Search

We study the following nonlinear elliptic problem { − Δ u + m u = g ( x , u ) + f ( x ) in R N ∖ B ¯ R , u = h on ∂ B R , u → 0 as ∣ x ∣ → + ∞ , with N ≥ 3 , m ∈ R + ∗ , under suitable conditions on the radial functions g , f and h . Multiplicity results are proved in the perturbed case f ≢ 0 and h ≢ 0 by using Bolle’s perturbation method and suitable growth estimates on min–max critical levels.

In this paper we consider the problem of prescribing the mean curvature on the boundary of the unit ball of R n , n ≥ 3 . Under the assumption that the prescribed function is flat near its critical point, we give precise estimates on the losses of the compactness and we provide new existence and multiplicity results of Bahri–Coron type.

We study global bifurcation diagrams and exact multiplicity of positive solutions for the one-dimensional prescribed mean curvature problem arising in MEMS { − ( u ′ ( x ) 1 + ( u ′ ( x ) ) 2 ) ′ = λ ( 1 − u ) p , u 1 , − L x L , u ( − L ) = u ( L ) = 0 , where λ > 0 is a bifurcation parameter, and p , L > 0 are two evolution parameters. We determine the exact number of positive solutions by the values of p , L and λ . Moreover, for p ≥ 1 , the bifurcation diagram undergoes fold and splitting bifurcations. While for 0 p 1 , the bifurcation diagram undergoes fold, splitting and segment-shrinking bifurcations. Our results extend and improve those of Brubaker and Pelesko [N.D. Brubaker, J.A. Pelesko, Analysis of a one-dimensional prescribed mean curvature equation with singular nonlinearity, Nonlinear Anal. 75 (2012) 5086–5102] and Pan and Xing [H. Pan, R. Xing, Exact multiplicity results for a one-dimensional prescribed mean curvature problem related to a MEMS model, Nonlinear Anal. RWA 13 (2012) 2432–2445] by generalizing the nonlinearity ( 1 − u ) − 2 to ( 1 − u ) − p with general p ∈ ( 1 , ∞ ) . We also answer an open question raised by Brubaker and Pelesko on the extension of (global) bifurcation diagram results to general p > 0 . Concerning this open question, we find and prove that global bifurcation diagrams for 0 p 1 are different to and more complicated than those for p ≥ 1 .

Variational techniques are used to get some existence and multiplicity results for some eigenvalue problems involving variable exponents.

In the present paper, we consider the following critical growth quasilinear elliptic system { − Δ p u i = 1 p ∗ D u i F ( u ) + λ i | u i | q − 2 u i in Ω , i = 1 , … , k , u i = 0 on ∂ Ω , i = 1 , … , k . By exploiting the Nehari manifold and variational methods, we obtain the existence and multiplicity results of nontrivial solutions for the system.

This paper is concerned with the periodic boundary value problem (1) { u ( t ) + λ u ( t − τ ) = ∇ F ( t , u ( t − τ ) ) a.e. t ∈ [ 0 , 2 τ ] u ( 0 ) − u ( 2 τ ) = u ( 0 ) − u ( 2 τ ) = 0 , where τ > 0 is a given constant, λ ∈ R is a parameter. The variational principle is given and some multiplicity results of periodic solutions of (1) are obtained directly via variational methods.

While the nonlinearities are bounded below, we establish the existence and multiplicity results of doubly periodic solutions for a coupled nonlinear telegraph system with a parameter. The proof is based on a well known fixed theorem in a cone.

In this paper we complete two tasks. We first extend the critical point theorem obtained by Li and Willem [S.J. Li, M. Willem, Applications of local linking to critical point theory, J. Math. Anal. Appl. 189 (1995), 6–32] to the nonsmooth case in which the energy functional is locally Lipschitz and satisfies the weaker nonsmooth Cerami condition. Then we study some semilinear elliptic systems with discontinuous nonlinearities and obtain some existence and multiplicity results.

In the present paper, we are concerned with some degenerate quasilinear equations involving variable exponents. Using various (variational and nonvariational) techniques, we prove existence, nonexistence and multiplicity results.

In this paper, we study a class of quasilinear elliptic exterior problems with nonlinear boundary conditions. Existence of ground states and multiplicity results are obtained via variational methods.

In this paper, we deal with the sublinear elliptic equation: − Δ u + V ( x ) u + a ( x ) | u | p s g n ( u ) = f on R N , N > 2 , 0 p 1 . Under suitable assumptions on the terms V and a , we prove some existence, uniqueness and multiplicity results. Continuity of solutions in the perturbation parameter f at 0 is also studied. Our main tools are the concentration-compactness principle and mountain-pass theorem.

We investigate various properties of time maps for one-dimensional prescribed mean curvature equations. Using these properties, we obtain some exact multiplicity results of positive solutions and sign-changing solutions. As it turned out, these quasilinear problems show many different phenomena from semilinear problems. Our methods are based on a detailed analysis of time maps.

Some multiplicity results of solutions are obtained for a class of elliptic systems which are near resonance with a nonprincipal eigenvalue by the classical saddle point theorem and a local saddle point theorem in critical point theory.

Existence and multiplicity results are obtained for superlinear p -Laplacian equations without the Ambrosetti and Rabinowitz condition. To overcome the difficulty that the Palais–Smale sequences of the Euler–Lagrange functional may be unbounded, we consider the Cerami sequences. Our results extend the recent results of Miyagaki and Souto [O. Miyagaki, M. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Differential Equations 245 (2008) 3628–3638].

In this paper we prove the existence of at least three solutions for a class of two-point boundary value systems of the type { − u i ″ + u i = λ F u i ( x , u 1 , … , u n ) + μ G u i ( x , u 1 , … , u n ) , u i ′ ( a ) = u i ′ ( b ) = 0 for 1 ≤ i ≤ n . The approach is fully based on a recent three critical points theorem of Ricceri [B. Ricceri, A three critical points theorem revisited, Nonlinear Anal., 70 (9) (2009) 3084–3089].

We establish some multiplicity results for a class of boundary value problems involving the Hardy-Sobolev operator using Morse theory. (C) 2009 Elsevier Ltd. All rights reserved.

A recently established three-critical-points theorem of Ricceri relating to continuously Gâteaux differentiable functionals, based on a minimax inequality and on a truncation argument, is extended to Motreanu–Panagiotopoulos functionals. As an application, multiplicity results, also yielding a uniform bound on the norms of solutions, are obtained for variational–hemivariational inequalities depending on two parameters.

The main purpose of this paper is to study the following damped vibration problems (1) { u ( t ) + ( q ( t ) I N × N + B ) u ( t ) + ∇ F ( t , u ( t ) ) = A ( t ) u ( t ) − 1 2 q ( t ) B u ( t ) , a.e. t ∈ [ 0 , T ] u ( 0 ) − u ( T ) = u ( t ) − u ( T ) = 0 where T > 0 , q ∈ L 1 ( 0 , T ; R ) with ∫ 0 T q ( t ) d t = 0 , A ( t ) = [ a i j ( t ) ] is a symmetric N × N matrix-valued function defined in [ 0 , T ] with a i j ∈ L ∞ ( [ 0 , T ] ) for all i , j = 1 , 2 , … , N , B = [ b i j ] is an antisymmetry N × N constant matrix. By establishing a proper variational set, one existence result and two multiplicity results of non-trivial periodic solutions of (1) are obtained.

We study the existence of multiple constant sign and nodal (sign-changing) solutions for semilinear elliptic equations. Our approach is based on a combination of variational techniques using critical point theory, of the method of upper-lower solutions and of Morse theory (critical groups). We prove several multiplicity results. Our analytical framework incorporates both coercive and superlinear parametric problems.

In this paper, we consider the nonlinear eigenvalue problems u ″ + λ h ( t ) f ( u ) = 0 , 0 t 1 , u ( 0 ) = 0 , u ( 1 ) = ∫ 0 1 u ( s ) d A ( s ) , where ∫ 0 1 u ( s ) d A ( s ) is a Stieltjes integral with A nondecreasing and A ( t ) is not a constant on ( 0 , 1 ) ; h ∈ C ( ( 0 , 1 ) , [ 0 , ∞ ) ) and h ( t ) ≢ 0 on any subinterval of ( 0 , 1 ) ; f ∈ C ( [ 0 , ∞ ) , [ 0 , ∞ ) ) and f ( s ) > 0 for s > 0 , and f 0 = f ∞ = 0 , f 0 = lim s → 0 + f ( s ) / s , f ∞ = lim s → + ∞ f ( s ) / s . We investigate the global structure of positive solutions by using global bifurcation techniques.

In this paper, we prove the existence of four nontrivial solutions of − Δ u − λ | x | 2 u = | u | 2 ∗ − 2 u + μ | x | α − 2 u + f ( x ) , x ∈ Ω ∖ { 0 } and show that at least one of them is sign changing. Our results extend some previous works, such as [G. Tarantello, Multiplicity results for an inhomogeneous Neumann problem with critical exponent, Manuscripta Math., 81 (1993) 51–78; D. Kang and Y. Deng, Multiple solutions for inhomogeneous elliptic problems involving critical Sobolev–Hardy exponents, J. Math. Anal. Appl., 60 (2005) 729–753; N. Hirano and N. Shioji, A multiplicty result including a sign changing solution for an inhomogeneous Neumann problem with critical exponent, Proc. Roy. Soc. Edinburgh, 137A (2007) 333–347.]

In this paper, we consider the existence and multiplicity of solutions for some Dirichlet impulsive problems and some existence and multiplicity results are obtained. The solutions are sought by means of Lax–Milgram theorem and some critical theorems.

We study existence and multiplicity of solutions for both Dirichlet and Neumann two-point boundary value problems at resonance. We obtain a detailed picture of the solution set, which, in particular, provides an effective way to compute all of the solutions. Our multiplicity results range from uniqueness to infinite multiplicity. Our approach can be seen as a dynamical version of the classical Liapunov–Schmidt procedure. After decomposing the space, we perform continuation in the subspace orthogonal to the kernel.

In this paper, we consider a quasilinear elliptic system with both concave–convex nonlinearities and critical growth terms in bounded domains. The existence and multiplicity results of positive solutions are obtained by variational methods.

In this paper we consider nonlinear periodic systems driven by the nonhomogeneous vector p -Laplacian differential operator and with a nonsmooth potential. Using variational techniques, based on the nonsmooth critical point theory, we prove existence and multiplicity results. Finally, we obtain an existence result for systems driven by the ordinary vector p -Laplacian and we show that it is still true in the more general framework of the previous results.

In this paper, we prove the existence of at least three weak solutions for the quasilinear elliptic systems { Δ p 1 u 1 + λ F u 1 ( x , u 1 , u 2 , … , u n ) = 0 in Ω , Δ p 2 u 2 + λ F u 2 ( x , u 1 , u 2 , … , u n ) = 0 in Ω , … Δ p n u n + λ F u n ( x , u 1 , u 2 , … , u n ) = 0 in Ω , u i = 0 for 1 ≤ i ≤ n on ∂ Ω . Our main tool is a recent three critical points theorem of Ricceri [B. Ricceri, On a three critical points theorem, Arch. Math. (Basel) 75 (2000) 220–226].

We establish the existence and multiplicity results for the one-dimensional prescribed mean curvature equation − ( u ′ / 1 + ( u ′ ) 2 ) ′ = λ e u , u ( 0 ) = 0 , u ( L ) = 0 . In particular, we characterize the precise dependence of the multiplicity of solutions with respect to both the length L of the interval and the value of the parameter λ .

In this paper, using B. Ricceri’s three critical points theorem, we prove the existence of at least three weak solutions for the problem { u ″ + λ h ( u ′ ) f ( x , u ) = 0 in ( a , b ) , u ( a ) = u ( b ) = 0 , under appropriate hypotheses.

In this paper, under the condition of two pairs of strict lower and upper solutions and using the concept of ( e 1 , B ) -limit increasing operator, some multiplicity results for an operator equation are obtained by the method of the fixed point index.

In this paper, existence and multiplicity results for solutions are obtained for the fourth-order boundary value problem ( BVP ) u ( 4 ) ( t ) + η u ″ ( t ) − ξ u ( t ) = λ f ( t , u ( t ) ) , 0 t 1 , u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 , where f : [ 0 , 1 ] × R → R is continuous, ξ , η ∈ R and λ ∈ R + are parameters. By using the critical point theory and Morse theory, we obtain that if ξ , η satisfy ξ π 4 + η π 2 1 , then the above BVP has solutions where λ is in some different intervals.

In this paper we obtain using Leray–Schauder degree theory some multiplicity results for sign-changing solutions of a four-point boundary value problem. We assume the existence of a pair of strict lower and upper solutions and some additional conditions on the nonlinear term f .

In this paper we consider the following problem: (⋆) { − Δ u ( x ) + u ( x ) = λ ( f ( x , u ) + h ( x ) ) in R N , u ∈ H 1 ( R N ) , u > 0 in R N , where λ > 0 is a parameter. We assume lim | x | → ∞ f ( x , u ) = f ( u ) uniformly on any compact subset of [ 0 , ∞ ) , but we do not require f ( x , u ) ≥ f ( u ) for all x ∈ R N . We prove that there exists + ∞ > λ ∗ > 0 such that (⋆) has exactly two positive solutions for λ ∈ ( 0 , λ ∗ ) , no solution for λ > λ ∗ , a unique positive solution u ∗ for λ = λ ∗ , and ( λ ∗ , u ∗ ) is a bifurcation point in C 2 , α ( R N ) ∩ W 2 , 2 ( R N ) .

In this paper, we first obtain some results on the structure of solution sets of some m -point boundary value problems. Then using these results, we obtain some multiplicity results for nodal solutions of m -point boundary value problems. The method to show our main results is based on the structure of solution sets due to P.H. Rabinowitz.

Using Leray-Schauder degree theory we obtain various existence and multiplicity results for nonlinear periodic boundary-value problems of the form (phi(u'))' = f(t, u, u'), u(0) - u(T) = 0 = u'(0) - u'(T), where phi : R -->] -a, a [ (0 R is a continuous function. (C) 2007 Elsevier Ltd. All rights reserved.

In this paper we consider nonlinear periodic systems driven by the ordinary p-Laplacian and having a nonsmooth potential function. Under minimal and natural hypotheses on the nonsmooth potential and using variational methods based on the nonsmooth critical point theory we prove four existence theorems and a multiplicity theorem. We conclude the paper with an existence theorem for the scalar problem, in which the energy functional is indefinite (i.e. it is unbounded both above and below).

In this paper we study the elliptic system of gradient type with nonstandard growth conditions in R N . Existence and multiplicity results, under two growth conditions on the reaction terms, are established.

The purpose of this work is to investigate the weakened Ambrosetti–Prodi type multiplicity results for weak doubly periodic solutions of damped beam equations. By using the topological degree theory, the author obtains a result which is similar to the result for damped wave equations in the literature.

In this work, we consider an elliptic system of two equations in dimension greater than one, with nonlinearities which are linear at − ∞ and superlinear at + ∞ . We prove, by variational techniques which involve a strongly indefinite functional, the existence of two solutions for suitable forcing terms, under a condition on the linear part which prevents resonance with the eigenvalues of the operator.

In the present paper, we consider a class of strong resonance problems for nonlinear elliptic Neumann BVPs, and establish existence and multiplicity results for positive, negative and sign-changing solutions by using a Morse theoretical approach.

In this paper we prove the existence of at least three weak solutions for the problem { Δ p u + λ f ( x , u ) = a ( x ) | u | p − 2 u in Ω , u = 0 on ∂ Ω . Our main tool is a recent three critical points Theorem of B. Ricceri [On a three critical points theorem, Arch. Math. (Basel) 75 (2000) 220–226].

Using the method of upper and lower solutions and some degree theory arguments, we establish the existence of at least three solutions of some second-order nonlinear differential equations of the type − x ″ = f ( t , x , x ′ ) , t ∈ I = [ 0 , 1 ] subject to nonlinear three-point boundary conditions x ( 0 ) = 0 , x ′ ( 1 ) = g ( x ( η ) ) , 0 η ≤ 1 . The growth of f with respect to x ′ is allowed to be quadratic.

We are concerned with determining values of r , for which there exist nodal solutions of the boundary value problems u ″ + r a ( t ) f ( u ) = 0 , 0 t 1 , u ( 0 ) = u ( 1 ) = 0 where r is a parameter, and a : ( 0 , 1 ) → [ 0 , ∞ ) is continuous and satisfies 0 ∫ 0 1 t ( 1 − t ) a ( t ) d t ∞ . The proof of our main result is based on bifurcation techniques.

In the present paper, a new critical point theorem for locally Lipschitz functions is given, and as applications, one existence and two multiplicity results for zero boundary value problems are obtained. Several important classical theorems are improved.

In the present paper, the following Neumann boundary value problem involving the p -Laplacian (1.1) { − Δ p ( u ) = f ( x , u ) , in Ω ∂ u ∂ n = 0 , on ∂ Ω is studied and some new existence and multiplicity results of solutions for the problem (1.1) are obtained.

We study the existence of nodal solutions of the m-point boundary value problem u ″ + f ( u ) = 0 , 0 t 1 , u ( 0 ) = 0 , u ( 1 ) = ∑ i = 1 m - 2 α i u ( η i ) , where η i ∈ Q ( i = 1 , 2 , … , m - 2 ) with 0 η 1 η 2 ⋯ η m - 2 1 , and α i ∈ R ( i = 1 , 2 , … , m - 2 ) with α i > 0 and 0 ∑ i = 1 m - 2 α i ⩽ 1 . We give conditions on the ratio f ( s ) / s at infinity and zero that guarantee the existence of nodal solutions. The proofs of our main results are based upon bifurcation techniques.

We study the global bifurcation of nonlinear Sturm-Liouville problems of the form -(pu')' + qu = lambda a(x)f(u), b(0)u(0) - c(0)u' (0) = 0, b(1)u(1) + c(1)u'(1) = 0 which are not linearizable in any neighborhood of the origin. (c) 2005 Published by Elsevier Ltd.

In this paper, the existence and multiplicity results of the solutions are obtained for the fourth-order boundary value problem u ( 4 ) ( t ) = f ( t , u ( t ) ) for all t ∈ [ 0 , 1 ] subject to u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 , where f is continuous. The monotone operator theory and critical point theory are employed to discuss this problem, respectively.

We consider the boundary value problem - u ′ ′ ( x ) = λ f ( u ( x ) ) , x ∈ ( - 1 , 1 ) , u ( - 1 ) = 0 = u ( 1 ) , where λ > 0 is a parameter and f ∈ C 2 ( 0 , ∞ ) is monotonically increasing and concave up such that f ( 0 ) 0 (i.e. is the semipositone). In this paper we study the case p = θ and p ∈ ( θ , + ∞ ) . (p is the supremum of the nonnegative solution and θ is such that F ( θ ) = ∫ 0 θ f ( s ) d s = 0 . ) We discuss existence and multiplicity results for nonnegative solutions.

This paper deals with the second-order four-point boundary value problem x ″ ( t ) = f ( t , x ( t ) ) , 0 t 1 , x ( 0 ) = ax ( ξ ) , x ( 1 ) = bx ( η ) , where f : [ 0 , 1 ] × R → R is continuous, 0 ξ , η 1 ; 0 a 1 / ( 1 - ξ ) ; 0 b 1 / η ; a ξ ( 1 - b ) + ( 1 - a ) s ( 1 - b η ) = 0 . By using the coincidence degree theory, the method of lower and upper solutions is established. As applications, some existence and multiplicity results are obtained.