Superlinear Elliptic Equations Research Articles

Ground state solutions for a kind of superlinear elliptic equations with variable exponent

AbstractIn this paper, we focus on the existence of ground state solutions for the $p(x)$ p ( x ) -Laplacian equation $$ \textstyle\begin{cases} -\Delta _{p(x)}u+\lambda \vert u \vert ^{p(x)-2}u=f(x,u)+h(x) \quad \text{in } \Omega , \\ u=0,\quad \text{on }\partial \Omega . \end{cases} $$ { − Δ p ( x ) u + λ | u | p ( x ) − 2 u = f ( x , u ) + h ( x ) in Ω , u = 0 , on ∂ Ω . Using the constraint variational method, quantitative deformation lemma, and strong maximum principle, we proved that the above problem admits three ground state solutions, especially speaking, one solution is sign-changing, one is positive, and one is negative. Our results improve on those existing in the literature.

A system of superlinear elliptic equations in a cylinder

The article is concerned with the existence of positive solutions of a semi-linear elliptic system defined in a cylinder Omega =Omega 'times (0,a)subset {mathbb {R}}^n, where Omega 'subset {mathbb {R}}^{n-1} is a bounded and smooth domain. The system couples a superlinear equation defined in the whole cylinder Omega with another superlinear (or linear) equation defined at the bottom of the cylinder Omega 'times {0}. Possible applications for such systems are interacting substances (gas in the cylinder and fluid at the bottom) or competing species in a cylindrical habitat (insects in the air and plants on the ground). We provide a priori L^infty bounds for all positive solutions of the system when the nonlinear terms satisfy certain growth conditions. It is interesting that due to the structure of the system our growth restrictions are weaker than those of the pioneering result by Brezis–Turner for a single equation. Using the a priori bounds and topological arguments, we prove the existence of positive solutions for these particular semi-linear elliptic systems.

Hopf reduction and orbit concentrating solutions for a class of superlinear elliptic equations

We consider singularly perturbed equations of the form{ε2Δu−u+up=0 in A⊂RN,u>0 in A,u=0 on ∂A, where A is a annulus and p>1. It has been conjectured for a long time that such problems possess solutions having m-dimensional concentration sets for every 0≤m≤N−1. For N=3 solutions with 2-dimensional and 0-dimensional concentration sets are known, while no result was available for 1-dimensional concentration sets. We answer positively this conjecture, proving the existence of solutions which concentrate on circles S1. The proof relies on work by Santra-Wei who proved the existence of solutions concentrating on a Clifford torus S1×S1 for an annulus in R4. We extend this result to equations with weights. Then we use the Hopf fibration to show that these solutions give rise to the S1-concentrating solutions in R3.

Superlinear Elliptic Equation with Mixed Boundary Value in Annular Domain

Superlinear Elliptic Equation with Mixed Boundary Value in Annular Domain

Solutions of Super-Linear Elliptic Equations and Their Morse Indices

We investigate the degenerate bi-harmonic equation $$\Delta_{m}^2 u=f(x,u)\quad \text{in} \Omega, \qquad u = \Delta u = 0\quad \text{on} \partial\Omega,$$ with $$m\ge 2$$ , and also the degenerate tri-harmonic equation: $$-\Delta_{m}^3 u=f(x,u)\quad \text{in} \Omega,\qquad u = \frac{\partial u}{\partial\nu}= \frac{\partial^{2} u}{\partial\nu^{2}} = 0\quad \text{on} \partial\Omega,$$ where $$\Omega\subset \mathbb{R}^{N}$$ is a bounded domain with smooth boundary $$N>4$$ or $$N>6$$ respectively, and $$f \in \mathrm{C}^{1}(\Omega\times \mathbb{R})$$ satisfies suitable m-superlinear and subcritical growth conditions. Our main purpose is to establish $$L^{p}$$ and $$L^{\infty}$$ explicit bounds for weak solutions via the Morse index. Our results extend previous explicit estimates obtained in [1]–[4].

Uniqueness of positive radial solutions of superlinear elliptic equations in annuli

We present a generalized comparison identity for a two point boundary value problem, and we apply it to show the uniqueness of positive radial solutions of{Δu(x)+f(u(x))=0in Aa,b,u(x)=0on ∂Aa,b, where n∈N with n≥2, 0<a<b<∞, Aa,b={x∈RN:a<|x|<b} and f∈C1[0,∞) is a superlinear function.

Nehari-Type Ground State Solutions for Superlinear Elliptic Equations with Variable Exponent in $${\mathbb {R}}^N$$

Nehari-Type Ground State Solutions for Superlinear Elliptic Equations with Variable Exponent in $${\mathbb {R}}^N$$

A priori estimates for super-linear elliptic equation: the Neumann boundary value problem

$$$\mbox{In this paper we study the nonexistence of finite Morse index solutions of the following} \\\mbox{Neumann boundary value problems} {(Eq.H)} \begin{cases} -\Delta u = (u^{+})^{p} \;\; \text{in $ \mathbb{R}_+^N$}, \frac{\partial u}{\partial x_{N}}=0 \quad\quad\;\; \text{ on $ \partial\mathbb{R}_+^N$}, u \in C^2(\overline{\mathbb{R}_+^N}) \mbox{ and sign-changing, }\\u^+ \mbox{ is bounded and } i(u)<\infty,\end{cases} \mbox{or} {(Eq.H')}\begin{cases}-\Delta u = |u|^{p-1}u \;\; \text{in $ \mathbb{R}_+^N$}, \frac{\partial u}{\partial x_{N}}=0 \;\;\;\;\;\;\;\;\quad\text{ on $ \partial\mathbb{R}_+^N$}, u \in C^2(\overline{\mathbb{R}_+^N}), u \mbox{ is bounded and } i(u) < \infty.\end{cases} \mbox{ As a consequence, we establish the relevant Bahri-Lions's }L^\infty\mbox{-estimate [3] via} \mbox{the boundedness of Morse index of solutions to} \begin{equation}\label{1.1} \left\{\begin{array}{lll} -\Delta u=f(x,u) &\text{in $ \frac{\partial u}{\partial \nu}=0 &\text{on $\partial \Omega,$} \end{array} \right. \end{equation} \mbox{where} f \mbox{ has an asymptotical behavior at in-nity} \mbox{which is not necessarily the same at} \pm\infty. \\\mbox{Our results complete previous Liouville type theorems and } L^\infty\mbox{-bounds via Morse index} \\\mbox{ obtained in [3, 6, 13, 16, 12, 21].}$$$

Asymptotically linear and superlinear elliptic equations with gradient terms

In this article we establish the existence of solutions for elliptic problem involving a gradient term. To handle the so-called non-variational problem, we use a variational methods. We assume that the nonlinear term satisfies an asymptotically linear growth condition or a superlinear growth condition. We show the existence of at least one positive solution and one negative solution.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2020/99/abstr.html

Existence of weak solutions to superlinear elliptic systems without the Ambrosetti-Rabinowitz condition

In this article, we study the existence of the weak solution for superlinear elliptic equations and systems without the Ambrosetti-Rabinowitz condition. The Ambrosetti-Rabinowitz condition guarantees the boundedness of the PS sequence of the functional I for the corresponding problem. We establish the existence of the weak solution for the superlinear elliptic equation by using \((PS)_c\) form of the Mountain pass lemma, and the existence of the weak solution for the superlinear elliptic system by using \((PS)_c^{*}\) form of the Linking theorem.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2020/52/abstr.html

Existence Results for Superlinear Elliptic Equations with Nonlinear Boundary Value Conditions

In this paper, we study the existence of solutions for the following superlinear elliptic equation with nonlinear boundary value condition $$ \left\{ \begin{array}{ll} -\Delta u+u=|u|^{r-2}u &\text{in} \; \Omega,\\ \\ \frac{\partial u}{\partial \nu}=|u|^{q-2}u &\text{on}\;\partial\Omega, \end{array} \right. $$ where $\Omega\subset \mathbb R^N, N\geq 3$ is a bounded domain with smooth boundary. We will prove the existence results for the above equation under four different cases: (i) Both $q$ and $r$ are subcritical; (ii) $r$ is critical and $q$ is subcritical; (iii) $r$ is subcritical and $q$ is critical; (iv) Both $q$ and $r$ are critical.

Superlinear Elliptic Equations with Variable Exponent via Perturbation Method

We are concerned with the following $p(x)$-Laplacian equations in $\mathbb{R}^{N}$$$ -\triangle _{p(x)} u+|u|^{p(x)-2}u= f(x,u)\quad \mbox{in } \mathbb{R} ^{N}. $$ The nonlinearity is superlinear but does not satisfy the Ambrosetti-Rabinowitz type condition. Our main difficulty is that the weak limit of (PS) sequence is not always the weak solution of this problem. To overcome this difficulty, by adding potential term and using mountain pass theorem, we get the weak solution $u_{\lambda }$ of perturbation equations. First, we prove that $u_{\lambda }\rightharpoonup u$ as $\lambda \rightarrow 0$. Second, by using vanishing lemma, we get that $u$ is a nontrivial solution of the original problem.

Positive solutions for nonlinear singular superlinear elliptic equations

We consider a nonlinear nonparametric elliptic Dirichlet problem driven by the p-Laplacian and reaction containing a singular term and a (p-1)-superlinear perturbation. Using variational tools together with suitable truncation and comparison techniques we produce two positive, smooth, ordered solutions.

Liouville results for elliptic equations in strips with finite Morse index

Consider the strip $$\varOmega =\mathbb {R}^{n-k}\times \omega $$ where $$n\ge 3,\;\;k\ge 1$$ and $$\omega $$ is a smooth bounded domain of $$\mathbb {R}^k$$ . We are concerned with the following superlinear elliptic equations: $$\begin{aligned} {\left\{ \begin{array}{ll} \Delta ^{2} u= |u|^{p-1}u &{}\quad \text {in} \; \varOmega \\ u =|\nabla u| = 0 &{}\quad \text {on} \; \partial \varOmega =\mathbb {R}^{n-k}\times \partial \omega \end{array}\right. } \end{aligned}$$ where $$p>1$$ and $$u \in C^4(\overline{\varOmega })$$ . We prove Liouville-type theorems for stable solutions or solutions which are stable outside a compact set of $$\varOmega $$ . We first provide an integral estimate from stability which combined with Pohozaev-type identity to obtain nonexistence results for $$p_\mathrm{s}(n,4)< p \le p_\mathrm{s}(n-k,4)$$ , where $$p_\mathrm{s}(z,4):=\frac{z+4}{z-4}$$ is the Sobolev exponent of $$\mathbb {R}^z$$ . Also, we establish monotonicity formula to prove the nonexistence of nontrivial stable solution for all $$p>1$$ and solution which is stable outside a compact set for $$p > p_\mathrm{s}(n-k,4)$$ . Our classification is close to a sharp result since in the subcritical case [i.e., $$1<p <p_\mathrm{s}(n,4) $$ ] we prove the existence of a mountain-pass solution with Morse index equal to 1 in the subspace of $$H_0^2(\varOmega )$$ having cylindrical symmetry.

Entire Solutions of Superlinear Problems with Indefinite Weights and Hardy Potentials

We provide the structure of regular/singular fast/slow decay radially symmetric solutions for a class of superlinear elliptic equations with an in- definite weight on the nonlinearity f (u, r). In particular we are interested in the case where f is positive in a ball and negative outside, or in the re- versed situation. We extend the approach to elliptic equations in presence of Hardy potentials. By the use of Fowler transformation we study the corresponding dynamical systems, presenting the construction of invariant manifolds when the global existence of solutions is not ensured.

A priori bounds for superlinear elliptic equations with semidefinite nonlinearity

We derive a priori bounds for positive solutions of the superlinear elliptic problems −Δu=a(x)up on a bounded domain Ω in RN, where a(x) is Hölder continuous in Ω. Our main motivation is to study the case where a(x)≥0, a(x)≢0 and a(x) has some zero sets in Ω. We show that, in this case, the scaling arguments reduce the problem of a priori bounds to the Liouville-type results for the equation −Δu=A(x′)up in RN, where A is the continuous function defined on the subspace Rk with 1≤k≤N and x′∈Rk. We also establish a priori bounds of global nonnegative solutions to the corresponding parabolic initial–boundary value problems.

On a diffusion model with absorption and production

We discuss the structure of radial solutions of some superlinear elliptic equations which model diffusion phenomena when both absorption and production are present. We focus our attention on solutions defined in R (regular) or in R∖{0} (singular) which are infinitesimal at infinity, discussing also their asymptotic behavior. The phenomena we find are present only if absorption and production coexist, i.e., if the reaction term changes sign. Our results are then generalized to include the case where Hardy potentials are considered.

On a concave–convex elliptic problem with a nonlinear boundary condition

We investigate an indefinite superlinear elliptic equation coupled with a sublinear Neumann boundary condition (depending on a positive parameter \(\lambda \)), which provides a concave–convex nature to the problem. We establish a global multiplicity result for positive solutions in the spirit of Ambrosetti–Brezis–Cerami and obtain their asymptotic profiles as \(\lambda \rightarrow 0\). Furthermore, we also analyse the case where the nonlinearity is concave. Our arguments are based on a bifurcation analysis, a comparison principle, and variational techniques.

Asymptotic behavior of positive solution in an annulus

This paper concerns the positive solutions of superlinear elliptic equation with Hardy potential:{−Δu+μ|x|2u=|u|p−2uin Ω,u>0in Ω,u=0on ∂Ω, where Ω is an annulus of RN, N⩾2. We obtain qualitative properties of solutions as p→+∞.

Ground state and multiple solutions via generalized Nehari manifold

By using variational methods and the generalized Nehari manifold due to Szulkin and Weth, the existence of the ground states and the multiplicity of solutions for a wide class of superlinear elliptic equations is studied.