the variational methods to study the following problem -?BNu = (d(x))α |u|p-2u,u ? H¹r(BN) in Hyperbolic space BN, where α > 0, d(x) = dBN (0,x), and H¹r (BN) denote the Laplace-Beltrami operator on BN, N ≥ 3. Unlike the corresponding problem in Euclidean space RN, we prove that there exists a positive solution of problem (1) provided that p ? (2, 2N+2a/N-2) which will be contrasted with a result due to Gidas and Spruck. [ABSTRACT FROM AUTHOR]

Abstract: We study the generalized Hénon equation in reflectionally symmetric or point symmetric domains and prove that a least energy solution is neither reflectionally symmetric nor even. Moreover, we prove the existence of a positive solution with prescribed exact symmetry. [Copyright &y& Elsevier]

Abstract: We investigate here the nonlinear elliptic equations and with , and . In particular, we prove some Liouville type theorems for weak solutions with finite Morse index in the low dimensional Euclidean spaces or half spaces. [Copyright &y& Elsevier]

Abstract: In this paper, we study the generalized Hénon equation with a radial coefficient function in the unit ball and show the existence of a positive non-radial solution. Our result is applicable to a wide class of coefficient functions. Our theorem ensures that if the ratio of the density of the coefficient function in to that in is small enough and a is sufficiently close to 1, then a least energy solution is not radially symmetric. [Copyright &y& Elsevier]

Abstract: Three algorithms based on the bifurcation theory are proposed to compute the symmetric positive solutions to the boundary value problem of the Henon equation on the unit disk. Taking l in the Henon equation as a bifurcation parameter, the symmetry-breaking bifurcation point on the branch of the symmetric positive solutions is found via the extended systems. Finally, other symmetric positive solutions are computed by the branch switching method based on the Lyapunov–Schmidt reduction. [Copyright &y& Elsevier]