Constant Sign Solutions Research Articles

Existence results of Brezis-Browder type for systems of Fredholm integral equations

In this article, we consider the following systems of Fredholm integral equations: Using an argument originating from Brezis and Browder [Bull. Am. Math. Soc. 81, 73-78 (1975)] and a fixed point theorem, we establish the existence of solutions of the first system in (C[0, T]) n , whereas for the second system, the existence criteria are developed separately in (C l [0,∞)) n as well as in (BC[0,∞)) n . For both systems, we further seek the existence of constant-sign solutions, which include positive solutions (the usual consideration) as a special case. Several examples are also included to illustrate the results obtained. 2010 Mathematics Subject Classification: 45B05; 45G15; 45M20.

Constant sign and nodal solutions for logistic-type equations with equidiffusive reaction

We consider a logistic-type equation driven by the p-Laplace differential operator with an equidiffusive reaction term. Combining variational methods based on critical point theory together with truncation techniques and Morse theory, we show that when λ > λ1, the problem has extremal solutions of constant sign and when λ > λ2 it has also a nodal (sign-changing) solution. Here λ1 < λ2 are the first two eigenvalues of the negative Dirichlet p-Laplacian. In the semilinear case (i.e. p = 2) we produce two nodal solutions.

Multiple solution results for elliptic Neumann problems involving set-valued nonlinearities

The main goal of this paper is to present multiple solution results for elliptic inclusions of Clarke's gradient type under nonlinear Neumann boundary conditions involving the p-Laplacian and set-valued nonlinearities. To be more precise, we study the inclusion−Δpu∈∂F(x,u)−|u|p−2uin Ω with the boundary condition|∇u|p−2∂u∂ν∈a(u+)p−1−b(u−)p−1+∂G(x,u)on ∂Ω. We prove the existence of two constant-sign solutions and one sign-changing solution depending on the parameters a and b. Our approach is based on truncation techniques and comparison principles for elliptic inclusions along with variational tools like the nonsmooth Mountain-Pass Theorem, the Second Deformation Lemma for locally Lipschitz functionals as well as comparison results of local C1(Ω¯)-minimizers and local W1,p(Ω)-minimizers of nonsmooth functionals.

Sign-changing and constant-sign solutions for p-Laplacian problems with jumping nonlinearities

By variational methods, we prove the existence of a sign-changing solution for the p-Laplacian equation under Dirichlet boundary condition with jumping nonlinearity having relation to the Fučík spectrum of p-Laplacian. We also provide the multiple existence results for the p-Laplacian problems.

Constant Sign and Sign Changing Solutions for Systems of Quasilinear Elliptic Equations

We establish the existence of two opposite constant sign solutions for a general noncoercive quasilinear elliptic system with homogeneous Dirichlet boundary conditions. In the case where the system has a variational structure, by strengthening the hypotheses, we obtain a third nontrivial solution which is sign changing in the sense that one cannot have both components of the new solution of the same constant sign. Our approach relies on a suitable method of sub-supersolutions combined with truncation and variational arguments that does not require a subcritical growth condition.

Multiple and sign-changing solutions for the multivalued p -Laplacian equation

We consider a class of elliptic inclusions under Dirichlet boundary conditions involving multifunctions of Clarke's generalized gradient. Under conditions given in terms of the first eigenvalue as well as the Fučik spectrum of the p -Laplacian we prove the existence of a positive, a negative and a sign-changing solution. Our approach is based on variational methods for nonsmooth functionals (nonsmooth critical point theory, second deformation lemma), and comparison principles for multivalued elliptic problems. In particular, the existence of extremal constant-sign solutions plays a key role in the proof of sign-changing solutions (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Constant-sign solutions for singular systems of Fredholm integral equations

We consider the system of Fredholm integral equations u i (t)= ∫ T 0 g i (t,s)[H i (s, u 1 (s), u 2 (s),...,u n (s))+K i (s, u 1 (s), u 2 (s),...,u n (s))]ds, t∈[0, T], 1≤i≤n where T>0 is fixed and the nonlinearities H i (t, u 1 , u 2 ,..., u n ) can be singular at t=0 and u j =0 where j ∈{1, 2,..., n}. Criteria are offered for the existence of constant-sign solutions, i.e. θ i u i (t)≥0 for t∈[0, 1] and 1 ≤i≤n, where θ i ∈{1, ―1} is fixed. We also include an example to illustrate the usefulness of the results obtained.

Maximum principles and nonexistence results for radial solutions to equations involvingp-Laplacian

We obtain the variant of maximum principle for radial solutions of, possibly singular, p-harmonic equations of the form as well as for solutions of the related ODE. We show that for the considered class of equations local maxima of |w| form a monotone sequence in |x| and constant sign solutions are monotone. The results are applied to nonexistence and nonlinear eigenvalue problems. We generalize our previous work for the case h≡0. Copyright © 2010 John Wiley & Sons, Ltd.

Sign-Changing and Extremal Constant-Sign Solutions of Nonlinear Elliptic Neumann Boundary Value Problems

Our aim is the study of a class of nonlinear elliptic problems under Neumann conditions involving the -Laplacian. We prove the existence of at least three nontrivial solutions, which means that we get two extremal constant-sign solutions and one sign-changing solution by using truncation techniques and comparison principles for nonlinear elliptic differential inequalities. We also apply the properties of the Fu ik spectrum of the -Laplacian and, in particular, we make use of variational and topological tools, for example, critical point theory, Mountain-Pass Theorem, and the Second Deformation Lemma.

Periodic constant-sign solutions for systems of Hill's equations

We consider the system of Hill's equations u i (t) + ai(t)ui(t) = Fit, u1(t), u2(t), ... , un(t) � ,1 i n,

Multiple solutions for inequality Dirichlet problems by the [formula omitted]-Laplacian

In this paper we study the nonlinear elliptic problem driven by p ( x ) -Laplacian with a nonsmooth locally Lipschitz potential (hemivariational inequality), that is (P) { − div ( ‖ ∇ u ( x ) ‖ R N p ( x ) − 2 ∇ u ( x ) ) ∈ ∂ j ( x , u ( x ) ) , a.a. x ∈ Ω , u = 0 , on ∂ Ω , where Ω ⊂ R N is a bounded domain and p : Ω ¯ → R is a continuous function satisfying some given assumptions. The approach used in this paper is the variational method for locally Lipschitz functions. More precisely, Weierstrass Theorem and Mountain Pass Theorem are used to prove the existence of at least two nontrivial solutions. Finally, we obtain the existence of at least two nontrivial solutions of constant sign.

Multiple positive solutions for a class of [formula omitted]-superlinear elliptic problems

We study nonlinear elliptic Dirichlet problems driven by the p -Laplacian and with a nonlinearity which consists of a p -superlinear term perturbed by a p -linear one. Using critical point theory, coupled with suitable truncation techniques, we show that the problem has two positive smooth solutions. When the conditions are symmetric in the two semiaxes, we have four nontrivial smooth solutions of constant sign. Finally for the semilinear problem, using Morse theory, we produce a fifth nontrivial smooth solution.

On a variant of the maximum principle involving radial $p$-Laplacian with applications to~nonlinear eigenvalue problems and nonexistence results

We obtain the variant of maximum principle for radial solutions of $p$-harmonic equation $-a\Delta_p(w)=\phi(w)$. As a consequence of this result we prove monotonicity of constant sign solutions, analyze the support of the solutions and study their oscillations. The results are applied to various type nonlinear eigenvalue problems and nonexistence theorems.

Existence of constant sign solutions for the p-Laplacian problems in the resonant case with respect to Fučík spectrum

We consider the following the p-Laplacian equation in a bounded domain Ω: { −Δpu=f(x,u)in Ω,u=0on ∂Ω. We treat the case of nonlinearity term f satisfying the following conditions f(x,t)={ a0t+p−1−b0t−p−1+o(|t|p−1)at 0,at+p−1−bt−p−1+o(|t|p−1)at ∞, for constants a0,b0,a and b. We prove the existence of a positive solution or a negative solution in the case of (a0−λ1)(a−λ1)=0 or (b0−λ1)(b−λ1)=0 respectively, where λ1 is the first eigenvalue of −Δp.

Nodal and multiple constant sign solutions for resonant [formula omitted]-Laplacian equations with a nonsmooth potential

In this paper we study a nonlinear Dirichlet elliptic differential equation driven by the p -Laplacian and with a nonsmooth potential (hemivariational inequality). Using a variational approach combined with suitable truncation techniques and the method of upper–lower solutions, we prove the existence of five nontrivial smooth solutions, two positive, two negative and the fifth nodal. Our hypotheses on the nonsmooth potential allow resonance at infinity with respect to the principal eigenvalue λ 1 > 0 of ( − Δ p , W 0 1 , p ( Z ) ) .

Constant-sign solutions for systems of singular integral equations of Hammerstein type

We consider the system of Hammerstein integral equations u i ( t ) = ∫ 0 T g i ( t , s ) f i ( s , u 1 ( s ) + ρ 1 ( s ) , u 2 ( s ) + ρ 2 ( s ) , … , u n ( s ) + ρ n ( s ) ) d s , t ∈ [ 0 , T ] , 1 ≤ i ≤ n where T > 0 is fixed, ρ i ’s are given functions and the nonlinearities f i ( t , x 1 , x 2 , … , x n ) can be singular at t = 0 and x j = 0 where j ∈ { 1 , 2 , ⋯ , n } . Criteria are offered for the existence of constant-sign solutions, i.e., θ i u i ( t ) ≥ 0 for t ∈ [ 0 , T ] and 1 ≤ i ≤ n , where θ i ∈ { 1 , − 1 } is fixed. The tools used are a nonlinear alternative of Leray–Schauder type, Krasnosel’skii’s fixed point theorem in a cone and Schauder’s fixed point theorem. We also include examples and applications to illustrate the usefulness of the results obtained.

Existence of five nonzero solutions with exact sign for a $p$-Laplacian equation

We consider nonlinear elliptic problems driven by the $p$-Laplacian with a nonsmooth potential depending on a parameter $\lambda > 0$. The main result guarantees the existence of two positive, two negative and a nodal (sign-changing) solution for the studied problem whenever $\lambda > 0$ belongs to a small interval (0, λ*) and $p$ ≥ 2. We do not impose any symmetry hypothesis on the nonlinear potential. The constant-sign solutions are obtained by using variational techniques based on nonsmooth critical point theory (minimization argument, Mountain Pass theorem, and a Brezis-Nirenberg type result for C1-minimizers), while the nodal solution is constructed by an upper-lower solutions argument combined with the Zorn lemma and a nonsmooth second deformation theorem.

Constant Sign and Nodal Solutions for Problems with the -Laplacian and a Nonsmooth Potential Using Variational Techniques

We consider a nonlinear elliptic equation driven by the -Laplacian with a nonsmooth potential (hemivariational inequality) and Dirichlet boundary condition. Using a variational approach based on nonsmooth critical point theory together with the method of upper and lower solutions, we prove the existence of at least three nontrivial smooth solutions: one positive, the second negative, and the third sign changing (nodal solution). Our hypotheses on the nonsmooth potential incorporate in our framework of analysis the so-called asymptotically -linear problems.

Multiple Constant Sign and Nodal Solutions for Superlinear Elliptic Equations

We consider semilinear elliptic problems with a superlinear right hand side nonlinearity, which however, need not satisfy the Ambrosetti-Rabinowitz condition. Using a combination of variational methods, with Morse theory (critical groups) and truncation techniques, we prove multiplicity theorems providing precise sign information for the solutions. We show that the problem can have seven and eight solutions, all with prescribed sign.

Constant-Sign Solutions of a System of Urysohn Integral Equations

We consider the following system of Urysohn integral equations By using fixed point theorems and fixed point index theory, criteria are established for the existence of single, double and multiple solutions of the system that are of constant signs, i.e., for each 1 ≤ i ≤ n, u i (t) ≥ 0 or u i (t) ≤ 0 for t ∈ [0,1]. We also consider the eigenvalue problem where criteria are derived such that for λ in some interval, the existence of a constant-sign solution is guaranteed.