Nontrivial Nonnegative Solutions Research Articles

Time Periodic Solutions for a Pseudo-parabolic Type Equation with Weakly Nonlinear Periodic Sources

In this paper, we prove the existence of nontrivial nonnegative time periodic solutions for a pseudo-parabolic type equation with weakly nonlinear periodic sources. Moreover, we investigate the asymptotic behavior of solutions as the viscous coefficient \(k\) tends to zero.

The effects of indefinite nonlinear boundary conditions on the structure of the positive solutions set of a logistic equation

We investigate a semilinear elliptic equation with a logistic nonlinearity and an indefinite nonlinear boundary condition, both depending on a parameter λ. Overall, we analyze the effect of the indefinite nonlinear boundary condition on the structure of the positive solutions set. Based on variational and bifurcation techniques, our main results establish the existence of three nontrivial non-negative solutions for some values of λ, as well as their asymptotic behavior. These results suggest that the positive solutions set contains an S-shaped component in some case, as well as a combination of a C-shaped and an S-shaped components in another case.

When does the heat equation have a solution with a sequence of similar level sets?

In this paper, we consider an overdetermined Cauchy problem for the heat equation. We prove that if the problem has a non-trivial nonnegative solution with a certain sequence of similar level sets, then the solution must be radially symmetric.

Cauchy problem for nonlinear parabolic equations with a gradient term

We consider the Cauchy problem of the nonlinear diffusive Hamilton–Jacobi equationut−Δum=|Duq|σ, where m>0. The existence and nonexistence of local and global solutions are studied by a priori estimates and compactness methods. We prove that the Cauchy problem of this equation has no nontrivial nonnegative global solutions if 0<σ<σ⁎ (here σ⁎=Nm+2qN+1); and there exist nontrivial global solutions for small initial values if σ>σ⁎. Moreover, we obtain the global existence of solutions in the limiting case and the exact L∞-estimates for local and global solutions.

Existence of entire solutions for a class of variable exponent elliptic equations

The paper deals with the existence of entire solutions for a quasilinear equation (Eλ) in RN, depending on a real parameter λ, which involves a general variable exponent elliptic operator A in divergence form and two main nonlinearities. The competing nonlinear terms combine each other. Under some conditions, we prove the existence of a critical value λ⁎>0 with the property that (Eλ) admits nontrivial nonnegative entire solutions if and only if λ≥λ⁎. Furthermore, under the further assumption that the potential A of A is uniform convex, we give the existence of a second independent nontrivial nonnegative entire solution of (Eλ), when λ>λ⁎. Our results extend the previous work of Autuori and Pucci (2013) [6] from the case of constant exponents p, q and r to the case of variable exponents. More interesting, we weaken the condition max{2,p}<q<min{r,p⁎} to the simple request that 1≪q≪r. Furthermore, we extend the previous work of Alama and Tarantello (1996) [2] from Dirichlet Laplacian problems in bounded domains of RN to the case of a general variable exponent differential equation in the entire RN, and also remove the assumption q>2. Hence the results of this paper are new even in the canonical case p(⋅)≡2.

Minimal energy solutions for repulsive nonlinear Schrödinger systems

In this paper we establish existence and nonexistence results concerning fully nontrivial minimal energy solutions of the nonlinear Schrödinger system−Δu+u=|u|2q−2u+b|u|q−2u|v|qin Rn,−Δv+ω2v=|v|2q−2v+b|u|q|v|q−2vin Rn. We consider the repulsive case b<0 and assume that the exponent q satisfies 1<q<nn−2 in case n≥3 and 1<q<∞ in case n=1 or n=2. For space dimensions n≥2 and arbitrary b<0 we prove the existence of fully nontrivial nonnegative solutions which converge to a solution of some optimal partition problem as b→−∞. In case n=1 we prove that minimal energy solutions exist provided the coupling parameter b has small absolute value whereas fully nontrivial solutions do not exist if 1<q≤2 and b has large absolute value.

Two solutions for an elliptic equation with fast increasing weight and concave–convex nonlinearities

We prove the existence of two nonnegative nontrivial solutions for the equation−div(K(x)∇u)=a(x)K(x)|u|q−2u+b(x)K(x)|u|p−2u,x∈RN, where N⩾3, K(x)=exp(|x|α/4), α⩾2 and the potentials a and b have indefinite sign and satisfy some mild integrability conditions. The results hold when a has small norm in a suitable weighted Lebesgue space.

MULTIPLE SOLUTIONS FOR A p-LAPLACIAN SYSTEM WITH NONLINEAR BOUNDARY CONDITIONS

A nonlinear elliptic problem involving p-Laplacian and nonlinear boundary condition is considered in this paper. By using the method of Nehari manifold, it is proved that the system possesses two nontrivial nonnegative solutions if the parameter is small enough.

Liouville type theorem for higher order Hardy-Hénon system of inequalities

In this paper, we prove some new Liouville type theorems for fourth order and more general higher order Hardy-Henon systems of inequalities. The test function method is applied to show the nonexistence of nontrivial nonnegative global solutions, which improve and extend some recent results. Mathematics subject classification (2010): 35K41, 35B53, 35B08.

On the Hénon-Lane-Emden conjecture

We consider Liouville-type theorems for the following Hénon-Lane-Emden system\begin{eqnarray*} \left\{ \begin{array}{lcl} -\Delta u&=& |x|^{a}v^p \ \ in\ \ \mathbb{R}^n,\\ -\Delta v&=& |x|^{b}u^q \ \ in\ \ \mathbb{R}^n,\end{array}\right. \end{eqnarray*}when $p,q \ge 1,$ $pq\neq1$, $a,b\ge0$. The main conjecture states that there is no non-trivial non-negative solution whenever $(p,q)$ is under the critical Sobolev hyperbola, i.e.$ \frac{n+a}{p+1}+\frac{n+b}{q+1}>{n-2}$. We show that this is indeed the case in dimension $n=3$ provided the solution is also assumed to be bounded, extending a result established recently by Phan-Souplet in the scalar case. Assuming stability of the solutions, we could then prove Liouville-type theorems in higher dimensions. For the scalar cases, albeit of second order ($a=b$ and $p=q$) or of fourth order ($a\ge 0=b$ and $p>1=q$), we show that for all dimensions $n\ge 3$ in the first case (resp., $n\ge 5$ in the second case), there is no positive solution with a finite Morse index, whenever $p$ is below the corresponding critical exponent, i.e $ 1< p < \frac{n+2+2a}{n-2}$ (resp., $ 1< p < \frac{n+4+2a}{n-4}$). Finally, we show that non-negative stable solutions of the full Hénon-Lane-Emden system are trivial provided \begin{equation*}\label{sysdim00} n < 2 + 2 (\frac{p(b+2)+a+2}{pq-1}) (\sqrt{\frac{pq(q+1)}{p+1}} + \sqrt{ \frac{pq(q+1)}{p+1} - \sqrt{\frac{pq(q+1)}{p+1}}}). \end{equation*}

Extinction properties of solutions for a fast diffusion equation with nonlocal source

In this paper, we investigate extinction properties of nonnegative nontrivial solutions for an initial boundary value problem of a fast diffusion equation with a nonlocal source in bounded domain. By using the super- and sub-solution and the energy methods, we obtain some sufficient conditions for extinction and non-extinction of the weak solutions and give corresponding decay estimates which depend on the initial data, coefficients, and domains.MSC:35K65, 35K20, 35B40.

Periodic solutions for a class of diffusive Nicholson’s blowflies model with Dirichlet boundary conditions

Abstract In this paper, we study the problem of periodic solutions for a class of diffusive Nicholson’s blowflies model with Dirichlet boundary conditions. By applying the Schauder fixed point theorem, the existence of nontrivial nonnegative periodic solutions of the considered model is established. Our results complement with some recent ones. MSC:35B05, 35B10.

Qualitative properties of a cooperative degenerate Lotka-Volterra model

Abstract This paper considers a kind of degenerate parabolic systems. First, we consider the initial boundary value problem of the two-species degenerate parabolic cooperative system. By using the method of a parabolic regularization and energy estimate, we establish the existence of the weak solution of the problem. Then we establish the comparison principle and discuss the uniqueness and the uniform bound. At last, we consider the periodic boundary value problem of the system. By constructing a pair of ordered upper and lower solutions, we establish the existence of nontrivial nonnegative periodic solutions.

Solution for a Class of Quasilinear Elliptic Equation with p-Laplacian

In this paper, a class of elliptic equation with convex-concave nonlinear term is considered. By the variational methods, we give a sufficient condition which insures that the problem considered in the present paper has at a nontrivial nonnegative solution.

The Nehari Manifold and its Application to a Fractional Boundary Value Problem

In this paper, we study the Nehari manifold and its application to the following fractional boundary value problem: $$\begin{aligned} {\left\{ \begin{array}{ll} - \frac{d}{d t} \Big (\frac{1}{2} {}_0D_t^{- \beta } (u^{\prime } (t)) + \frac{1}{2} {}_tD_T^{- \beta } (u^{\prime } (t)) \Big )\\ \quad = \lambda f (t) (u (t))^{p - 1} + g (t) (u(t))^{q - 1}, &{} \mathrm{a.e.}\;\; t \in [0, T],\\ u (0) = u (T) = 0, \end{array}\right. } \end{aligned}$$ where \({}_0D_t^{- \beta }\) and \({}_tD_T^{- \beta }\) are the left and right Riemann–Liouville fractional integrals of order \(0 \le \beta < 1\), respectively. We prove that the problem has at least two nontrivial non-negative solutions when the parameter \(\lambda \) belongs to a certain subset of \(\mathbb{R }\).

Elliptic problems involving the fractional Laplacian in [formula omitted

We study the existence and multiplicity of solutions for elliptic equations in RN, driven by a non-local integro-differential operator, which main prototype is the fractional Laplacian. The model under consideration, denoted by (Pλ), depends on a real parameter λ and involves two superlinear nonlinearities, one of which could be critical or even supercritical. The main theorem of the paper establishes the existence of three critical values of λ which divide the real line in different intervals, where (Pλ) admits no solutions, at least one nontrivial non-negative entire solution and two nontrivial non-negative entire solutions.

N-Laplacian Equations in ℝN with Subcritical and Critical GrowthWithout the Ambrosetti-Rabinowitz Condition

Abstract Let Ω be a bounded domain in ℝN. In this paper, we consider the following nonlinear elliptic equation of N-Laplacian type: when f is of subcritical or critical exponential growth. This nonlinearity is motivated by the Moser-Trudinger inequality. In fact, we will prove the existence of a nontrivial nonnegative solution to (0.1) without the Ambrosetti-Rabinowitz (AR) condition. Earlier works in the literature on the existence of nontrivial solutions to N−Laplacian in ℝN when the nonlinear term f has the exponential growth only deal with the case when f satisfies the (AR) condition. Our approach is based on a suitable version of the Mountain Pass Theorem introduced by G. Cerami [9, 10, 21]. Examples of f and comparison with earlier assumptions in the literature are given.

Periodic solution of a quasilinear parabolic equation with nonlocal terms and Neumann boundary conditions

Abstract In this article, we study the periodic solution of a quasilinear parabolic equation with nonlocal terms and Neumann boundary conditions. By using the theory of Leray-Schauder degree, we obtain the existence of a nontrivial nonnegative time periodic solution.

Finite-time quenching of competing species with constrained boundary evaporation

We consider a class of SKT type reaction-cross diffusion models with vanishing random diffusion coefficients.For homogeneous Dirichlet boundary conditions we prove non-existence of global-in-time non-trivial non-negative smooth solutions. Some numerical results are also presented, suggesting the possibility of finite-time extinction.

Multiple Solutions for a Class of Concave-Convex Quasilinear Elliptic Systems with Nonlinear Boundary Condition

In this paper, a quasilinear elliptic system is investigated, which involves concave-convex nonlinearities and nonlinear boundary condition. By Nehari manifold, fibering method and analytic techniques, the existence of multiple nontrivial nonnegative solutions to this equation is verified.