In this paper, we first prove some propositions of Sobolev spaces defined on a locally finite graph $G=(V,E)$, which are fundamental when dealing with equations on graphs under the variational framework. Then we consider a nonlinear biharmonic equation $$ Δ^{2} u -Δu+(λa+1)u= |u|^{p-2}u $$ on $G=(V,E)$. Under some suitable assumptions, we prove that for any $λ>1$ and $p>2$, the equation admits a ground state solution $u_λ$. Moreover, we prove that as $λ\rightarrow +\infty$, the solutions $u_λ$ converge to a solution of the equation \begin{align*} \begin{cases} Δ^{2}u -Δu+u = |u|^{p-2}u, &\text{in}\ \ Ω, u=0, &\text{on}\ \ \partialΩ, \end{cases} \end{align*} where $Ω=\{x\in V: a(x)=0\}$ is the potential well and $\partialΩ$ denotes the the boundary of $Ω$.
We study the existence and the regularity of the biharmonic Green kernel in a Brelot biharmonic space whose associated harmonic spaces have Green kernels. We show by some examples that this kernel does not always exist. We then introduce and study the adjoint of the given biharmonic space. This study was initiated by Smyrnelis, however, it seems that several results were incomplete and we clarify them here.
We investigate the non-existence and existence of positive solutions to biharmonic elliptic inequalities on manifolds. Using Green function and volume growth conditions, we establish the critical exponent for biharmonic problem.
In this paper, we investigate the existence of ground state solutions and non-existence of non-trivial weak solution of biharmonic equation with some nonlocal terms and critical Sobolev exponent. Firstly, we prove the non-existence by establishing Pohozaev type of identity. Next, we study the existence of ground state solutions by using the minimization method on the associated Nehari manifold.
Using variational methods, we establish existence of multi-bump solutions for the following class of problems $$ \left\{ \begin{array}{l} Δ^2 u +(λV(x)+1)u = f(u), \quad \mbox{in} \quad \mathbb{R}^{N}, u \in H^{2}(\mathbb{R}^{N}), \end{array} \right. $$ where $N \geq 1$, $Δ^2$ is the biharmonic operator, $f$ is a continuous function with subcritical growth and $V : \mathbb{R}^N \rightarrow \mathbb{R}$ is a continuous function verifying some conditions.
Some superlinear fourth order elliptic equations are considered. Ground states are proved to exist and to concentrate at a point in the limit. The proof relies on variational methods, where the existence and concentration of nontrivial solutions are related to a suitable truncated equation.
Using variational methods, we establish existence of multi-bump solutions for the following class of problems $$ \left\{ \begin{array}{l} Δ^2 u +(λV(x)+1)u = f(u), \quad \mbox{in} \quad \mathbb{R}^{4}, u \in H^{2}(\mathbb{R}^{4}), \end{array} \right. $$ where $Δ^2$ is the biharmonic operator, $f$ is a continuous function with critical exponential growth and $V : \mathbb{R}^4 \rightarrow \mathbb{R}$ is a continuous function verifying some conditions.
We study supercritical $O(d)$-equivariant biharmonic maps with a focus on $d = 5$, where $d$ is the dimension of the domain. We give a characterisation of non-trivial equivariant biharmonic maps from $\mathbf{R}^5$ into $S^5$ as heteroclinic orbits of an associated dynamical system. Moreover, we prove the existence of such non-trivial equivariant biharmonic maps. Finally, in stark contrast to the harmonic map analogue, we show the existence of an equivariant biharmonic map from $B^5(0, 1)$ into $S^5$ that winds around $S^5$ infinitely many times.
We consider a two obstacle problem for the parabolic biharmonic equation in a bounded domain. We prove long time existence of solutions via an implicit time discretization scheme, and we investigate the regularity properties of solutions.
In this paper, we propose an existence result pertaining to a nontrivial solution to the problem \begin{align*} \Bigg\{\begin{split} & Δ^2_p u -Δ_p u + λV(x)|u|^{p-2}u = f(x,u)\,,\,x\in \mathbb{R}^N, & u \in W^{2,p}(\mathbb{R}^N), \end{split} \end{align*} where $λ>0$, $p>1, N>2p$ and $V\in C(\mathbb{R}^N, \mathbb{R}^+)$, $f\in C(\mathbb{R}^N \times \mathbb{R},\mathbb{R})$ with certain properties. We also investigate the concentration of solutions to the problem on the set $V^{-1}(0)$ as $λ\rightarrow \infty$.
In this paper, the following critical biharmonic elliptic problem \begin{eqnarray*} \begin{cases} Δ^2u= λu+μu\ln u^2+|u|^{2^{**}-2}u, &x\inΩ,\\ u=\dfrac{\partial u}{\partial ν}=0, &x\in\partialΩ\end{cases} \end{eqnarray*} is considered, where $Ω\subset \mathbb{R}^{N}$ is a bounded smooth domain with $N\geq5$. Some interesting phenomenon occurs due to the uncertainty of the sign of the logarithmic term. It is shown, mainly by using Mountain Pass Lemma, that the problem admits at lest one nontrivial weak solution under some appropriate assumptions of $λ$ and $μ$. Moreover, a nonexistence result …
\ In this paper, the following biharmonic elliptic problem \begin{eqnarray*} \begin{cases} Δ^2u-λ\frac{|u|^{q-2}u}{|x|^s}=|u|^{2^{**}-2}u+ f(x,u), &x\inΩ,\\ u=\dfrac{\partial u}{\partial n}=0, &x\in\partialΩ\end{cases} \end{eqnarray*} is considered. The main feature of the equation is that it involves a Hardy term and a nonlinearity with critical Sobolev exponent. By combining a careful analysis of the fibering maps of the energy functional associated with the problem with the Mountain Pass Lemma, it is shown, for some positive parameter $λ$ depending on $s$ and $q$, that the problem admits at…
We study the biharmonic equation $Δ^2 u =u^{-α}$, $0<α<1$, in a smooth and bounded domain $Ω\subset\RR^n$, $n\geq 2$, subject to Dirichlet boundary conditions. Under some suitable assumptions on $ø$ related to the positivity of the Green function for the biharmonic operator, we prove the existence and uniqueness of a solution.
Let $(M,g)$ be a four dimensional compact Riemannian manifold with boundary and $(N,h)$ be a compact Riemannian manifold without boundary. We show the existence of a unique, global weak solution of the heat flow of extrinsic biharmonic maps from $M$ to $N$ under the Dirichlet boundary condition, which is regular with the exception of at most finitely many time slices. We also discuss the behavior of solution near the singular times. As an immediate application, we prove the existence of a smooth extrinsic biharmonic map from $M$ to $N$ under any Dirichlet boundary condition.
In this paper, exploiting variational methods, the existence of multiple weak solutions for a class of elliptic Navier boundary problems involving the $p$-biharmonic operator is investigated. Moreover, a concrete example of an application is presented.
The aim of this work is to present results about existence of solutions for a class of biharmonic elliptic problems with homogeneous Navier conditions. The problem is symmetric and has linear behavior on -\infty and superlinear on +\infty. To obtain these results we apply topological degree theory.
We study existence, multiplicity and qualitative properties of entire solutions for a noncompact problem related to second-order interpolation inequalities with weights.
In this paper, we consider the existence of nontrivial solutions to the following critical biharmonic problem with a logarithmic term \begin{equation*} \begin{cases} Δ^2 u=μΔu+λu+|u|^{2^{**}-2}u+τu\log u^2, \ \ x\inΩ, u|_{\partial Ω}=\frac{\partial u}{\partial n}|_{\partialΩ}=0, \end{cases} \end{equation*} where $μ,λ,τ\in \mathbb{R}$, $|μ|+|τ|\ne 0$, $Δ^2=ΔΔ$ denotes the iterated N-dimensional Laplacian, $Ω\subset \mathbb{R}^{N}$ is a bounded domain with smooth boundary $\partial Ω$, $2^{**}=\frac{2N}{N-4}(N\ge5)$ is the critical Sobolev exponent for the embedding $H_{0}^{2}(Ω)\hookrightarrow …
This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. One has eventual local positivity for positive initial data, but on short time scales, one will in general have also regions of negativity. The first goal of this paper is to find sufficient conditions on initial data which ensure the existence …
We study the existence of nontrivial weak solutions for a class of generalized $p(x)$-biharmonic equations with singular nonlinearity and Navier boundary condition. The proofs combine variational and topological arguments. The approach developed in this paper allows for the treatment of several classes of singular biharmonic problems with variable growth arising in applied sciences, including the capillarity equation and the mean curvature problem.
In this paper, we study the relaxed energy for biharmonic maps from a $m$-dimensional domain into spheres. By an approximation method, we prove the existence of a minimizer of the relaxed energy of the Hessian energy, and that the minimizer is biharmonic and smooth outside a singular set $Σ$ of finite $(m-4)$-dimensional Hausdorff measure. Moreover, when $m=5$, we prove that the singular set $Σ$ is 1-rectifiable.
We study the existence and non-existence of nontrivial weak solution of $$ {Δ^2u-μ\frac{u}{|x|^{4}} = \frac{|u|^{q_β-2}u}{|x|^β}+|u|^{q-2}u\quad\textrm{in ${\mathbb R}^N$,}} $$ where $N\geq 5$, $q_β=\frac{2(N-β)}{N-4}$, $0<β<4$, $1<q\leq 2^{**}$ and $μ<μ_1:=\big(\frac{N(N-4)}{4}\big)^2$. Using Pohozaev type of identity, we prove the non-existence result when $1<q< 2^{**}$. On the other hand when the equation has multiple critical nonlinearities i.e. $q=2^{**}$ and $-(N-2)^2\leqμ<μ_1$, we establish the existence of nontrivial solution using the Mountain-Pass theorem by Ambrosetti and Rabinowitz…
Using dual method we establish the existence of nodal ground state solution for the following class of problems $$ \left\{ \begin{array}{l} Δ^2 u = f(u), \quad \mbox{in} \quad Ω, \\ u =Bu=0,\quad\mbox{on} \quad \partial Ω\end{array} \right. $$ where $Δ^2$ is the biharmonic operator, $B=Δ$ or $B=\dfrac{\partial}{\partial ν}$ and $f$ is a $C^1$-function having subcritical growth.
We study the focusing mass-subcritical biharmonic nonlinear Schrödinger equation (BNLS) on the product space $\mathbb{R}_x^d\times\mathbb{T}_y^n$. Following the crucial scaling arguments introduced in \cite{TTVproduct2014} we establish existence and stability results for the normalized ground states of BNLS. Moreover, in the case where lower order dispersion is absent, we prove the existence of a critical mass number $c_0\in(0,\infty)$ that sharply determines the $y$-dependence of the deduced ground states. In the mixed dispersion case, we encounter a major challenge as the BNLS is no longer s…
In this note, we establish the existence of a positive solution and its stability to the following problem $$Δ_{\mathbb{H}^n}^2u=a(ξ)u-f(ξ,u)\text{ in }Ω, \,\,\, u|_{\partialΩ} = 0 =\left.Δ_{\mathbb{H}^n} u|_{\partialΩ},$$ on Heisenberg group.
We study the following semilinear biharmonic equation $$ \left\{\begin{array}{lllllll} Δ^{2}u=\fracλ{1-u}, &\quad \mbox{in}\quad \B, u=\frac{\partial u}{\partial n}=0, &\quad \mbox{on}\quad \partial\B, \end{array} \right. %\eqno(M_λ) $$ where $\B$ is the unit ball in $\R^{n}$ and $n$ is the exterior unit normal vector. We prove the existence of $λ^{*}>0$ such that for $λ\in (0,λ^{*})$ there exists a minimal (classical) solution $\underline{u}_λ$, which satisfies $0<\underline{u}_λ<1$. In the extremal case $λ=λ^{*}$, we prove the existence of a weak solution which is unique solution even in a v…
We consider the nonlinear biharmonic Schrödinger equation $$i\partial_tu+(Δ^2+μΔ)u+f(u)=0,\qquad (\text{BNLS})$$ in the critical Sobolev space $H^s(\R^N)$, where $N\ge1$, $μ=0$ or $-1$, $0<s<\min\{\fc N2,8\}$ and $f(u)$ is a nonlinear function that behaves like $λ\left|u\right|^αu$ with $λ\in\mathbb{C},α=\frac{8}{N-2s}$. We prove the existence and uniqueness of the global solutions to (BNLS) for the small initial data.
In this paper we prove local well-posedness in Orlicz spaces for the biharmonic heat equation $\partial_{t} u+ Δ^2 u=f(u),\;t>0,\;x\in\R^N,$ with $f(u)\sim \mbox{e}^{u^2}$ for large $u.$ Under smallness condition on the initial data and for exponential nonlinearity $f$ such that $f(u)\sim u^m$ as $u\to 0,$ $m$ integer and $N(m-1)/4\geq 2$, we show that the solution is global. Moreover, we obtain a decay estimates for large time for the nonlinear biharmonic heat equation as well as for the nonlinear heat equation. Our results extend to the nonlinear polyharmonic heat equation.
We study here a singular perturbation problem of biLaplacian type, which can be seen as the biharmonic counterpart of classical combustion models. We provide different results, that include the convergence to a free boundary problem driven by a biharmonic operator, as introduced in a previous paper of ours, and a monotonicity formula in the plane. For the latter result, an important tool is provided by an integral identity that is satisfied by solutions of the singular perturbation problem. We also investigate the quadratic behaviour of solutions near the zero level set, at least for small val…
We study existence, multiplicity and qualitative properties of entire solutions for a noncompact problem related to p-biharmonic type equations with weights. More precisely, we deal with the following family of equations $$ Δ_{p}^2 u = λ|x|^{-2p}|u|^{p-2}u + |x|^{-β}|u|^{q-2} u\quad\text{in} \quad \mathbb R^N, $$ where $N> 2p$, $p>1$, $q>p$, $β= N - \frac{q}{p}(N-2p)$ and $λ\in\mathbb R$ is smaller than the Rellich constant.
We prove a Brezis-Kato-type regularity result for weak solutions to the biharmonic nonlinear equation $$ Δ^2 u = g(x,u)\qquad\text{in }\mathbb{R}^N$$ with a Carathéodory function $g:\mathbb{R}^N\times \mathbb{R}\to \mathbb{R}$, $N\geq 5$. The regularity results give rise to the existence of ground state solutions provided that $g$ has a general subcritical growth at infinity. We also conceive a new biharmonic logarithmic Sobolev inequality $$ \int_{\mathbb{R}^N}|u|^2\log |u|\,dx\leq\frac{N}{8}\log \Big(C\int_{\mathbb{R}^N}|Δu|^2\,dx \Big), \quad\text{for } u \in H^2(\mathbb{R}^N), \; \int_{\ma…
In this paper, we study the following biharmonic Choquard type equation \begin{align*} \begin{split} \left\{ \begin{array}{ll} γΔ^2u-βΔu=λu+(I_μ*F(u))f(u), \quad\mbox{in}\ \ \mathbb{R}^4, \displaystyle\int_{\mathbb{R}^4}|u|^2dx=c^2>0,\quad u\in H^2(\mathbb{R}^4), \end{array} \right. \end{split} \end{align*} where $γ>0$, $β\geq0$, $λ\in \mathbb{R}$, $I_μ=\frac{1}{|x|^μ}$ with $μ\in (0,4)$, $F(u)$ is the primitive function of $f(u)$, and $f$ is a continuous function with exponential critical growth. We can prove the existence of ground state normalized solutions for the above problem when the no…
This paper is devoted to studying the following nonlinear biharmonic Schrödinger equation with combined power-type nonlinearities \begin{equation*} \begin{aligned} Δ^{2}u-λu=μ|u|^{q-2}u+|u|^{4^*-2}u\quad\mathrm{in}\ \mathbb{R}^{N}, \end{aligned} \end{equation*} where $N\geq5$, $μ>0$, $2<q<2+\frac{8}{N}$, $4^*=\frac{2N}{N-4}$ is the $H^2$-critical Sobolev exponent, and $λ$ appears as a Lagrange multiplier. By analyzing the behavior of the ground state energy with respect to the prescribed mass, we establish the existence of normalized ground state solutions. Furthermore, all ground states are p…
In this paper, we deal with the logarithmic weighted fourth order elliptic equation in the unit disk of $B\subset\R^{4}$: $$\displaystyle(P_λ)~~Δ(w(x) Δu) = λ f(x,u) \quad\mbox{ in }\quad B, \quad u=\frac{\partial u}{\partial n} \quad\mbox{ on } \quad\partial B,$$ where the nonlinearity $f$ is assumed to have exponential critical growth in view of Adam's type inequalities. By using the constrained minimization in Nehari set combined with the quantitative deformation lemma and degree theory, we prove the existence of nodal solutions to the problem $(P_λ)$ .
We consider the following problem: \begin{eqnarray*} ( P)\qquad \displaystyle\left\{\begin{array} {ll} & Δ^2 u = K(x)u^{-α} \quad \mbox{ in }\,Ω, \\ &u> 0\quad \mbox{ in }\,Ω, \;\;u\vert_{\partialΩ}=0, \,Δu\vert_{\partialΩ} = 0. \end{array}\right. \end{eqnarray*} We prove the main existence result: Assume that $α+β<2$. Then there exists a unique solution $u$ to $(P)$. Furthermore, there exist $c_1, c_2>0$ such that \begin{eqnarray}\label{behaviour-bound} c_1 ρ(x)\leq u(x)\leq c_2 ρ(x) \end{eqnarray} where $ρ(x)=d(x,\partialΩ)$. This result is sharp: Assume that $α+β\geq 2$. Then, there is no s…
In this work, we investigate the IVP for a time-fractional fourth-order equation with nonlinear source terms. More specifically, we consider the time-fractional biharmonic with exponential nonlinearity and the time-fractional Cahn-Hilliard equation. By using the Fourier transform concept, the generalized formula for the mild solution as well as the smoothing effects of resolvent operators are proved. For the IVP associated with the first one, by using the Orlicz space with the function $ Ξ(z)=e^{|z|^p}-1$ and some embeddings between it and the usual Lebesgue spaces, we prove that the solution …
In this paper, we are concerned with the existence of least energy solutions for the following biharmonic equations: $$Δ^2 u+(λV(x)-δ)u=|u|^{p-2}u \quad in\quad \mathbb{R}^N$$ where $N\geq 5, 2<p\leq\frac{2N}{N-4}, λ>0$ is a parameter, $V(x)$ is a nonnegative potential function with nonempty zero sets $\mbox{int} V^{-1}(0)$, $0<δ<μ_0$ and $μ_0$ is the principle eigenvalue of $Δ^2$ in the zero sets $\mbox{int} V^{-1}(0)$ of $V(x)$. Here $\mbox{int} V^{-1}(0)$ denotes the interior part of the set $V^{-1}(0):=\{x\in \mathbb{R}^N: V(x)=0\}$. We prove that the above equation admits a least energy s…
The goal of this paper is to establish singular Adams type inequality for biharmonic operator on Heisenberg group. As an application, we establish the existence of a solution to \begin{equation*} Δ_{\mathbb{H}^n}^2 u=\frac{f(ξ,u)}{ρ(ξ)^a}\,\,\text{ in }Ω,\,\, u|_{\partialΩ}=0=\left.\frac{\partial u}{\partial ν}\right|_{\partialΩ}, \end{equation*} where $0\in Ω\subseteq \mathbb{H}^4$ is a smooth bounded domain, $0\leq a<Q,\,(Q=10).$ The special feature of this problem is that it contains an exponential nonlinearity and singular potential.
In this work we study the nonnegative solutions of the elliptic system Δu=|x|^{a}v^δ, Δv=|x|^{b}u^μ in the superlinear case μδ>1, which blow up near the boundary of a domain of R^{N}, or at one isolated point. In the radial case we give the precise behavior of the large solutions near the boundary in any dimension N. We also show the existence of infinitely many solutions blowing up at 0. Furthermore, we show that there exists a global positive solution in R^{N}\{0}, large at 0, and we describe its behavior. We apply the results to the sign changing solutions of the biharmonic equation Δ^2 u=|…
This work presents the construction of the existence theory of radial solutions to the elliptic equation \begin{equation}\nonumber Δ^2 u = (-1)^k S_k[u] + λf(x), \qquad x \in B_1(0) \subset \mathbb{R}^N, \end{equation} provided either with Dirichlet boundary conditions \begin{eqnarray}\nonumber u = \partial_n u = 0, \qquad x \in \partial B_1(0), \end{eqnarray} or Navier boundary conditions \begin{equation}\nonumber u = Δu = 0, \qquad x \in \partial B_1(0), \end{equation} where the $k-$Hessian $S_k[u]$ is the $k^{\mathrm{th}}$ elementary symmetric polynomial of eigenvalues of the Hessian matrix…
We show the local wellposedness of biharmonic wave maps with initial data of sufficiently high Sobolev regularity and a blow-up criterion in the sup-norm of the gradient of the solutions. In contrast to the wave maps equation we use a vanishing viscosity argument and an appropriate parabolic regularization in order to obtain the existence result. The geometric nature of the equation is exploited to prove convergence of approximate solutions, uniqueness of the limit, and continuous dependence on initial data.
