MULTIPLE SOLUTIONS FOR THE p-laplace EQUATION WITH NONLINEAR BOUNDARY CONDITIONS

Size: px
Start display at page:

Download "MULTIPLE SOLUTIONS FOR THE p-laplace EQUATION WITH NONLINEAR BOUNDARY CONDITIONS"

Transcription

1 Electronic Journal of Differential Equations, Vol. 2006(2006), No. 37, pp ISSN: URL: or ftp ejde.math.txstate.edu (login: ftp) MULTIPLE SOLUTIONS FOR THE p-laplace EQUATION WITH NONLINEAR BOUNDARY CONDITIONS JULIÁN FERNÁNDEZ BONDER Abstract. In this note, we show the existence of at least three nontrivial solutions to the quasilinear elliptic equation pu + u p 2 u = f(x, u) in a smooth bounded domain of R N with nonlinear boundary conditions u p 2 u = g(x, u) on. The proof is based on variational arguments. ν 1. Introduction Let us consider the nonlinear elliptic problem p u + u p 2 u = f(x, u) in p 2 u u ν = g(x, u) on, (1.1) where is a bounded smooth domain in R N, p u = div( u p 2 u) is the p- laplacian and / ν is the outer unit normal derivative. Problem (1.1) appears naturally in several branches of pure and applied mathematics, such as the study of optimal constants for the Sobolev trace embedding (see [5, 10, 12, 11]); the theory of quasiregular and quasiconformal mappings in Riemannian manifolds with boundary (see [7, 16]); non-newtonian fluids, reaction diffusion problems, flow through porus media, nonlinear elasticity, glaciology, etc. (see [1, 2, 3, 6]). The purpose of this note, is to prove the existence of at least three nontrivial solutions for (1.1) under adequate assumptions on the sources terms f and g. This result extends previous work by the author [8, 9]. Here, no oddness condition is imposed in f or g and a positive, a negative and a sign-changing solution are found. The proof relies on the Lusternik Schnirelman method for non-compact manifolds (see [14]). For a related result with Dirichlet boundary conditions, see [15] and more recently [4, 17]. The approach in this note follows the one in [15] Mathematics Subject Classification. 35J65, 35J20. Key words and phrases. p-laplace equations; nonlinear boundary conditions; variational methods. c 2006 Texas State University - San Marcos. Submitted January 10, Published March 21, Supported by grant TX066 from Universidad de Buenos Aires, grants and from ANPCyT PICT, project from Fundacion Antorchas and CONICET (Argentina). 1

2 2 J. FERNÁNDEZ B. EJDE-2006/37 Throughout this work, by (weak) solutions of (1.1) we understand critical points of the associated energy functional acting on the Sobolev space W 1,p (): Φ(v) = 1 v p + v p dx F (x, v) dx G(x, v) ds, (1.2) p where F (x, u) = u 0 f(x, z) dz, G(x, u) = u g(x, z) dz and ds is the surface measure. 0 We will denote F(v) = F (x, v) dx and G(v) = G(x, v) ds, (1.3) so the functional Φ can be rewritten as Φ(v) = 1 p v p W 1,p () F(v) G(v). 2. Assumptions and statement of the results The precise assumptions on the source terms f and g are as follows: (F1) f : R R, is a measurable function with respect to the first argument and continuously differentiable with respect to the second argument for almost every x. Moreover, f(x, 0) = 0 for every x. (F2) There exist constants p < q < p = Np/(N p), s > p /(p q), t = sq/(2 + (q 2)s) > p /(p 2) and functions a L s (), b L t (), such that for x, u, v R, f u (x, u) a(x) u q 2 + b(x), (f u (x, u) f u (x, v))u (a(x)( u q 2 + v q 2 ) + b(x)) u v. (F3) There exist constants c 1 (0, 1/(p 1)), c 2 > p, 0 < c 3 < c 4, such that for any u L q () c 3 u q L q () c 2 F (x, u) dx f(x, u)u dx c 1 f u (x, u)u 2 dx c 4 u q L q (). (G1) g : R R is a measurable function with respect to the first argument and continuously differentiable with respect to the second argument for almost every y. Moreover, g(y, 0) = 0 for every y. (G2) There exist constants p < r < p = (N 1)p/(N p), σ > p /(p r), τ = σr/(2+(r 2)σ) > p /(p 2) and functions α L σ (), β L τ (), such that for y, u, v R, g u (y, u) α(y) u r 2 + β(y), (g u (y, u) g u (y, v))u (α(y)( u r 2 + v r 2 ) + β(y)) u v. (G3) There exist constants k 1 (0, 1/(p 1)), k 2 > p, 0 < k 3 < k 4, such that for any u L r () k 3 u r L r () k 2 G(x, u) ds g(x, u)u ds k 1 g u (x, u)u 2 dx k 4 u r L r ().

3 EJDE-2006/37 p-laplacian WITH NONLINEAR BOUNDARY CONDITIONS 3 Remark 2.1. Assumptions (F1) (F3) imply, since the immersion W 1,p () L q () with 1 < q < p is compact, that F is C 1 with compact derivative. Analogously, (G1) (G3) implies the same facts for G by the compactness of the immersion W 1,p () L r () for 1 < r < p. The main result of the paper reads as follows. Theorem 2.2. Under assumptions (F1) (F3), (G1) (G3), there exist three different, nontrivial, (weak) solutions of (1.1). Moreover these solutions are, one positive, one negative and the other one has non-constant sign. 3. Proof of the Theorem The proof uses the same approach as in [15]. That is, we will construct three disjoint sets K i not containing 0 such that Φ has a critical point in K i. These sets will be subsets of smooth manifolds M i W 1,p () that will be constructed by imposing a sign restriction and a normalizing condition. In fact, let M 1 = {u W 1,p () : u + ds > 0, u + p W 1,p () = F (u), u + + G (u), u + }, M 2 = {u W 1,p () : u ds > 0, u p W 1,p () = F (u), u + G (u), u }, M 3 = M 1 M 2, where u + = max{u, 0}, u = max{ u, 0} are the positive and negative parts of u, and, is the duality pairing of W 1,p (). Finally we define K 1 = {u M 1 u 0}, K 2 = {u M 2 u 0}, K 3 = M 3. For the proof of the main theorem, we need the following Lemmas. Lemma 3.1. There exist c j > 0 such that, for every u K i, i = 1, 2, 3, ( ) u p W 1,p () c 1 f(x, u)u dx + g(x, u)u ds c 2 Φ(u) c 3 u p W 1,p (). Proof. Since u K i, we have u p W 1,p () = f(x, u)u dx + g(x, u)u ds. This proves the first inequality. Now, by (F3) and (G3) F (x, u) dx 1 f(x, u)u dx, k 2 G(x, u) ds 1 g(x, u)u ds. c 2 So, for C = max{ 1 k 2 ; 1 c 2 } < 1 p, we have This proves the third inequality. Φ(u) ( 1 p C) u p W 1,p ().

4 4 J. FERNÁNDEZ B. EJDE-2006/37 To prove the middle inequality we proceed as follows: Φ(u) = 1 p u p W 1,p () F (x, u) dx G(x, u) ds = 1 ( ) ( f(x, u)u dx + g(x, u)u ds F (x, u) dx + p ( 1 ( ) p C) f(x, u)u dx + g(x, u)u ds. This completes the proof. Lemma 3.2. There exists c > 0 such that u + W 1,p () c for u K 1, u W 1,p () c for u K 2, u + W 1,p (), u W 1,p () c for u K 3. Proof. By the definition of K i, by (F3) and (G3), we have u ± p W 1,p () = f(x, u)u ± dx + g(x, u)u ± ds c( u ± q L q () + u ± r L r () ). ) G(x, u) ds Now the proof follows by the Sobolev immersion Theorem and by the Sobolev trace Theorem, as p < q, r. Lemma 3.3. There exists c > 0 such that Φ(u) c u p W 1,p () W 1,p () such that u W 1,p () c. Proof. By (F3), (G3) and the Sobolev immersions we have Φ(u) = 1 p u p W 1,p () F(u) G(u) 1 p u p W 1,p () c( u q L q () + u r L r () ) 1 p u p W 1,p () c( u q W 1,p () + u r W 1,p () ) c u p W 1,p (), for every u if u W 1,p () is small enough, as p < q, r. The following lemma describes the properties of the manifolds M i. Lemma 3.4. M i is a C 1,1 sub-manifold of W 1,p () of co-dimension 1 (i = 1, 2), 2 (i = 3) respectively. The sets K i are complete. Moreover, for every u M i we have the direct decomposition T u W 1,p () = T u M i span{u +, u }, where T u M is the tangent space at u of the Banach manifold M. Finally, the projection onto the first component in this decomposition is uniformly continuous on bounded sets of M i.

5 EJDE-2006/37 p-laplacian WITH NONLINEAR BOUNDARY CONDITIONS 5 Proof. Let us denote M 1 = M 2 = { u W 1,p () : { u W 1,p () : M 3 = M 1 M 2. } u + ds > 0, } u ds > 0, Observe that M i M i. By the Sobolev trace Theorem, the set Mi is open in W 1,p (), therefore it is enough to prove that M i is a smooth sub-manifold of Mi. In order to do this, we will construct a C 1,1 function ϕ i : Mi R d with d = 1 (i = 1, 2), d = 2 (i = 3) respectively and M i will be the inverse image of a regular value of ϕ i. In fact, we define: For u M 1, For u M 2, ϕ 1 (u) = u + p W 1,p () F (u), u + G (u), u +. ϕ 2 (u) = u p W 1,p () F (u), u G (u), u. For u M 3, ϕ 3 (u) = (k 1 (u), k 2 (u)). Obviously, we have M i = ϕ 1 i (0). We need to show that 0 is a regular value for ϕ i. To this end we compute, for u M 1, ϕ 1 (u), u + =p u + p W 1,p () f u (x, u)u f(x, u)u + dx g u (x, u)u g(x, u)u + ds =(p 1) f(x, u)u + dx f u (x, u)u 2 + dx + (p 1) g(x, u)u + ds g u (x, u)u 2 + ds. By (F3) and (G3) the last term is bounded by (p 1 c 1 1 ) f(x, u)u + dx + (p 1 k1 1 ) g(x, u)u + ds. Recall that c 1, k 1 < 1/(p 1). Now, by Lemma 3.1, this is bounded by c u + p W 1,p () which is strictly negative by Lemma 3.2. Therefore, M 1 is a smooth sub-manifold of W 1,p (). The exact same argument applies to M 2. Since trivially ϕ 1 (u), u = ϕ 2 (u), u + = 0 for u M 3, the same conclusion holds for M 3. To see that K i is complete, let u k be a Cauchy sequence in K i, then u k u in W 1,p (). Moreover, (u k ) ± u ± in W 1,p (). Now it is easy to see, by Lemma 3.2 and by continuity that u K i. Finally, by the first part of the proof we have the decomposition T u W 1,p () = T u M i span{u +, u }.

6 6 J. FERNÁNDEZ B. EJDE-2006/37 Now let v T u W 1,p () be a unit tangential vector, then v = v 1 + v 2 where v i are given by v 2 = ( ϕ i (u) span{u+,u }) 1 ϕ i (u), v span{u +, u }, v 1 = v v 2 T u M i. From these formulas and from the estimates given in the first part of the proof, the uniform continuity follows. Now, we need to check the Palais-Smale condition for the functional Φ restricted to the manifold M i. Lemma 3.5. The functional Φ Ki satisfies the Palais-Smale condition. Proof. Let {u k } K i be a Palais-Smale sequence, that is Φ(u k ) is uniformly bounded and Φ Ki (u k ) 0 strongly. We need to show that there exists a subsequence u kj that converges strongly in K i. Let v j T uj W 1,p () be a unit tangential vector such that Φ(u j ), v j = Φ(u j ) (W 1,p ()). Now, by Lemma 3.4, v j = w j + z j with w j T uj M i and z j span{(u j ) +, (u j ) }. Since Φ(u j ) is uniformly bounded, by Lemma 3.1, u j is uniformly bounded in W 1,p () and hence w j is uniformly bounded in W 1,p (). Therefore Φ(u j ) (W 1,p ()) = Φ(u j), v j = Φ Ki (u j ), v j 0. As u j is bounded in W 1,p (), there exists u W 1,p () such that u j u, weakly in W 1,p (). As it is well known that the unrestricted functional Φ satisfies the Palais-Smale condition (cf. [9] and [13]), the lemma follows. See [15] for the details. We obtain immediately the following result. Lemma 3.6. Let u K i be a critical point of the restricted functional Φ Ki. Then u is also a critical point of the unrestricted functional Φ and hence a weak solution to (1.1). With all this preparatives, the proof of the Theorem follows easily. Proof of the Theorem. The proof now is a standard application of the Lusternik Schnirelman method for non-compact manifolds. See [14]. References [1] D. Arcoya and J.I. Diaz. S-shaped bifurcation branch in a quasilinear multivalued model arising in climatology. J. Differential Equations, 150 (1998), [2] C. Atkinson and K. El Kalli. Some boundary value problems for the Bingham model. J. Non-Newtonian Fluid Mech. 41 (1992), [3] C. Atkinson and C.R. Champion. On some boundary value problems for the equation (F ( w ) w) = 0. Proc. R. Soc. London A, 448 (1995), [4] T. Bartsch and Z. Liu. On a superlinear elliptic p-laplacian equation. J. Differential Equations, 198 (2004), [5] M. del Pino and C. Flores. Asymptotic behavior of best constants and extremals for trace embeddings in expanding domains. Comm. Partial Differential Equations, 26 (11-12) (2001), [6] J.I. Diaz. Nonlinear partial differential equations and free boundaries. Pitman Publ. Program 1985.

7 EJDE-2006/37 p-laplacian WITH NONLINEAR BOUNDARY CONDITIONS 7 [7] J. F. Escobar, Uniqueness theorems on conformal deformations of metrics, Sobolev inequalities, and an eigenvalue estimate. Comm. Pure Appl. Math., 43 (1990), [8] J. Fernández Bonder. Multiple positive solutions for quasilinear elliptic problems with signchanging nonlinearities. Abstr. Appl. Anal., 2004 (2004), no. 12, [9] J. Fernández Bonder and J.D. Rossi. Existence results for the p-laplacian with nonlinear boundary conditions. J. Math. Anal. Appl., 263 (2001), [10] J. Fernández Bonder and J.D. Rossi. Asymptotic behavior of the best Sobolev trace constant in expanding and contracting domains. Comm. Pure Appl. Anal. 1 (2002), no. 3, [11] J. Fernández Bonder, E. Lami-Dozo and J.D. Rossi. Symmetry properties for the extremals of the Sobolev trace embedding. Ann. Inst. H. Poincaré Anal. Non Linèaire, 21 (2004), no. 6, [12] J. Fernández Bonder, S. Martínez and J.D. Rossi. The behavior of the best Sobolev trace constant and extremals in thin domains. J. Differential Equations, 198 (2004), no. 1, [13] P. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conf. Ser. in Math., no. 65, Amer. Math. Soc., Providence, R.I. (1986). [14] J.T. Schwartz. Generalizing the Lusternik-Schnirelman theory of critical points. Comm. Pure Appl. Math., 17 (1964), [15] M. Struwe. Three nontrivial solutions of anticoercive boundary value problems for the Pseudo- Laplace operator. J. Reine Angew. Math. 325 (1981), [16] P. Tolksdorf. Regularity for a more general class of quasilinear elliptic equations. J. Differential Equations, 51 (1984), [17] Z. Zhang, J. Chen and S. Li. Construction of pseudo-gradient vector field and sign-changing multiple solutions involving p-laplacian. J. Differential Equations, 201 (2004), Julián Fernández Bonder Departamento de Matemática, FCEyN, UBA (1428) Buenos Aires, Argentina address: jfbonder@dm.uba.ar URL: jfbonder

Multiple positive solutions for a class of quasilinear elliptic boundary-value problems

Multiple positive solutions for a class of quasilinear elliptic boundary-value problems Electronic Journal of Differential Equations, Vol. 20032003), No. 07, pp. 1 5. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu login: ftp) Multiple positive

More information

EXISTENCE OF SOLUTIONS FOR A RESONANT PROBLEM UNDER LANDESMAN-LAZER CONDITIONS

EXISTENCE OF SOLUTIONS FOR A RESONANT PROBLEM UNDER LANDESMAN-LAZER CONDITIONS Electronic Journal of Differential Equations, Vol. 2008(2008), No. 98, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) EXISTENCE

More information

SEMILINEAR ELLIPTIC EQUATIONS WITH DEPENDENCE ON THE GRADIENT

SEMILINEAR ELLIPTIC EQUATIONS WITH DEPENDENCE ON THE GRADIENT Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 139, pp. 1 9. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SEMILINEAR ELLIPTIC

More information

AN ESTIMATE FOR THE BLOW-UP TIME IN TERMS OF THE INITIAL DATA

AN ESTIMATE FOR THE BLOW-UP TIME IN TERMS OF THE INITIAL DATA AN ESTIMATE FOR THE BLOW-UP TIME IN TERMS OF THE INITIAL DATA JULIO D. ROSSI Abstract. We find an estimate for the blow-up time in terms of the initial data for solutions of the equation u t = (u m ) +

More information

EXISTENCE OF NONTRIVIAL SOLUTIONS FOR A QUASILINEAR SCHRÖDINGER EQUATIONS WITH SIGN-CHANGING POTENTIAL

EXISTENCE OF NONTRIVIAL SOLUTIONS FOR A QUASILINEAR SCHRÖDINGER EQUATIONS WITH SIGN-CHANGING POTENTIAL Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 05, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE OF NONTRIVIAL

More information

NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian UNDER NONHOMOGENEOUS NEUMANN BOUNDARY CONDITION

NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian UNDER NONHOMOGENEOUS NEUMANN BOUNDARY CONDITION Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 210, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian

More information

POTENTIAL LANDESMAN-LAZER TYPE CONDITIONS AND. 1. Introduction We investigate the existence of solutions for the nonlinear boundary-value problem

POTENTIAL LANDESMAN-LAZER TYPE CONDITIONS AND. 1. Introduction We investigate the existence of solutions for the nonlinear boundary-value problem Electronic Journal of Differential Equations, Vol. 25(25), No. 94, pp. 1 12. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) POTENTIAL

More information

EXISTENCE OF WEAK SOLUTIONS FOR A NONUNIFORMLY ELLIPTIC NONLINEAR SYSTEM IN R N. 1. Introduction We study the nonuniformly elliptic, nonlinear system

EXISTENCE OF WEAK SOLUTIONS FOR A NONUNIFORMLY ELLIPTIC NONLINEAR SYSTEM IN R N. 1. Introduction We study the nonuniformly elliptic, nonlinear system Electronic Journal of Differential Equations, Vol. 20082008), No. 119, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu login: ftp) EXISTENCE

More information

SYMMETRY IN REARRANGEMENT OPTIMIZATION PROBLEMS

SYMMETRY IN REARRANGEMENT OPTIMIZATION PROBLEMS Electronic Journal of Differential Equations, Vol. 2009(2009), No. 149, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SYMMETRY IN REARRANGEMENT

More information

Nonlinear elliptic systems with exponential nonlinearities

Nonlinear elliptic systems with exponential nonlinearities 22-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 9, 22, pp 139 147. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu

More information

MULTIPLE SOLUTIONS FOR CRITICAL ELLIPTIC PROBLEMS WITH FRACTIONAL LAPLACIAN

MULTIPLE SOLUTIONS FOR CRITICAL ELLIPTIC PROBLEMS WITH FRACTIONAL LAPLACIAN Electronic Journal of Differential Equations, Vol. 016 (016), No. 97, pp. 1 11. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu MULTIPLE SOLUTIONS

More information

EQUAZIONI A DERIVATE PARZIALI. STEKLOV EIGENVALUES FOR THE -LAPLACIAN

EQUAZIONI A DERIVATE PARZIALI. STEKLOV EIGENVALUES FOR THE -LAPLACIAN EQUAZIONI A DERIVATE PARZIALI. STEKLOV EIGENVALUES FOR THE -LAPLACIAN J. GARCIA-AZORERO, J. J. MANFREDI, I. PERAL AND J. D. ROSSI Abstract. We study the Steklov eigenvalue problem for the - laplacian.

More information

MULTIPLE POSITIVE SOLUTIONS FOR KIRCHHOFF PROBLEMS WITH SIGN-CHANGING POTENTIAL

MULTIPLE POSITIVE SOLUTIONS FOR KIRCHHOFF PROBLEMS WITH SIGN-CHANGING POTENTIAL Electronic Journal of Differential Equations, Vol. 015 015), No. 0, pp. 1 10. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu MULTIPLE POSITIVE SOLUTIONS

More information

Variational eigenvalues of degenerate eigenvalue problems for the weighted p-laplacian

Variational eigenvalues of degenerate eigenvalue problems for the weighted p-laplacian Variational eigenvalues of degenerate eigenvalue problems for the weighted p-laplacian An Lê Mathematics Sciences Research Institute, 17 Gauss Way, Berkeley, California 94720 e-mail: anle@msri.org Klaus

More information

Existence of solutions to a superlinear p-laplacian equation

Existence of solutions to a superlinear p-laplacian equation Electronic Journal of Differential Equations, Vol. 2001(2001), No. 66,. 1 6. ISSN: 1072-6691. URL: htt://ejde.math.swt.edu or htt://ejde.math.unt.edu ft ejde.math.swt.edu (login: ft) Existence of solutions

More information

EXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM

EXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM EXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM JENICĂ CRÎNGANU We derive existence results for operator equations having the form J ϕu = N f u, by using

More information

MULTIPLE SOLUTIONS FOR AN INDEFINITE KIRCHHOFF-TYPE EQUATION WITH SIGN-CHANGING POTENTIAL

MULTIPLE SOLUTIONS FOR AN INDEFINITE KIRCHHOFF-TYPE EQUATION WITH SIGN-CHANGING POTENTIAL Electronic Journal of Differential Equations, Vol. 2015 (2015), o. 274, pp. 1 9. ISS: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu MULTIPLE SOLUTIOS

More information

EXISTENCE OF POSITIVE SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS

EXISTENCE OF POSITIVE SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS Electronic Journal of Differential Equations, Vol. 013 013, No. 180, pp. 1 8. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE OF POSITIVE

More information

A REMARK ON MINIMAL NODAL SOLUTIONS OF AN ELLIPTIC PROBLEM IN A BALL. Olaf Torné. 1. Introduction

A REMARK ON MINIMAL NODAL SOLUTIONS OF AN ELLIPTIC PROBLEM IN A BALL. Olaf Torné. 1. Introduction Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 24, 2004, 199 207 A REMARK ON MINIMAL NODAL SOLUTIONS OF AN ELLIPTIC PROBLEM IN A BALL Olaf Torné (Submitted by Michel

More information

p-laplacian problems with critical Sobolev exponents

p-laplacian problems with critical Sobolev exponents Nonlinear Analysis 66 (2007) 454 459 www.elsevier.com/locate/na p-laplacian problems with critical Sobolev exponents Kanishka Perera a,, Elves A.B. Silva b a Department of Mathematical Sciences, Florida

More information

EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS WITH UNBOUNDED POTENTIAL. 1. Introduction In this article, we consider the Kirchhoff type problem

EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS WITH UNBOUNDED POTENTIAL. 1. Introduction In this article, we consider the Kirchhoff type problem Electronic Journal of Differential Equations, Vol. 207 (207), No. 84, pp. 2. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS

More information

Existence and Multiplicity of Solutions for a Class of Semilinear Elliptic Equations 1

Existence and Multiplicity of Solutions for a Class of Semilinear Elliptic Equations 1 Journal of Mathematical Analysis and Applications 257, 321 331 (2001) doi:10.1006/jmaa.2000.7347, available online at http://www.idealibrary.com on Existence and Multiplicity of Solutions for a Class of

More information

Elliptic equations with one-sided critical growth

Elliptic equations with one-sided critical growth Electronic Journal of Differential Equations, Vol. 00(00), No. 89, pp. 1 1. ISSN: 107-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Elliptic equations

More information

FUNCTIONAL COMPRESSION-EXPANSION FIXED POINT THEOREM

FUNCTIONAL COMPRESSION-EXPANSION FIXED POINT THEOREM Electronic Journal of Differential Equations, Vol. 28(28), No. 22, pp. 1 12. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) FUNCTIONAL

More information

PERIODIC SOLUTIONS FOR NON-AUTONOMOUS SECOND-ORDER DIFFERENTIAL SYSTEMS WITH (q, p)-laplacian

PERIODIC SOLUTIONS FOR NON-AUTONOMOUS SECOND-ORDER DIFFERENTIAL SYSTEMS WITH (q, p)-laplacian Electronic Journal of Differential Euations, Vol. 24 (24), No. 64, pp. 3. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu PERIODIC SOLUIONS FOR NON-AUONOMOUS

More information

MAXIMIZATION AND MINIMIZATION PROBLEMS RELATED TO A p-laplacian EQUATION ON A MULTIPLY CONNECTED DOMAIN. N. Amiri and M.

MAXIMIZATION AND MINIMIZATION PROBLEMS RELATED TO A p-laplacian EQUATION ON A MULTIPLY CONNECTED DOMAIN. N. Amiri and M. TAIWANESE JOURNAL OF MATHEMATICS Vol. 19, No. 1, pp. 243-252, February 2015 DOI: 10.11650/tjm.19.2015.3873 This paper is available online at http://journal.taiwanmathsoc.org.tw MAXIMIZATION AND MINIMIZATION

More information

NONHOMOGENEOUS ELLIPTIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT AND WEIGHT

NONHOMOGENEOUS ELLIPTIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT AND WEIGHT Electronic Journal of Differential Equations, Vol. 016 (016), No. 08, pp. 1 1. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NONHOMOGENEOUS ELLIPTIC

More information

HOMOCLINIC SOLUTIONS FOR SECOND-ORDER NON-AUTONOMOUS HAMILTONIAN SYSTEMS WITHOUT GLOBAL AMBROSETTI-RABINOWITZ CONDITIONS

HOMOCLINIC SOLUTIONS FOR SECOND-ORDER NON-AUTONOMOUS HAMILTONIAN SYSTEMS WITHOUT GLOBAL AMBROSETTI-RABINOWITZ CONDITIONS Electronic Journal of Differential Equations, Vol. 010010, No. 9, pp. 1 10. ISSN: 107-6691. UL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu HOMOCLINIC SOLUTIONS FO

More information

arxiv: v1 [math.ap] 16 Dec 2014

arxiv: v1 [math.ap] 16 Dec 2014 Non-existence of positive weak solutions for some nonlinear (p, q)-laplacian systems arxiv:1412.5913v1 [math.ap] 16 Dec 2014 Salah. A. Khafagy Mathematics Department, Faculty of Science, Al-Azhar University,

More information

SELF-ADJOINTNESS OF SCHRÖDINGER-TYPE OPERATORS WITH SINGULAR POTENTIALS ON MANIFOLDS OF BOUNDED GEOMETRY

SELF-ADJOINTNESS OF SCHRÖDINGER-TYPE OPERATORS WITH SINGULAR POTENTIALS ON MANIFOLDS OF BOUNDED GEOMETRY Electronic Journal of Differential Equations, Vol. 2003(2003), No.??, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) SELF-ADJOINTNESS

More information

PICONE S IDENTITY FOR A SYSTEM OF FIRST-ORDER NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

PICONE S IDENTITY FOR A SYSTEM OF FIRST-ORDER NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 143, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu PICONE S IDENTITY

More information

EXISTENCE OF SOLUTIONS FOR CROSS CRITICAL EXPONENTIAL N-LAPLACIAN SYSTEMS

EXISTENCE OF SOLUTIONS FOR CROSS CRITICAL EXPONENTIAL N-LAPLACIAN SYSTEMS Electronic Journal of Differential Equations, Vol. 2014 (2014), o. 28, pp. 1 10. ISS: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTECE OF SOLUTIOS

More information

STEKLOV PROBLEMS INVOLVING THE p(x)-laplacian

STEKLOV PROBLEMS INVOLVING THE p(x)-laplacian Electronic Journal of Differential Equations, Vol. 204 204), No. 34, pp.. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu STEKLOV PROBLEMS INVOLVING

More information

Existence of Multiple Positive Solutions of Quasilinear Elliptic Problems in R N

Existence of Multiple Positive Solutions of Quasilinear Elliptic Problems in R N Advances in Dynamical Systems and Applications. ISSN 0973-5321 Volume 2 Number 1 (2007), pp. 1 11 c Research India Publications http://www.ripublication.com/adsa.htm Existence of Multiple Positive Solutions

More information

EXISTENCE OF SOLUTIONS TO P-LAPLACE EQUATIONS WITH LOGARITHMIC NONLINEARITY

EXISTENCE OF SOLUTIONS TO P-LAPLACE EQUATIONS WITH LOGARITHMIC NONLINEARITY Electronic Journal of Differential Equations, Vol. 2009(2009), No. 87, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE OF SOLUTIONS

More information

Non-homogeneous semilinear elliptic equations involving critical Sobolev exponent

Non-homogeneous semilinear elliptic equations involving critical Sobolev exponent Non-homogeneous semilinear elliptic equations involving critical Sobolev exponent Yūki Naito a and Tokushi Sato b a Department of Mathematics, Ehime University, Matsuyama 790-8577, Japan b Mathematical

More information

NOTE ON THE NODAL LINE OF THE P-LAPLACIAN. 1. Introduction In this paper we consider the nonlinear elliptic boundary-value problem

NOTE ON THE NODAL LINE OF THE P-LAPLACIAN. 1. Introduction In this paper we consider the nonlinear elliptic boundary-value problem 2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 155 162. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

More information

THE NEUMANN PROBLEM FOR THE -LAPLACIAN AND THE MONGE-KANTOROVICH MASS TRANSFER PROBLEM

THE NEUMANN PROBLEM FOR THE -LAPLACIAN AND THE MONGE-KANTOROVICH MASS TRANSFER PROBLEM THE NEUMANN PROBLEM FOR THE -LAPLACIAN AND THE MONGE-KANTOROVICH MASS TRANSFER PROBLEM J. GARCÍA-AZORERO, J. J. MANFREDI, I. PERAL AND J. D. ROSSI Abstract. We consider the natural Neumann boundary condition

More information

ON THE EXISTENCE OF THREE SOLUTIONS FOR QUASILINEAR ELLIPTIC PROBLEM. Paweł Goncerz

ON THE EXISTENCE OF THREE SOLUTIONS FOR QUASILINEAR ELLIPTIC PROBLEM. Paweł Goncerz Opuscula Mathematica Vol. 32 No. 3 2012 http://dx.doi.org/10.7494/opmath.2012.32.3.473 ON THE EXISTENCE OF THREE SOLUTIONS FOR QUASILINEAR ELLIPTIC PROBLEM Paweł Goncerz Abstract. We consider a quasilinear

More information

MAXIMUM PRINCIPLE AND EXISTENCE OF POSITIVE SOLUTIONS FOR NONLINEAR SYSTEMS ON R N

MAXIMUM PRINCIPLE AND EXISTENCE OF POSITIVE SOLUTIONS FOR NONLINEAR SYSTEMS ON R N Electronic Journal of Differential Equations, Vol. 20052005, No. 85, pp. 1 12. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu login: ftp MAXIMUM

More information

ELLIPTIC EQUATIONS WITH MEASURE DATA IN ORLICZ SPACES

ELLIPTIC EQUATIONS WITH MEASURE DATA IN ORLICZ SPACES Electronic Journal of Differential Equations, Vol. 2008(2008), No. 76, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) ELLIPTIC

More information

BLOW-UP AND EXTINCTION OF SOLUTIONS TO A FAST DIFFUSION EQUATION WITH HOMOGENEOUS NEUMANN BOUNDARY CONDITIONS

BLOW-UP AND EXTINCTION OF SOLUTIONS TO A FAST DIFFUSION EQUATION WITH HOMOGENEOUS NEUMANN BOUNDARY CONDITIONS Electronic Journal of Differential Equations, Vol. 016 (016), No. 36, pp. 1 10. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu BLOW-UP AND EXTINCTION OF SOLUTIONS TO A FAST

More information

HOW TO APPROXIMATE THE HEAT EQUATION WITH NEUMANN BOUNDARY CONDITIONS BY NONLOCAL DIFFUSION PROBLEMS. 1. Introduction

HOW TO APPROXIMATE THE HEAT EQUATION WITH NEUMANN BOUNDARY CONDITIONS BY NONLOCAL DIFFUSION PROBLEMS. 1. Introduction HOW TO APPROXIMATE THE HEAT EQUATION WITH NEUMANN BOUNDARY CONDITIONS BY NONLOCAL DIFFUSION PROBLEMS CARMEN CORTAZAR, MANUEL ELGUETA, ULIO D. ROSSI, AND NOEMI WOLANSKI Abstract. We present a model for

More information

Changing sign solutions for the CR-Yamabe equation

Changing sign solutions for the CR-Yamabe equation Changing sign solutions for the CR-Yamabe equation Ali Maalaoui (1) & Vittorio Martino (2) Abstract In this paper we prove that the CR-Yamabe equation on the Heisenberg group has infinitely many changing

More information

NONLINEAR SCHRÖDINGER ELLIPTIC SYSTEMS INVOLVING EXPONENTIAL CRITICAL GROWTH IN R Introduction

NONLINEAR SCHRÖDINGER ELLIPTIC SYSTEMS INVOLVING EXPONENTIAL CRITICAL GROWTH IN R Introduction Electronic Journal of Differential Equations, Vol. 014 (014), No. 59, pp. 1 1. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NONLINEAR SCHRÖDINGER

More information

ON THE EXISTENCE OF SIGNED SOLUTIONS FOR A QUASILINEAR ELLIPTIC PROBLEM IN R N. Dedicated to Luís Adauto Medeiros on his 80th birthday

ON THE EXISTENCE OF SIGNED SOLUTIONS FOR A QUASILINEAR ELLIPTIC PROBLEM IN R N. Dedicated to Luís Adauto Medeiros on his 80th birthday Matemática Contemporânea, Vol..., 00-00 c 2006, Sociedade Brasileira de Matemática ON THE EXISTENCE OF SIGNED SOLUTIONS FOR A QUASILINEAR ELLIPTIC PROBLEM IN Everaldo S. Medeiros Uberlandio B. Severo Dedicated

More information

EXISTENCE AND REGULARITY OF WEAK SOLUTIONS FOR SINGULAR ELLIPTIC PROBLEMS. 1. Introduction In this article we study the quasilinear elliptic problem

EXISTENCE AND REGULARITY OF WEAK SOLUTIONS FOR SINGULAR ELLIPTIC PROBLEMS. 1. Introduction In this article we study the quasilinear elliptic problem 204 Madrid Conference on Applied Mathematics in honor of Alfonso Casal, Electronic Journal of Differential Equations, Conference 22 (205), pp. 9 30. ISSN: 072-669. URL: http://ejde.math.txstate.edu or

More information

INFINITELY MANY SOLUTIONS FOR A CLASS OF SUBLINEAR SCHRÖDINGER EQUATIONS. Jing Chen and X. H. Tang 1. INTRODUCTION

INFINITELY MANY SOLUTIONS FOR A CLASS OF SUBLINEAR SCHRÖDINGER EQUATIONS. Jing Chen and X. H. Tang 1. INTRODUCTION TAIWANESE JOURNAL OF MATHEMATICS Vol. 19, No. 2, pp. 381-396, April 2015 DOI: 10.11650/tjm.19.2015.4044 This paper is available online at http://journal.taiwanmathsoc.org.tw INFINITELY MANY SOLUTIONS FOR

More information

Martin Luther Universität Halle Wittenberg Institut für Mathematik

Martin Luther Universität Halle Wittenberg Institut für Mathematik Martin Luther Universität Halle Wittenberg Institut für Mathematik Constant-Sign and Sign-Changing Solutions for Nonlinear Elliptic Equations with Neumann Boundary Values Patrick Winkert Report No. 20

More information

Nonhomogeneous Neumann problem for the Poisson equation in domains with an external cusp

Nonhomogeneous Neumann problem for the Poisson equation in domains with an external cusp J. Math. Anal. Appl. 31 (5) 397 411 www.elsevier.com/locate/jmaa Nonhomogeneous Neumann problem for the Poisson equation in domains with an external cusp Gabriel Acosta a,1, María G. Armentano b,,, Ricardo

More information

EXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC SCHRÖDINGER EQUATIONS

EXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC SCHRÖDINGER EQUATIONS Electronic Journal of Differential Equations, Vol. 017 (017), No. 15, pp. 1 7. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC

More information

UNIQUENESS OF SELF-SIMILAR VERY SINGULAR SOLUTION FOR NON-NEWTONIAN POLYTROPIC FILTRATION EQUATIONS WITH GRADIENT ABSORPTION

UNIQUENESS OF SELF-SIMILAR VERY SINGULAR SOLUTION FOR NON-NEWTONIAN POLYTROPIC FILTRATION EQUATIONS WITH GRADIENT ABSORPTION Electronic Journal of Differential Equations, Vol. 2015 2015), No. 83, pp. 1 9. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu UNIQUENESS OF SELF-SIMILAR

More information

Computations of Critical Groups at a Degenerate Critical Point for Strongly Indefinite Functionals

Computations of Critical Groups at a Degenerate Critical Point for Strongly Indefinite Functionals Journal of Mathematical Analysis and Applications 256, 462 477 (2001) doi:10.1006/jmaa.2000.7292, available online at http://www.idealibrary.com on Computations of Critical Groups at a Degenerate Critical

More information

EXISTENCE AND MULTIPLICITY OF PERIODIC SOLUTIONS GENERATED BY IMPULSES FOR SECOND-ORDER HAMILTONIAN SYSTEM

EXISTENCE AND MULTIPLICITY OF PERIODIC SOLUTIONS GENERATED BY IMPULSES FOR SECOND-ORDER HAMILTONIAN SYSTEM Electronic Journal of Differential Equations, Vol. 14 (14), No. 11, pp. 1 1. ISSN: 17-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE AND MULTIPLICITY

More information

SIMULTANEOUS AND NON-SIMULTANEOUS BLOW-UP AND UNIFORM BLOW-UP PROFILES FOR REACTION-DIFFUSION SYSTEM

SIMULTANEOUS AND NON-SIMULTANEOUS BLOW-UP AND UNIFORM BLOW-UP PROFILES FOR REACTION-DIFFUSION SYSTEM Electronic Journal of Differential Euations, Vol. 22 (22), No. 26, pp. 9. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SIMULTANEOUS AND NON-SIMULTANEOUS

More information

NONCLASSICAL SHOCK WAVES OF CONSERVATION LAWS: FLUX FUNCTION HAVING TWO INFLECTION POINTS

NONCLASSICAL SHOCK WAVES OF CONSERVATION LAWS: FLUX FUNCTION HAVING TWO INFLECTION POINTS Electronic Journal of Differential Equations, Vol. 2006(2006), No. 149, pp. 1 18. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) NONCLASSICAL

More information

CONSEQUENCES OF TALENTI S INEQUALITY BECOMING EQUALITY. 1. Introduction

CONSEQUENCES OF TALENTI S INEQUALITY BECOMING EQUALITY. 1. Introduction Electronic Journal of ifferential Equations, Vol. 2011 (2011), No. 165, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu CONSEQUENCES OF

More information

SHARP BOUNDARY TRACE INEQUALITIES. 1. Introduction

SHARP BOUNDARY TRACE INEQUALITIES. 1. Introduction SHARP BOUNDARY TRACE INEQUALITIES GILES AUCHMUTY Abstract. This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region R N. The inequalities bound (semi-)norms

More information

SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS. Massimo Grosi Filomena Pacella S. L. Yadava. 1. Introduction

SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS. Massimo Grosi Filomena Pacella S. L. Yadava. 1. Introduction Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 21, 2003, 211 226 SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS Massimo Grosi Filomena Pacella S.

More information

EXISTENCE OF POSITIVE SOLUTIONS OF A NONLINEAR SECOND-ORDER BOUNDARY-VALUE PROBLEM WITH INTEGRAL BOUNDARY CONDITIONS

EXISTENCE OF POSITIVE SOLUTIONS OF A NONLINEAR SECOND-ORDER BOUNDARY-VALUE PROBLEM WITH INTEGRAL BOUNDARY CONDITIONS Electronic Journal of Differential Equations, Vol. 215 (215), No. 236, pp. 1 7. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE OF POSITIVE

More information

Stability properties of positive solutions to partial differential equations with delay

Stability properties of positive solutions to partial differential equations with delay Electronic Journal of Differential Equations, Vol. 2001(2001), No. 64, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Stability properties

More information

BIFURCATION AND MULTIPLICITY RESULTS FOR CRITICAL p -LAPLACIAN PROBLEMS. Kanishka Perera Marco Squassina Yang Yang

BIFURCATION AND MULTIPLICITY RESULTS FOR CRITICAL p -LAPLACIAN PROBLEMS. Kanishka Perera Marco Squassina Yang Yang TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS Vol. 47, No. 1 March 2016 BIFURCATION AND MULTIPLICITY RESULTS FOR CRITICAL p -LAPLACIAN PROBLEMS Kanishka Perera Marco Squassina Yang Yang Topol. Methods Nonlinear

More information

SUPER-QUADRATIC CONDITIONS FOR PERIODIC ELLIPTIC SYSTEM ON R N

SUPER-QUADRATIC CONDITIONS FOR PERIODIC ELLIPTIC SYSTEM ON R N Electronic Journal of Differential Equations, Vol. 015 015), No. 17, pp. 1 11. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SUPER-QUADRATIC CONDITIONS

More information

MIXED BOUNDARY-VALUE PROBLEMS FOR QUANTUM HYDRODYNAMIC MODELS WITH SEMICONDUCTORS IN THERMAL EQUILIBRIUM

MIXED BOUNDARY-VALUE PROBLEMS FOR QUANTUM HYDRODYNAMIC MODELS WITH SEMICONDUCTORS IN THERMAL EQUILIBRIUM Electronic Journal of Differential Equations, Vol. 2005(2005), No. 123, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) MIXED

More information

DECAY ESTIMATES FOR THE KLEIN-GORDON EQUATION IN CURVED SPACETIME

DECAY ESTIMATES FOR THE KLEIN-GORDON EQUATION IN CURVED SPACETIME Electronic Journal of Differential Equations, Vol. 218 218), No. 17, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu DECAY ESTIMATES FOR THE KLEIN-GORDON EQUATION

More information

MULTIPLE SOLUTIONS FOR A KIRCHHOFF EQUATION WITH NONLINEARITY HAVING ARBITRARY GROWTH

MULTIPLE SOLUTIONS FOR A KIRCHHOFF EQUATION WITH NONLINEARITY HAVING ARBITRARY GROWTH MULTIPLE SOLUTIONS FOR A KIRCHHOFF EQUATION WITH NONLINEARITY HAVING ARBITRARY GROWTH MARCELO F. FURTADO AND HENRIQUE R. ZANATA Abstract. We prove the existence of infinitely many solutions for the Kirchhoff

More information

SOLUTION OF AN INITIAL-VALUE PROBLEM FOR PARABOLIC EQUATIONS VIA MONOTONE OPERATOR METHODS

SOLUTION OF AN INITIAL-VALUE PROBLEM FOR PARABOLIC EQUATIONS VIA MONOTONE OPERATOR METHODS Electronic Journal of Differential Equations, Vol. 214 (214), No. 225, pp. 1 1. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SOLUTION OF AN INITIAL-VALUE

More information

COMBINED EFFECTS FOR A STATIONARY PROBLEM WITH INDEFINITE NONLINEARITIES AND LACK OF COMPACTNESS

COMBINED EFFECTS FOR A STATIONARY PROBLEM WITH INDEFINITE NONLINEARITIES AND LACK OF COMPACTNESS Dynamic Systems and Applications 22 (203) 37-384 COMBINED EFFECTS FOR A STATIONARY PROBLEM WITH INDEFINITE NONLINEARITIES AND LACK OF COMPACTNESS VICENŢIU D. RĂDULESCU Simion Stoilow Mathematics Institute

More information

A NOTE ON THE EXISTENCE OF TWO NONTRIVIAL SOLUTIONS OF A RESONANCE PROBLEM

A NOTE ON THE EXISTENCE OF TWO NONTRIVIAL SOLUTIONS OF A RESONANCE PROBLEM PORTUGALIAE MATHEMATICA Vol. 51 Fasc. 4 1994 A NOTE ON THE EXISTENCE OF TWO NONTRIVIAL SOLUTIONS OF A RESONANCE PROBLEM To Fu Ma* Abstract: We study the existence of two nontrivial solutions for an elliptic

More information

HOMOLOGICAL LOCAL LINKING

HOMOLOGICAL LOCAL LINKING HOMOLOGICAL LOCAL LINKING KANISHKA PERERA Abstract. We generalize the notion of local linking to include certain cases where the functional does not have a local splitting near the origin. Applications

More information

arxiv: v1 [math.ap] 24 Oct 2014

arxiv: v1 [math.ap] 24 Oct 2014 Multiple solutions for Kirchhoff equations under the partially sublinear case Xiaojing Feng School of Mathematical Sciences, Shanxi University, Taiyuan 030006, People s Republic of China arxiv:1410.7335v1

More information

A COUNTEREXAMPLE TO AN ENDPOINT BILINEAR STRICHARTZ INEQUALITY TERENCE TAO. t L x (R R2 ) f L 2 x (R2 )

A COUNTEREXAMPLE TO AN ENDPOINT BILINEAR STRICHARTZ INEQUALITY TERENCE TAO. t L x (R R2 ) f L 2 x (R2 ) Electronic Journal of Differential Equations, Vol. 2006(2006), No. 5, pp. 6. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) A COUNTEREXAMPLE

More information

On periodic solutions of superquadratic Hamiltonian systems

On periodic solutions of superquadratic Hamiltonian systems Electronic Journal of Differential Equations, Vol. 22(22), No. 8, pp. 1 12. ISSN: 172-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) On periodic solutions

More information

PERIODIC SOLUTIONS OF THE FORCED PENDULUM : CLASSICAL VS RELATIVISTIC

PERIODIC SOLUTIONS OF THE FORCED PENDULUM : CLASSICAL VS RELATIVISTIC LE MATEMATICHE Vol. LXV 21) Fasc. II, pp. 97 17 doi: 1.4418/21.65.2.11 PERIODIC SOLUTIONS OF THE FORCED PENDULUM : CLASSICAL VS RELATIVISTIC JEAN MAWHIN The paper surveys and compares some results on the

More information

LERAY LIONS DEGENERATED PROBLEM WITH GENERAL GROWTH CONDITION

LERAY LIONS DEGENERATED PROBLEM WITH GENERAL GROWTH CONDITION 2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 73 81. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

More information

THE L 2 -HODGE THEORY AND REPRESENTATION ON R n

THE L 2 -HODGE THEORY AND REPRESENTATION ON R n THE L 2 -HODGE THEORY AND REPRESENTATION ON R n BAISHENG YAN Abstract. We present an elementary L 2 -Hodge theory on whole R n based on the minimization principle of the calculus of variations and some

More information

Nonexistence of solutions for quasilinear elliptic equations with p-growth in the gradient

Nonexistence of solutions for quasilinear elliptic equations with p-growth in the gradient Electronic Journal of Differential Equations, Vol. 2002(2002), No. 54, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Nonexistence

More information

TRACES FOR FRACTIONAL SOBOLEV SPACES WITH VARIABLE EXPONENTS

TRACES FOR FRACTIONAL SOBOLEV SPACES WITH VARIABLE EXPONENTS Adv. Oper. Theory 2 (2017), no. 4, 435 446 http://doi.org/10.22034/aot.1704-1152 ISSN: 2538-225X (electronic) http://aot-math.org TRACES FOR FRACTIONAL SOBOLEV SPACES WITH VARIABLE EXPONENTS LEANDRO M.

More information

BLOW-UP FOR PARABOLIC AND HYPERBOLIC PROBLEMS WITH VARIABLE EXPONENTS. 1. Introduction In this paper we will study the following parabolic problem

BLOW-UP FOR PARABOLIC AND HYPERBOLIC PROBLEMS WITH VARIABLE EXPONENTS. 1. Introduction In this paper we will study the following parabolic problem BLOW-UP FOR PARABOLIC AND HYPERBOLIC PROBLEMS WITH VARIABLE EXPONENTS JUAN PABLO PINASCO Abstract. In this paper we study the blow up problem for positive solutions of parabolic and hyperbolic problems

More information

On a Nonlocal Elliptic System of p-kirchhoff-type Under Neumann Boundary Condition

On a Nonlocal Elliptic System of p-kirchhoff-type Under Neumann Boundary Condition On a Nonlocal Elliptic System of p-kirchhoff-type Under Neumann Boundary Condition Francisco Julio S.A Corrêa,, UFCG - Unidade Acadêmica de Matemática e Estatística, 58.9-97 - Campina Grande - PB - Brazil

More information

CRITICAL POINT METHODS IN DEGENERATE ANISOTROPIC PROBLEMS WITH VARIABLE EXPONENT. We are interested in discussing the problem:

CRITICAL POINT METHODS IN DEGENERATE ANISOTROPIC PROBLEMS WITH VARIABLE EXPONENT. We are interested in discussing the problem: STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume LV, Number 4, December 2010 CRITICAL POINT METHODS IN DEGENERATE ANISOTROPIC PROBLEMS WITH VARIABLE EXPONENT MARIA-MAGDALENA BOUREANU Abstract. We work on

More information

A Semilinear Elliptic Problem with Neumann Condition on the Boundary

A Semilinear Elliptic Problem with Neumann Condition on the Boundary International Mathematical Forum, Vol. 8, 2013, no. 6, 283-288 A Semilinear Elliptic Problem with Neumann Condition on the Boundary A. M. Marin Faculty of Exact and Natural Sciences, University of Cartagena

More information

Lecture Note III: Least-Squares Method

Lecture Note III: Least-Squares Method Lecture Note III: Least-Squares Method Zhiqiang Cai October 4, 004 In this chapter, we shall present least-squares methods for second-order scalar partial differential equations, elastic equations of solids,

More information

EXISTENCE OF CONJUGACIES AND STABLE MANIFOLDS VIA SUSPENSIONS

EXISTENCE OF CONJUGACIES AND STABLE MANIFOLDS VIA SUSPENSIONS Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 172, pp. 1 11. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF CONJUGACIES AND STABLE MANIFOLDS

More information

BOUNDED WEAK SOLUTION FOR THE HAMILTONIAN SYSTEM. Q-Heung Choi and Tacksun Jung

BOUNDED WEAK SOLUTION FOR THE HAMILTONIAN SYSTEM. Q-Heung Choi and Tacksun Jung Korean J. Math. 2 (23), No., pp. 8 9 http://dx.doi.org/.568/kjm.23.2..8 BOUNDED WEAK SOLUTION FOR THE HAMILTONIAN SYSTEM Q-Heung Choi and Tacksun Jung Abstract. We investigate the bounded weak solutions

More information

SUBSOLUTIONS: A JOURNEY FROM POSITONE TO INFINITE SEMIPOSITONE PROBLEMS

SUBSOLUTIONS: A JOURNEY FROM POSITONE TO INFINITE SEMIPOSITONE PROBLEMS Seventh Mississippi State - UAB Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conf. 17 009), pp.13 131. ISSN: 107-6691. URL: http://ejde.math.txstate.edu

More information

ASYMPTOTIC THEORY FOR WEAKLY NON-LINEAR WAVE EQUATIONS IN SEMI-INFINITE DOMAINS

ASYMPTOTIC THEORY FOR WEAKLY NON-LINEAR WAVE EQUATIONS IN SEMI-INFINITE DOMAINS Electronic Journal of Differential Equations, Vol. 004(004), No. 07, pp. 8. ISSN: 07-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) ASYMPTOTIC

More information

MULTIPLE POSITIVE SOLUTIONS FOR P-LAPLACIAN EQUATION WITH WEAK ALLEE EFFECT GROWTH RATE

MULTIPLE POSITIVE SOLUTIONS FOR P-LAPLACIAN EQUATION WITH WEAK ALLEE EFFECT GROWTH RATE Differential and Integral Equations Volume xx, Number xxx,, Pages xx xx MULTIPLE POSITIVE SOLUTIONS FOR P-LAPLACIAN EQUATION WITH WEAK ALLEE EFFECT GROWTH RATE Chan-Gyun Kim 1 and Junping Shi 2 Department

More information

THE CAUCHY PROBLEM AND STEADY STATE SOLUTIONS FOR A NONLOCAL CAHN-HILLIARD EQUATION

THE CAUCHY PROBLEM AND STEADY STATE SOLUTIONS FOR A NONLOCAL CAHN-HILLIARD EQUATION Electronic Journal of Differential Equations, Vol. 24(24), No. 113, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) THE CAUCHY

More information

Closed geodesics in stationary manifolds with strictly convex boundary 1

Closed geodesics in stationary manifolds with strictly convex boundary 1 Differential Geometry and its Applications 13 (2) 251 266 251 North-Holland www.elsevier.com/locate/difgeo Closed geodesics in stationary manifolds with strictly convex boundary 1 A.M. Candela and A. Salvatore

More information

FRACTIONAL SOBOLEV SPACES WITH VARIABLE EXPONENTS AND FRACTIONAL P (X)-LAPLACIANS. 1. Introduction

FRACTIONAL SOBOLEV SPACES WITH VARIABLE EXPONENTS AND FRACTIONAL P (X)-LAPLACIANS. 1. Introduction FRACTIONAL SOBOLEV SPACES WITH VARIABLE EXPONENTS AND FRACTIONAL P (X-LAPLACIANS URIEL KAUFMANN, JULIO D. ROSSI AND RAUL VIDAL Abstract. In this article we extend the Sobolev spaces with variable exponents

More information

RELATIONSHIP BETWEEN SOLUTIONS TO A QUASILINEAR ELLIPTIC EQUATION IN ORLICZ SPACES

RELATIONSHIP BETWEEN SOLUTIONS TO A QUASILINEAR ELLIPTIC EQUATION IN ORLICZ SPACES Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 265, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu RELATIONSHIP

More information

ATTRACTORS FOR SEMILINEAR PARABOLIC PROBLEMS WITH DIRICHLET BOUNDARY CONDITIONS IN VARYING DOMAINS. Emerson A. M. de Abreu Alexandre N.

ATTRACTORS FOR SEMILINEAR PARABOLIC PROBLEMS WITH DIRICHLET BOUNDARY CONDITIONS IN VARYING DOMAINS. Emerson A. M. de Abreu Alexandre N. ATTRACTORS FOR SEMILINEAR PARABOLIC PROBLEMS WITH DIRICHLET BOUNDARY CONDITIONS IN VARYING DOMAINS Emerson A. M. de Abreu Alexandre N. Carvalho Abstract Under fairly general conditions one can prove that

More information

EXISTENCE RESULTS FOR QUASILINEAR HEMIVARIATIONAL INEQUALITIES AT RESONANCE. Leszek Gasiński

EXISTENCE RESULTS FOR QUASILINEAR HEMIVARIATIONAL INEQUALITIES AT RESONANCE. Leszek Gasiński DISCRETE AND CONTINUOUS Website: www.aimsciences.org DYNAMICAL SYSTEMS SUPPLEMENT 2007 pp. 409 418 EXISTENCE RESULTS FOR QUASILINEAR HEMIVARIATIONAL INEQUALITIES AT RESONANCE Leszek Gasiński Jagiellonian

More information

WELL-POSEDNESS OF WEAK SOLUTIONS TO ELECTRORHEOLOGICAL FLUID EQUATIONS WITH DEGENERACY ON THE BOUNDARY

WELL-POSEDNESS OF WEAK SOLUTIONS TO ELECTRORHEOLOGICAL FLUID EQUATIONS WITH DEGENERACY ON THE BOUNDARY Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 13, pp. 1 15. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu WELL-POSEDNESS OF WEAK SOLUTIONS TO ELECTRORHEOLOGICAL

More information

AN EIGENVALUE PROBLEM FOR THE SCHRÖDINGER MAXWELL EQUATIONS. Vieri Benci Donato Fortunato. 1. Introduction

AN EIGENVALUE PROBLEM FOR THE SCHRÖDINGER MAXWELL EQUATIONS. Vieri Benci Donato Fortunato. 1. Introduction Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume, 998, 83 93 AN EIGENVALUE PROBLEM FOR THE SCHRÖDINGER MAXWELL EQUATIONS Vieri Benci Donato Fortunato Dedicated to

More information

Numerical Solutions to Partial Differential Equations

Numerical Solutions to Partial Differential Equations Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Sobolev Embedding Theorems Embedding Operators and the Sobolev Embedding Theorem

More information

Spectrum of one dimensional p-laplacian Operator with indefinite weight

Spectrum of one dimensional p-laplacian Operator with indefinite weight Spectrum of one dimensional p-laplacian Operator with indefinite weight A. Anane, O. Chakrone and M. Moussa 2 Département de mathématiques, Faculté des Sciences, Université Mohamed I er, Oujda. Maroc.

More information

CONDITIONS FOR HAVING A DIFFEOMORPHISM BETWEEN TWO BANACH SPACES

CONDITIONS FOR HAVING A DIFFEOMORPHISM BETWEEN TWO BANACH SPACES Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 99, pp. 1 6. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu CONDITIONS FOR

More information

Local mountain-pass for a class of elliptic problems in R N involving critical growth

Local mountain-pass for a class of elliptic problems in R N involving critical growth Nonlinear Analysis 46 (2001) 495 510 www.elsevier.com/locate/na Local mountain-pass for a class of elliptic problems in involving critical growth C.O. Alves a,joão Marcos do O b; ;1, M.A.S. Souto a;1 a

More information