Nontrivial Nonnegative Solutions Research Articles

On generalized logistic equations with non-local term of feedback control type

We consider a problem of finding functions λ(x),u(x) such that −Δpu=λf(x,u,∇u)−g(x,u) in Ω and u=0 on ∂Ω, λ(x)∈F(x,u(x)) a.e. in Ω. Assume that, the single-valued functions f,g and the multi-valued function F satisfy certain growth conditions. Using the fixed point index theory for multi-valued operators with non-necessary convex values, we prove the existence of one or two nontrivial non-negative solutions of the problem.

On Liouville-type theorems for 𝑘-Hessian equations with gradient terms

In this paper, we investigate several Liouville-type theorems related to k k -Hessian equations with non-linear gradient terms. More specifically, we study non-negative solutions to S k [ D 2 u ] ≥ h ( u , | D u | ) S_k[D^2u]\ge h(u,|Du|) in R n \mathbb {R}^n . The results depend on some qualified growth conditions of h h at infinity. A Liouville-type result to subsolutions of a prototype equation S k [ D 2 u ] = f ( u ) + g ( u ) ϖ ( | D u | ) S_k[D^2u]=f(u)+g(u)\varpi (|Du|) is investigated. A necessary and sufficient condition for the existence of a non-trivial non-negative entire solution to S k [ D 2 u ] = f ( u ) + g ( u ) | D u | q S_k[D^2u]=f(u)+g(u)|Du|^q for 0 ≤ q > k + 1 0\le q>k+1 is also given.

On a fractional (p,q)-Laplacian equation with critical nonlinearities

In this paper, we consider the following fractional ( p , q ) -Laplacian equations with critical Hardy-Sobolev exponents { ( − Δ ) p s 1 u + ( − Δ ) q s 2 u = λ | u | r − 2 u + μ | u | p s 1 ∗ ( α ) − 2 u | x | α in Ω , u = 0 in R N ∖Ω , where 0 < s 2 < s 1 < 1 < q < r ≤ p < N s 1 , 0 $ ]]> λ , μ > 0 are two parameters, 0 ≤ α < p s 1 and p s 1 ∗ ( α ) = p ( N − α ) N − p s 1 is the fractional Hardy-Sobolev critical exponent, Ω ⊂ R N is an open bounded domain with smooth boundary. By using variational methods, we show that the problem has a nontrivial nonnegative weak solution.

Degenerate Schrödinger--Kirchhoff {(p,N)}-Laplacian problem with singular Trudinger--Moser nonlinearity in ℝ N

Abstract In this paper, we deal with the existence of nontrivial nonnegative solutions for a ( p , N ) {(p,N)} -Laplacian Schrödinger–Kirchhoff problem in ℝ N {\mathbb{R}^{N}} with singular exponential nonlinearity. The main features of the paper are the ( p , N ) {(p,N)} growth of the elliptic operators, the double lack of compactness, and the fact that the Kirchhoff function is of degenerate type. To establish the existence results, we use the mountain pass theorem, the Ekeland variational principle, the singular Trudinger–Moser inequality, and a completely new Brézis–Lieb-type lemma for singular exponential nonlinearity.

Existence of solutions to modified nonlinear Schrödinger equations on non-compact Riemannian manifolds

We study the existence of nontrivial solutions to a class of modified nonlinear Schrödinger equation on a complete non-compact N-dimensional (N≥3) Riemannian manifold with asymptotically non-negative Ricci curvature. Using the critical point theory, in a general Sobolev space instead of an Orlicz one, we prove the existence of a nontrivial non-negative solution to a class of modified Schrödinger equation with coercive potential. We also estimate the decay rate of the solution and show that the solution is exponentially decay.

The Multiplicity of Nonnegative Nontrivial Solutions for p(x)-Kirchhoff Equation with Concave–Convex Nonlinearities

The Multiplicity of Nonnegative Nontrivial Solutions for p(x)-Kirchhoff Equation with Concave–Convex Nonlinearities

A priori bounds and multiplicity results for slightly superlinear and sublinear elliptic [formula omitted]-Laplacian equations

We consider the following problem −Δpu=h(x,u)inΩ, u∈W01,p(Ω), where Ω is a bounded domain in RN, 1<p<N, with a smooth boundary. In this paper, we assume that h(x,u)=a(x)f(u)+b(x)g(u) such that f is regularly varying of index p−1 and p-superlinear at infinity. The function g is a p-sublinear function at zero. The coefficients a and b belong to Lk(Ω) for some k>Np and they are without sign condition. Firstly, we show an a priori bound on the nonnegative solutions, then by using variational arguments, we prove the existence of at least two nonnegative nontrivial solutions. One of the main difficulties is that the nonlinearity term h(x,u) does not satisfy the standard Ambrosetti and Rabinowitz condition.

Fujita type results for a quasilinear parabolic inequality of Hardy-Hénon type with time forcing terms

We investigate a quasilinear parabolic inequality of Hardy-Hénon type with a time forcing term ut−divA(x,u,Du)≥tγ|x|σuq,x∈RN,t>0,u(x,t)≥0,x∈RN,t>0,u(x,0)=u0(x)≥0,x∈RN,where Du represents the gradient of u, |x| represents the length of x∈RN; the function A is a Carathéodory function such that A(x,z,0)=0, A(x,z,w)⋅w≥0 for every (x,z,w)∈RN×[0,∞)×RN, and on the operator A we require weak m-coercivity, namely A(x,z,w)⋅w≥c0|x|β|A(x,z,w)|mm−1,(x,z,w)∈RN×[0,∞)×RNholds for some c0>0, m≥2 , and 0≤β<Nm−1; γ>−1, σ>−N, and q>N(m−1)N−β(m−1). Based on nonlinear capacity estimates, we obtain the Fujita type results to the above problem, i.e., the conditions on the parameters γ,σ,q such that the above problem admits no nontrivial nonnegative solutions for arbitrary initial value u0.

A singular Liouville equation on planar domains

AbstractWe show the existence of a solution for an equation where the nonlinearity is logarithmically singular at the origin, namely, in with Dirichlet boundary condition. The function f has exponential growth, which can be subcritical or critical with respect to the Trudinger–Moser inequality. We study the energy functional corresponding to the perturbed equation , where is well defined at 0 and approximates . We show that has a critical point in , which converges to a legitimate nontrivial nonnegative solution of the original problem as . We also investigate the problem with replaced by , when the parameter is sufficiently large.

Nonnegative nontrivial solutions for a class of p(x)-Kirchhoff equation involving concave-convex nonlinearities

In this paper, we study the existence of a class of p(x)-Kirchhoff equation involving concave-convex nonlinearities. The main tools used are the perturbation technique, variational method, and a priori estimation.

Fujita-type theorems for a quasilinear parabolic differential inequality with weighted nonlocal source term

Abstract This work is concerned with the nonexistence of nontrivial nonnegative weak solutions for a quasilinear parabolic differential inequality with weighted nonlocal source term in the whole space, which involves weighted polytropic filtration operator or generalized mean curvature operator. We establish the new critical Fujita exponents containing the first and second types. The key ingredient of the technique in proof is the test function method developed by Mitidieri and Pohozaev. No use of comparison and maximum principles or assumptions on symmetry or behavior at infinity of the solutions are required.

The concentration-compactness principle for the nonlocal anisotropic [formula omitted]-Laplacian of mixed order

In this paper, we study the existence of minimizers of the Sobolev quotient for a class of nonlocal operators with an orthotropic structure having different exponents of integrability and different orders of differentiability. Our method is based on the concentration-compactness principle which we extend to this class of operators. One consequence of our main result is the existence of a nontrivial nonnegative solution to the corresponding critical problem.

Existence of nonnegative nontrivial solutions for Kirchhoff type problems with variable exponent

This paper is devoted to study a class of Kirchhoff type problems with variable exponent. By means of perturbation technique, variational methods and a priori estimation, the existence of nonnegative nontrivial solutions to this problem is obtained.

Existence of solutions for a double-phase variable exponent equation without the Ambrosetti-Rabinowitz condition

Abstract The article deals with the existence of a pair of nontrivial nonnegative and nonpositive solutions for a nonlinear weighted quasilinear equation in R N {{\mathbb{R}}}^{N} , which involves a double-phase general variable exponent elliptic operator A {\bf{A}} . More precisely, A {\bf{A}} has behaviors like ∣ ξ ∣ q ( x ) − 2 ξ {| \xi | }^{q\left(x)-2}\xi if ∣ ξ ∣ | \xi | is small and like ∣ ξ ∣ p ( x ) − 2 ξ {| \xi | }^{p\left(x)-2}\xi if ∣ ξ ∣ | \xi | is large. Existence is proved by the Cerami condition instead of the classical Palais-Smale condition, so that the nonlinear term f ( x , u ) f\left(x,u) does not necessarily have to satisfy the Ambrosetti-Rabinowitz condition.

Multiplicity and concentration of solutions to fractional anisotropic Schrödinger equations with exponential growth

In this paper, we consider the Schrödinger equation involving the fractional (p,p_1,dots ,p_m)-Laplacian as follows (-Δ)psu+∑i=1m(-Δ)pisu+V(εx)(|u|(N-2s)/2su+∑i=1m|u|pi-2u)=f(u)inRN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} (-\\Delta )_{p}^{s}u+\\sum _{i=1}^{m}(-\\Delta )_{p_i}^{s}u+V(\\varepsilon x)(|u|^{(N-2s)/2s}u+\\sum _{i=1}^{m}|u|^{p_i-2}u)=f(u)\\;\ ext{ in }\\; {\\mathbb {R}}^{N}, \\end{aligned}$$\\end{document}where varepsilon is a positive parameter, N=ps, sin (0,1), 2le p<p_1< dots< p_m<+infty , mge 1. The nonlinear function f has the exponential growth and potential function V is continuous function satisfying some suitable conditions. Using the penalization method and Ljusternik–Schnirelmann theory, we study the existence, multiplicity and concentration of nontrivial nonnegative solutions for small values of the parameter. In our best knowledge, it is the first time that the above problem is studied.

A Nonexistence Result for Choquard-Type Hamiltonian System

In this article, we establish a nonexistence result of nontrivial non-negative solutions for the following Choquard-type Hamiltonian system by the Pohožaev identity , when , , , , , and , where and denotes the convolution in .

Note on a parabolic problem with multi-coupled nonlinearity and space-time weighted functions

This paper deals with two semilinear parabolic equations involving multi-coupled nonlinearity, where the weighted functions depend on space and time variables. We determine the exponent regions where the nonnegative nontrivial solutions blow up for any initial data. Moreover, some exponent regions on the global solutions are given with some conditions on the initial data and the ten exponents.

Multiplicity Results for Weak Solutions of a Semilinear Dirichlet Elliptic Problem with a Parametric Nonlinearity

This paper deals with the existence of weak solutions to a Dirichlet problem for a semilinear elliptic equation involving the difference of two main nonlinearities functions that depends on a real parameter λ . According to the values of λ , we give both nonexistence and multiplicity results by using variational methods. In particular, we first exhibit a critical positive value such that the problem admits at least a nontrivial non-negative weak solution if and only if λ is greater than or equal to this critical value. Furthermore, for λ greater than a second critical positive value, we show the existence of two independent nontrivial non-negative weak solutions to the problem.

Hardy–Littlewood–Sobolev inequality and existence of the extremal functions with extended kernel

In this paper, we consider the following Hardy–Littlewood–Sobolev inequality with extended kernel (0.1)\begin{equation} \int_{\mathbb{R}_+^{n}}\int_{\partial\mathbb{R}^{n}_+} \frac{x_n^{\beta}}{|x-y|^{n-\alpha}}f(y)g(x) {\rm d}y{\rm d}x\leq C_{n,\alpha,\beta,p}\|f\|_{L^{p}(\partial\mathbb{R}_+^{n})} \|g\|_{L^{q'}(\mathbb{R}_+^{n})}, \end{equation}for any nonnegative functions $f\in L^{p}(\partial \mathbb {R}_+^{n})$, $g\in L^{q'}(\mathbb {R}_+^{n})$ and $p,\,\ q'\in (1,\,\infty )$, $\beta \geq 0$, $\alpha +\beta &gt;1$ such that $\frac {n-1}{n}\frac {1}{p}+\frac {1}{q'}-\frac {\alpha +\beta -1}{n}=1$.We prove the existence of all extremal functions for (0.1). We show that if $f$ and $g$ are extremal functions for (0.1) then both of $f$ and $g$ are radially decreasing. Moreover, we apply the regularity lifting method to obtain the smoothness of extremal functions. Finally, we derive the sufficient and necessary condition of the existence of any nonnegative nontrivial solutions for the Euler–Lagrange equations by using Pohozaev identity.

A Choquard‐type equation with a singular absorption nonlinearity in two dimensions

In this article, we show the existence of a nonnegative solution to the singular problem posed in a bounded domain in (see below). We achieve this by approximating the singular function by a function , which pointwisely converges to as . Using variational techniques, the perturbed equation is shown to have a solution when the parameter is small enough. Letting and proving a pointwise gradient estimate, we show that the solution converges to a nontrivial nonnegative solution of the original problem .