Nontrivial Nonnegative Solutions Research Articles

Combined effects in coupled quasilinear elliptic systems via a “like” Gagliardo–Nirenberg inequality

In this paper, we study the existence, nonexistence and multiplicity results to a class of quasilinear elliptic system involving ‐Laplacian by exploiting the competition of nonlinearities. We mainly propose a “like” Gagliardo–Nirenberg inequality for a pair of functions, and use it to combine the variational method to prove the existence, nonexistence and multiplicity of nontrivial nonnegative solutions for a quasilinear elliptic system involving ‐Laplacian under different conditions between parameters.

Nonuniqueness for fractional parabolic equations with sublinear power-type nonlinearity

We show that the parabolic equation ut+(−Δ)su=q(x)|u|α−1u posed in a time-space cylinder (0,T)×RN and coupled with zero initial condition and zero nonlocal Dirichlet condition in (0,T)×(RN∖Ω), where Ω is a bounded domain, has at least one nontrivial nonnegative finite energy solution provided α∈(0,1) and the nonnegative bounded weight function q is separated from zero on an open subset of Ω. This fact contrasts with the (super)linear case α≥1 in which the only bounded finite energy solution is identically zero.

Double phase anisotropic variational problems involving critical growth

Abstract In this study, we investigate some existence results for double phase anisotropic variational problems involving critical growth. We first establish a Lions-type concentration-compactness principle and its variant at infinity for the solution space, which are our independent interests. Using these results, we obtain a nontrivial nonnegative solution to problems of generalized concave-convex type. We also obtain infinitely many solutions when the nonlinear term is symmetric. Our results are new even for the p ( ⋅ ) p\left(\cdot ) -Laplace equations.

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.