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

Size: px
Start display at page:

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

Transcription

1 Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 21, 2003, SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS Massimo Grosi Filomena Pacella S. L. Yadava Abstract. In this paper we prove a symmetry result for positive solutions of the Dirichlet problem (0.1) ( u = f(u) in D, u = 0 on D, when f satisfies suitable assumptions and D is a small symmetric perturbation of a domain for which the Gidas Ni Nirenberg symmetry theorem applies. We consider both the case when f has subcritical growth and f(s) = s (N+2)/(N 2) + λs, N 3, λ suitable positive constant. 1. Introduction Let us consider the following problem u = f(u) in, (1.1) u > 0 in, u = 0 on, where f: R R is a C 1 -function and is a bounded smooth domain in R N, N 2, which contains the origin and is symmetric with respect to the hyperplane 2000 Mathematics Subject Classification. 35B38, 35B50, 35J10, 35J60. Key words and phrases. Elliptic equations, symmetry of solutions. First and second authors are supported by M.U.R.S.T., project Variational methods and nonlinear differential equations. c 2003 Juliusz Schauder Center for Nonlinear Studies 211

2 212 M. Grosi F. Pacella S. L. Yadava T 0 = {x = (x 1,..., x N ) R N, x 1 = 0} and convex in the x 1 -direction. Under this hypothesis it is well known that classical C 2 () C 1 () solutions are even in x 1 and strictly increasing in the x 1 -variable in the cap = {x, x 1 < 0}. This is the famous symmetry result of Gidas, Ni and Nirenberg (see [7]) which is based on the method of moving planes which goes back to Alexandrov and Serrin in [11]. Now we would like to consider the same problem in some domains n which are suitable approximations of but are not any more convex in the x 1 -direction. The typical example is obtained by making one or more holes in, i.e. n = \ k i=1 B i where B i are small balls in whose radius tends to zero, as n. The question we address in this paper is whether the symmetry of the solutions of the same problem as in (1.1), but with replaced by n, is preserved. Let us note immediately that the moving planes method cannot be applied in n if the convexity in the x 1 -direction is destroyed but can only give information on the monotonicity of the solutions in x 1, in some subsets of n. Nevertheless in an interesting paper ([3]) Dancer proved, among other results, that for some subcritical nonlinearities, as for example f(s) = s p, p > 1 if N = 2, 1 < p < (N + 2)/(N 2) if N 3, if the solution u is unique and nondegenerate in then also the approximating problems in n have only one solution which is necessarily symmetric, if n is symmetric. So in this case the symmetry comes from the uniqueness of the solution. Thus the question is whether the symmetry is preserved even if the solution is not unique. Here we answer positively this question in two different cases: first we consider nonlinear terms f(s) with subcritical growth and then we take f(s) = s (N+2)/(N 2) + λs, with λ (0, λ 1 ) if N 4, λ 1 being the first eigenvalue of the Laplace operator with zero Dirichlet boundary condition, or λ (λ, λ 1 ) if N = 3, for a certain number λ > 0. In the subcritical case we prove that all solutions of the approximating problems are symmetric if n is sufficiently close to. In particular they are radial if n is an annulus with a small hole. In the critical case we prove the same result for least energy solutions which exist by the well-known Brezis Nirenberg result ([2]). We also prove that the least energy solution is unique and nondegenerate if the domains n are approximations of which is a ball. This last result is not contained in the uniqueness theorem of Dancer. Let us note that in the recent paper [10] it is shown, among other results, that if f is strictly convex, A is an annulus and u is a solution of (1.1) in A of index one, then u is axially symmetric. Thus, in the case when n is an annulus and f is convex our result extend that of [10] because it states that if the hole is sufficiently small the solution of index one are not only axially symmetric but actually radially symmetric.

3 Symmetry Results for Perturbed Problems and Related Questions 213 To prove our result the main idea is to exploit the sign of the first eigenvalue of the linearized operator L = f (u)i at a solution u in the cap = {x R N, x 1 < 0} or + = {x R N, x 1 > 0}. In fact, if f(0) 0 it was shown in a lecture by L. Nirenberg (see [6]) that this eigenvalue is indeed positive; in particular zero is not an eigenvalue of L in or +. From this and some results on the convergence of the solutions, we deduce the symmetry of the solutions u n in n, with respect to the x 1 -variable when n is symmetric with respect to the hyperplane T 0 = {x 1 = 0}. We also get the radial symmetry of the solutions when the domains n are annuli with a small hole. The outline of the paper is the following. In Section 2 we prove some preliminary theorems on the convergence of the eigenvalues of linear operators and we also show the uniform L -estimates in the subcritical case, already proved by Dancer (see [5]). In Section 3 we study the subcritical case while in Section 4 we deal with the critical case. Finally let us remark that the perturbed domains we consider are obtained by the initial domain by removing from it a finite number of symmetric subdomains, i.e. making finitely many holes whose size tends to zero. It could be possible analyze other kind of perturbations but, for simplicity, we will not consider them. Acknowledgements. This research started while the third author was visiting the Department of Mathematics of the University of Roma La Sapienza. He would like to thank the department for the financial support and the warm hospitality. 2. Preliminary results Let be a bounded domain in R N, N 2 with smooth C 2 -boundary and D i, i = 1,..., k smooth open subsets of, star shaped with respect to an interior point y i and such that D i D j =, for i j. Then we define the homothetic domains D i n = [ε i n(d i y i )] + y i with respect to y i, with the sequences ε i n converging to zero as n. Our approximating domains will be (2.1) n = \ k Dn. i In n we consider the linear operators L n = a n (x)i where is the Laplace operator, I is the identity and a n L ( n ). Analogously we consider in the linear operator L = a(x)i with a L (). For these operators we would like to prove some results about the convergence of the first and second eigenvalue. i=1

4 214 M. Grosi F. Pacella S. L. Yadava Proposition 2.1. Assume that a n converges to the function a in L N/2 (), N 3. Then the first eigenvalues λ 1 (L n, n ) in n with zero Dirichlet boundary conditions, converge to the first eigenvalue λ 1 (L, ) analogously defined. Proof. Let us set λ 1,n = λ 1 (L n, n ), λ 1 = λ 1 (L, ) and show that the sequence {λ 1,n } is bounded. By the variational characterization we have that (2.2) λ 1,n φ 2 dx n a n φ 2 dx n for a function φ C0 (B), B φ2 dx = 1, where B is a ball contained in any n. Hence by (2.2) (2.3) λ 1,n φ 2 dx + (a a n )φ 2 dx aφ 2 dx B B ( ) 2/N ( φ 2 dx + a a n N/2 dx φ 2 dx B B B + a φ 2 dx C B B ) 2/2 for a suitable constant C, having denoted by 2 the critical Sobolev exponent, 2 = 2N/(N 2). So {λ 1,n } is bounded from above. Let φ n > 0 be a first eigenfunction of L n in n with n φ 2 n dx = 1. Then, extending φ n by zero to the whole domain we get (2.4) λ 1,n = φ2 n dx a nφ 2 n dx φ2 n dx The sequence φ n converges weakly in H 1 0 () and strongly in L 2 () to a function φ H 1 0 () which cannot be zero otherwise from (2.4) and the convergence of a n to a in L N/2 () we would get that λ 1,n against what we proved. Then, from (2.4) we deduce (2.5) λ 1,n 1 + ( a n N/2 dx) 2/N ( φ2 φ2 n dx n dx) 2/2 for a suitable constant C. Hence λ 1,n is bounded from below and thus converges, up to a subsequence, to a number λ. Moreover, from the weak convergence of φ n to φ in H0 1 () we get that φ is nonnegative and solves the problem C (2.6) { φ aφ = λφ in, φ = 0 on, Since we already proved that φ 0, by the strong maximum principle we get that φ > 0 and hence is a first eigenfunction of L in, i.e. λ = λ 1.

5 Symmetry Results for Perturbed Problems and Related Questions 215 In the sequel we shall need to estimate the sign of the second eigenvalue of L n in n. Therefore, with the same notations as in Proposition 2.1 we prove Proposition 2.2. If a n converges to a in L N/2 (), N 3 and λ 2 = λ 2 (L, ) > 0 then also λ 2,n = λ 2 (L n, n ) is positive for n sufficiently large. Proof. Arguing by contradiction let us assume that for a subsequence that we still denote in the same way, λ 2,n 0. Then, since λ 1,n < λ 2,n and the sequence {λ 1,n } is bounded by Proposition 2.1, the same holds for {λ 2,n } and hence it converges, up to another subsequence, to a number λ 0. Then, considering a sequence of second eigenfunctions φ 2,n in n with φ2 2,n dx = 1 and extending them to zero to the whole domain, we have that φ 2,n φ weakly in H0 1 (). As in the previous proposition, using the variational characterization of λ 2,n we get that φ 0 and is a solution of (2.7) { φ a φ = λ φ in, φ = 0 on. So φ is an eigenfunction of L corresponding to the eigenvalue λ 0. Since, by hypothesis, λ 2 > 0, the only possibility is that λ = λ 1 (L, ) and hence φ is a first eigenfunction of L in. If φ 1,n is a sequence of first eigenfunctions of L n in n we have the orthogonality condition (2.8) φ 1,n φ 2,n dx = 0. Since, by the previous proposition, we have that also φ 1,n φ weakly in H0 1 (), from (2.8), passing to the limit we get (2.9) φ 2 dx = 0 which is impossible. Hence λ 2,n > 0 for n sufficiently large. Now we consider the following semilinear elliptic problem in n u = f(u) in n, (2.10) u > 0 on n, u = 0 on n, where f: R R is of class C 1. We would like to get some uniform L -estimates for the solutions of (2.10) when f is subcritical. This was already shown by Dancer in [5] using the Gidas Spruck approach (see [8]). For the reader s convenience we sketch the proof here.

6 216 M. Grosi F. Pacella S. L. Yadava Proposition 2.3. Let u C 2 ( n ) be a classical solution of (2.10) with f satisfying f(s) (2.11) lim s s p = a > 0, where 1 < p < (N + 2)/(N 2) if N 3, p > 1 if N = 2. Then there exists a number C > 0, independent of u and n such that (2.12) u L ( n) C Proof. Arguing by contradiction we assume that for a sequence {u n } of solutions of (2.10) we have (2.13) u n L ( n) = u n (x n ) for some sequence of points x n n. Let us set ( 1 x v n (x) = u n u n u n (p 1)/2 These functions satisfy + x n ). (2.14) 1 u = u n p f( u n v n ) in n, v n (0) = 1 on n, 0 < v n 1 on n, where n = ( n x n ) u n (p 1)/2. From (2.11) we get (2.15) f( u n v n ) u n p = vp nf( u n v n ) ( u n v n ) p C for some positive constant C. Now let us fix any compact set K n, such that (2.16) d(k, n ) α > 0 where d(k, n ) is the distance of K from n. Since the right hand side in the equation (2.14) is uniformly bounded, by the standard regularity theory we have that v n converges to a function v in C 1 (K). Therefore, by (2.11) and (2.14) we get (2.17) { v = av p in D, 0 < v 1 in D,

7 Symmetry Results for Perturbed Problems and Related Questions 217 where D is the limit domain whose shape depends on the comparison between the rate of divergence of u n (p 1)/2 and of convergence to zero of the parameters ε i n (which define the size of the holes). As shown in [5] there are four possibilities (2.18) (i) D = R N, (ii) D = R N + = {x = (x 1,..., x N ) R N, x 1 > 0}, (iii) D = R N \ {0}, (iv) D = R N \ αd i for some i {1, 2,..., k} and α = lim n εi n u n (p 1)/2. The first two cases (i) and (ii) are excluded as in [8] because of the nonexistence of nontrivial solutions of (2.17) in R N or R N +. The case (iii) is excluded because if D = R N \ {0} only singular solutions at zero can exist while v is bounded (see [9] and [5]). Hence, the only possibility is that v 0 a.e. in R N, but, this can be excluded arguing exactly as in [5] (see Theorem 2 and proof of (ii) of Theorem 1 therein). Finally, by Theorem 1 of [5], also case (iv) is excluded, since the subdomains D i are star shaped. Hence (2.12) holds. Remark 2.4. The hypothesis that the subdomains D i are star shaped is only used in the proof of (iv) of the previous proposition. We would like to point out that it is not needed when p N/(N 2) (see Theorem 1 of [5]). Now we assume that contains the origin and is symmetric with respect to the hyperplane T 0 = {x = (x 1,..., x N ) R N, x 1 = 0} and convex in the x 1 -direction. We also define the caps = {x, x 1 < 0} and + = {x, x 1 > 0}. We end this section by recalling the following result. Proposition 2.5. Let u be a positive solution of the semilinear problem (2.19) { u = f(u) in, u = 0 on, where f: R R is a C 1 -function with f(0) 0. Then the first eigenvalue of the linearized operator L u = f (u)i in (or + ) with zero Dirichlet boundary conditions is positive. Proof. The statement was proved in a lecture by L. Nirenberg (see also [6, Theorem 2.1]). We recall here the simple proof. By the symmetry result of [7] any positive solution of (2.19) is symmetric and u/ x 1 > 0 in. Deriving (2.19) we have that the function u/ x 1 solves

8 218 M. Grosi F. Pacella S. L. Yadava the linearized equation, i.e. ( ) u (2.20) f (u) u = 0 x 1 x 1 in and, by the condition f(0) 0, the Hopf s Lemma applied to the solution u implies that u/ x 1 0 on. Since u/ x 1 > 0 in this yelds the validity of the maximum principle in which, in turns, is equivalent to claim that the first eigenvalue of L u is positive in. Since u is symmetric the same holds in The subcritical case As at the beginning of the previous section we assume that is a smooth bounded domain containing the origin, symmetric with respect to the hyperplane T 0 and convex in the x 1 -direction. We also take the smooth star shaped subdomains D i, i = 1,..., k in such a way that the domains n = \ k i=1 Di n, ε i n 0 (see Section 2 for the precise definition) are also symmetric with respect to T 0, but, of course, they are not any more convex in the x 1 -direction. Let f: R R be a C 1 -function such that f(s) (3.1) lim s s p = a > 0, where 1 < p < (N + 2)/(N 2) if N 3, p > 1 if N = 2 and (3.2) f(0) 0, f (0) λ 1 where λ 1 is the first eigenvalue of the Laplace operator, with zero Dirichlet boundary conditions in. With this nonlinearity we consider the following semilinear problem u = f(u) in D, (3.3) u > 0 on D, u = 0 on D, where D is either n or. Remark 3.1. If f is convex at zero and f(0) = 0 it is easy to see that a necessary condition to have positive solutions of (3.3) in is to require f (0) < λ 1 and the same is true in n. Since the first eigenvalue λ 1,n of the Laplace operator in H 1 0 ( n ) converges to λ 1 if we have f (0) < λ 1 then also f (0) < λ 1,n for sufficiently large n. Hence the requirement f (0) λ 1 in (3.2) is often not a real hypothesis. Now we show the convergence of the solutions of (3.3) in n to the solution of (3.3) in. This was already proved by Dancer ([3]); since some steps of the proof will be also used later we give all details here.

9 Symmetry Results for Perturbed Problems and Related Questions 219 Theorem 3.2. Let u n be a solution of (3.3) in n. Then the sequence {u n } converges weakly in H 1 0 () to a solution u 0 of (3.3) in. Proof. Let us extend the functions u n to the whole domain giving the value zero outside of n. By Proposition 2.3 we know that the functions {u n } are uniformly bounded in the L -norm and by (3.3) they are bounded in H0 1 (), so that, up to a subsequence that we still denote by u n, we have that u n u 0 weakly in H0 1 (). Let us show that u 0 is a weak solution of (3.3) in. To do this, it is enough to use as test functions those belonging to the set V = {ψ H0 1 () such that ψ H0 1 ( n ), for some n N}. In fact the set V is dense in H0 1 (), because the subdomains Dn i reduce to a finite number of points, as n. So, fixed ψ V we have that there exists n N such that ψ H0 1 ( n ) and hence ψ H0 1 ( n ), for any n n. Then, by (3.3), we have (3.4) u n ψ dx = f(u n )ψ dx for all n n. Since u n u 0 in H0 1 () we have that u n ψ dx u 0 ψ dx and also f(u n ) f(u 0 ) a.e. in. Thus, since the sequence {u n } is uniformly bounded, using the dominated convergence theorem, we get f(u n)ψ dx f(u 0)ψ dx. By this and (3.4) we deduce that u 0 is a weak solution of (3.3) in and hence, by standard regularity theorems, u 0 is also a classical C 2 () solution. Obviously u 0 0 so that, by the hypothesis f(0) 0 we can apply the strong maximum principle claiming that either u 0 > 0, as we wanted prove, or u 0. In the last case f(0) must be zero. So, considering the functions v n = u n / u n L 2 (), we have that v n > 0 and ( 1 ) (3.5) v n = f (tu n ) dt v n in n. 0 Moreover, up to a subsequence, v n v 0 in H0 1 () while v n v 0 strongly in L 2 (). The limit function v 0 cannot be zero because we have ( 1 (3.6) 1 = vn 2 dx = f (tu n ) dt )v 2n dx C 1 vn 2 dx 0 n n where C 1 = f (s) L ([0,C]), C being the constant which appears in (2.12). Passing to the limit in (3.5), arguing as for (3.4), we get that v 0 is a solution of { v0 = f (0)v 0 in, (3.7) v 0 = 0 on. Since v 0 0 and v 0 0, by the strong maximum principle we have v 0 > 0 which implies that f (0) = λ 1, against the hypothesis. So u 0 is a positive solution of (3.3) in.

10 220 M. Grosi F. Pacella S. L. Yadava Now we show the symmetry of the solutions of (3.3) when the domains D i n are sufficiently small. Theorem 3.3. For any nonlinearity f satisfying (3.1) and (3.2) there exists n 0 N such that for any n n 0 all solutions of problem (3.3) in n are even in the x 1 -variable. Proof. Arguing by contradiction let us assume that there exists a sequence {u n } of solutions of (3.3) in n such that u n are not even in x 1. This implies that, denoting by n the set {x n, such that x 1 < 0} the functions w n (x) = u n (x 1,..., x N ) u n ( x 1, x 2,..., x N ) are not identically zero in n and actually, since they are continuous, they are not zero in a set of positive measure. Therefore, extending w n by zero to the whole set = {x, x 1 < 0} we can define in the functions v n = w n / w n L 2 ( ) which belong to H0 1 ( ) since they are zero on. It is easy to see that v n satisfy ( 1 ) (3.8) v n = f (tu n (x) + (1 t)u n ( x 1, x 2,..., x N )) dt in 0 and converge weakly in H0 1 ( ) to a function v 0, while v n v 0 in L 2 ( ). Exactly as in Theorem 3.2 we prove that v 0 0 and v 0 is a solution of v 0 = f (u 0 )v 0 in, (3.9) v 0 = 0 on, v 0 0 in, where u 0 is the limit of the solutions u n that, by Theorem 3.2, is a positive solution of (3.3) in. Then, by Proposition 2.5, we know that the first eigenvalue of the linearized operator f (u 0 )I in, with zero Dirichlet boundary conditions, is positive. This contradicts (3.9) which implies that zero is an eigenvalue of f (u 0 )I in. Thus the assertion holds We also get the radial symmetry of the solutions when n are annuli and n is sufficiently large. Corollary 3.4. Let n be annuli. Then for any nonlinearity f satisfying (3.1) and (3.2) there exists n N such that for any n n all solutions of problem (3.3) in n are radial. Proof. Since n is symmetric with respect to any hyperplane T ν passing through the origin and orthogonal to a direction ν S N 1, (S N 1 being the unit sphere in R N ), to each T ν the previous theorem applies and gives an integer n ν such that, for any n ν, all solutions of (3.3) in n are symmetric with respect to the hyperplane T ν. To prove the statement we need to show that there exists a positive integer n such that n ν n, for any ν S N 1. v n

11 Symmetry Results for Perturbed Problems and Related Questions 221 Arguing by contradiction we construct a sequence of directions ν k such that n νk and, up to a subsequence ν k ν 0 S N 1. This means that we have a sequence of solutions {u k } in k, k which are not symmetric with respect to the hyperplane T νk = {x R N : x ν k = 0}. This implies that the functions (3.10) w k = u k (x) u k (x νk ), x k = {x k : x ν k < 0} where x νk is the reflected point of x with respect to T νk, are not zero in a set of positive measure. Hence considering the limit symmetry hyperplane T ν0 = {x R N : x ν 0 = 0} and extending w k by zero to the domain we can define the functions w k v k = w k L2 () which satisfy the equation ( 1 ) v k = f (tu k (x) + (1 t)u k (x νk )) dt v k in k. 0 Exactly, as in Theorems 3.2 and 3.3, we prove that v k converges weakly in H0 1 () to a function v 0 0 such that { v0 = f (u 0 )v 0 in 0, (3.11) v 0 = 0 on 0, where u 0 is the limit of the solutions u k, which is symmetric with respect to T ν0. Therefore, by Proposition 2.5, we reach the same contradiction as in Theorem The critical case Keeping the previous notation we consider a bounded smooth domain containing the origin, symmetric with respect to the hyperplane T 0 and convex in the x 1 -direction and denote by n the approximating domains as defined in Sections 2 and 3. We study the following critical semilinear problem u = u (N+2)/(N 2) + λu in D, (4.1) u > 0 in D, u = 0 on D, where D is either or n, N 4 and λ (0, λ 1 ) where λ 1 is the first eigenvalue of the Laplace operator in with Dirichlet boundary conditions. Note that, by the continuity of this eigenvalue with respect to the domain, we have that λ (0, λ 1,n ) for n sufficiently large, having denoted by λ 1,n the first eigenvalue of in n with the same boundary conditions.

12 222 M. Grosi F. Pacella S. L. Yadava Thus by a famous result of Brezis and Nirenberg, we have that, for every λ (0, λ 1 ), there exists a solution of (4.1), both in or n which, up to a multiplier, minimizes the functional (4.2) Q λ (u) = u 2 λ u 2 D D among the functions u H 1 0 (D) with D u2 = 1 where 2 = 2N/(N 2) = (N + 2)/(N 2) + 1 and D is either or n. If N = 3, the same existence result holds in a suitable neighbourhood of λ 1, say (λ, λ 1 ), (see [2]). As a consequence for N = 3, it will understood that we will take λ (λ, λ 1 ). In particular, if is a ball λ = λ 1 /4. Let us denote by S λ and S λ n the infimum of (4.2) in H 1 0 () and H 1 0 ( n ) respectively. We start by proving the following result. Proposition 4.1. S λ n S λ as n. Proof. Since n we have that S λ S λ n for any n 1. So if we prove that (4.3) lim n Sλ n S λ the claim follows. Let us consider the function η r : B(0, 2r) R defined as 2 n 2 ( ) r n 2 (4.4) η r (x) = 1 2 n 2 x n 2 1 if r < x < 2r, 0 if x r. Recalling the definition of D i n let us consider the smallest number r i n such that D i n B(y i, r i n) and set (4.5) ζ n (x) = { ηr i n (x y i) if x y i < 2r i n, 1 elsewhere. Finally we consider the function v n H 1 0 ( n ), ζ n v 0 v n =, ζ n v 0 L p+1 () where v 0 is the function which minimizes Q λ (u) in. Let us compute Q λ (v n ). By Lebesgue dominated convergence theorem we get (4.6) vn 2 dx = ζ2 nv0 2 dx ζ n v 0 2 v2 0 dx L p+1 () ( vp+1 0 ). 2/(p+1)

13 Symmetry Results for Perturbed Problems and Related Questions 223 Concerning the integral v n 2 dx we have the following estimate (4.7) (ζ n v 0 ) 2 dx = v 0 2 ζn 2 dx + ζn 2 2 v0 2 dx + 2 v 0 ζ n v 0 ζ n dx = The estimates (4.6) and (4.7) imply that ( k v 0 2 dx + O i=0 r i(n 2) n ). (4.8) lim n Q λ(v n ) = Q λ (v 0 ) = S λ which proves (4.3). Let v n be a sequence of minima of (4.2) in H 1 0 ( n ) corresponding to a fixed value of λ belonging to (0, λ 1 ) if N 4 or to (λ, λ 1 ) if N = 3. We extend v n by zero to the whole domain and we have Proposition 4.2. The sequence v n converges strongly in H 1 0 () to a function v 0 which is a minimizer of (4.2) in H 1 0 (). Proof. By definition we have that (4.9) vn 2 dx = 1 and v n 2 dx λ v 2 n = S λ n. Since by the previous proposition Sn λ S λ we have that {v n } is a minimizing sequence for the functional Q λ defined in (4.2), in the space H0 1 (). By the result of Brezis and Nirenberg ([2]) we know that for the value of λ considered, S λ is smaller than S which is the best Sobolev constant for the imbedding of H0 1 () into L 2 (). Again by a result of [2] we have that this implies that v n converges strongly in H0 1 () to a function v 0 which is a minimizer of (4.2) in H0 1 (). The minimizer v 0 (respectively v n ) of (4.2) are obviously solutions of the problem v = µv (N+2)/(N 2) + λv in D, (4.10) v > 0 in D, v = 0 on D, where D is either or n and µ is a suitable Lagrange multiplier, namely µ = S λ or µ = Sn. λ Then it is easy to see that the function u = µ 1/(p 1) v, p = (N +2)/(N 2) is a solution of (4.1). These solutions, obtained through the minimization procedure, will be called least-energy solutions of (4.1). To prove the symmetry of these solutions we can either argue as in the subcritical case or, since the nonlinearity f(s) = s (N+2)/(N 2) + λs is strictly convex, exploit the following proposition which is proved in [10].

14 224 M. Grosi F. Pacella S. L. Yadava Proposition 4.3. Let us denote by λ 1 (L n, n ) and λ 1 (L n, + n ) the first eigenvalues of the linearized operators L n = f (u n )I in the domains n = {x n, x 1 < 0}, + n = {x n, x 1 > 0}. If they are both nonnegative then u n is even in x 1. Proof. See Proposition 1.1 in [10]. Theorem 4.4. For every λ (0, λ 1 ) if N 4, (λ, λ 1 ) if N = 3, there exists n 0 N such that for every n n 0 all least-energy solutions of (4.1) in n are even in the x 1 -variable. Proof. Arguing by contradiction let us assume that there is a sequence {u n } of least-energy solutions of (4.1) in n which are not even in x 1. By Proposition 4.2 {u n } converges strongly in H0 1 () to a least energy solution u 0 of (4.1) in and hence a n (x) = (pun p 1 + λ)x, p = (N + 2)/(N 2) converges in the space L N/2 () to the function a(x) = (pu p λ)x. By Proposition 2.1 applied to the linearized operators L n = a n (x)i and L = a(x)i we have that the eigenvalues λ 1 (L n, n ) λ 1 (L, ) and λ 1 (L n, + n ) λ 1 (L, + ). By Proposition 2.5 we know that λ 1 (L, ) and λ 1 (L, + ) are both positive so that also λ 1 (L n, n ) and λ 1 (L n, + n ) are positive for n sufficiently large. This, in turns, implies, by Proposition 4.3, that the functions u n are even in x 1 against what we assumed. Alternatively, arguing as in the proof of Theorem 3.3 we could consider the functions w n (x) = u n (x 1, x 2,..., x N ) u n ( x 1, x 2,..., x N ) in n and then set v n = w n / w n L 2 ( ). The functions v n satisfy (3.8) in n with f (s) = s 4/(N 2) λ and converge weakly in H 1 0 ( ) to a function v 0. To prove that v 0 0, in Theorem 3.3 we used the uniform L -estimates which are not known when the nonlinearity has a critical growth. In our case we observe that, by Proposition 4.2, u n u 0 in H 1 0 () and hence in L 2N/(N 2) (). Therefore (4.11) u 4/(N 2) n u 4/(N 2) 0 in L N/2 () and this implies that v 0 is a solution of (3.11) and v 0 0. In fact (4.12) 1 = vn 2 dx = n ( 1 0 ) (tu n (x) + (1 t)u n ( x 1, x 2,..., x N )) 4/(N 2) dt vn 2 dx. If v 0 0 then v n 0 weakly in H0 1 () and using (4.11) we have that ( 1 ) (4.13) (tu n (x) + (1 t)u n ( x 1, x 2,..., x N )) 4/(N 2) dt vn 2 dx 0 n 0

15 Symmetry Results for Perturbed Problems and Related Questions 225 and it gives a contradiction with (4.12). So v 0 0. After this we can repeat exactly the same proof as in Theorem 3.3 and then the same contradiction arises. Corollary 4.5. Let n be annuli. Then for every λ in the intervals considered there exists n N such that for any n n all solutions of problem (4.1) in n are radial. Proof. It is similar to that of Corollary 3.4, using (4.11) instead of the uniform L -estimates. If is a ball we know that problem (4.1) has only one solution which is also non degenerate (see [1] or [12]). We end by showing that the same is true for the approximating domains n. Theorem 4.6. Let be a ball. Then, for every λ in the intervals considered there exists n N such that, for any n n, problem (4.1) in n has only one least-energy solution which is also nondegenerate. Proof. By contradiction let us assume that there exist two sequences of least-energy solutions of (4.11) in n, say {u 1,n }, {u 2,n }, with {u 1,n } {u 2,n }. As usual we extend them by zero in and define the functions u 1,n u 2,n (4.14) v n = (u 1,n u 2,n ) L 2 () which satisfy ( 1 ) v n = p (tu 1,n + (1 t)u 2,n ) p 1 dt v n + λv n in n, (4.15) 0 v n = 0 on n, where p = (N + 2)/(N 2). Since v n 2 dx = 1 we get that v n v 0 in H0 1 (). By Proposition 4.2 both {u 1,n } and {u 2,n } converge strongly in H0 1 () to the unique least-energy solution u 0 of (4.1) in the ball. Using this and arguing as in the second part of the Proof of Theorem 4.4 we deduce that v 0 0. Then, passing to the limit in (4.15), we get that v 0 is a solution of the linearized equation at u 0 in, i.e. { v0 pu p 1 0 v 0 λv 0 = 0 in, (4.16) v 0 = 0 on. Since u 0 is a nondegenerate solution of (4.1) (see [1], [12]) we reach a contradiction which shows that {u 1,n } {u 2,n } for n sufficiently large. Finally, since u 0 is a least-energy solution of (4.1) in the ball we also have that it has index one, i.e. the second eigenvalue of the linearized operator at u 0 is positive in. Therefore, using the strong convergence of the least-energy solutions u n of (4.1) in n and Proposition 2.2, we get that the second eigenvalue

16 226 M. Grosi F. Pacella S. L. Yadava of the linearized operators at u n in n are also positive. This implies that the solutions u n are nondegenerate. From the proof of Theorem 4.6 it is obvious that the same result holds if is not a ball but any domain where (4.1) has only one non degenerate least-energy solution. References [1] Adimurthi and S. L. Yadava, An elementary proof of the uniqueness of positive radial solutions of a quasilinear Dirichlet problem, Arch. Rational Mech. Anal. 127 (1994), [2] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), [3] E. N. Dancer, The effect of domain shape on the number of positive solutions of certain weakly nonlinear equations, J. Differential Equations 74 (1988), [4], On the number of positive solutions of some weakly nonlinear equations on annular regions, Math. Z. 206 (1991), [5], Superlinear problems on domains with holes of asymptotic shape and exterior problems, Math. Z. 229 (1998), [6] L. Damascelli, M. Grossi and F. Pacella, Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle, Ann. Inst. H. Poincaré, Anal. Non Linéaire 16 (1999), [7] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), [8] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), [9], Global and local behaviour of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), [10] F. Pacella, Symmetry results for solutions of semilinear elliptic equations with convex nonlinearities, J. Funct. Anal. 192 (2002), [11] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), [12] P. N. Srikanth, Uniqueness of solutions of nonlinear Dirichlet problems, Differential Integral Equations 6 (1993), Massimo Grosi and Filomena Pacella Dipartimento di Matematica Università di Roma La Sapienza P. le Aldo Moro Roma, ITALY address: grossi@mat.uniroma1.it, pacella@mat.uniroma1.it S. L. Yadava T.I.F.R. Centre P.O. Box 1234 Bangalore, , INDIA address: yadava@math.tifrbng.in TMNA : Volume N o 2 Manuscript received September 16, 2001

SYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS. M. Grossi S. Kesavan F. Pacella M. Ramaswamy. 1. Introduction

SYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS. M. Grossi S. Kesavan F. Pacella M. Ramaswamy. 1. Introduction Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 12, 1998, 47 59 SYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS M. Grossi S. Kesavan F. Pacella M. Ramaswamy

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

Symmetry of nonnegative solutions of elliptic equations via a result of Serrin

Symmetry of nonnegative solutions of elliptic equations via a result of Serrin Symmetry of nonnegative solutions of elliptic equations via a result of Serrin P. Poláčik School of Mathematics, University of Minnesota Minneapolis, MN 55455 Abstract. We consider the Dirichlet problem

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

SYMMETRY OF SOLUTIONS TO SEMILINEAR ELLIPTIC EQUATIONS VIA MORSE INDEX

SYMMETRY OF SOLUTIONS TO SEMILINEAR ELLIPTIC EQUATIONS VIA MORSE INDEX PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 135, Number 6, June 2007, Pages 1753 1762 S 0002-9939(07)08652-2 Article electronically published on January 31, 2007 SYMMETRY OF SOLUTIONS TO SEMILINEAR

More information

Existence of Positive Solutions to Semilinear Elliptic Systems Involving Concave and Convex Nonlinearities

Existence of Positive Solutions to Semilinear Elliptic Systems Involving Concave and Convex Nonlinearities Journal of Physical Science Application 5 (2015) 71-81 doi: 10.17265/2159-5348/2015.01.011 D DAVID PUBLISHING Existence of Positive Solutions to Semilinear Elliptic Systems Involving Concave Convex Nonlinearities

More information

Uniqueness of ground states for quasilinear elliptic equations in the exponential case

Uniqueness of ground states for quasilinear elliptic equations in the exponential case Uniqueness of ground states for quasilinear elliptic equations in the exponential case Patrizia Pucci & James Serrin We consider ground states of the quasilinear equation (.) div(a( Du )Du) + f(u) = 0

More information

I Results in Mathematics

I Results in Mathematics Result.Math. 45 (2004) 293-298 1422-6383/04/040293-6 DOl 10.1007/s00025-004-0115-3 Birkhauser Verlag, Basel, 2004 I Results in Mathematics Further remarks on the non-degeneracy condition Philip Korman

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

ON NONHOMOGENEOUS BIHARMONIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT

ON NONHOMOGENEOUS BIHARMONIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT PORTUGALIAE MATHEMATICA Vol. 56 Fasc. 3 1999 ON NONHOMOGENEOUS BIHARMONIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT M. Guedda Abstract: In this paper we consider the problem u = λ u u + f in, u = u

More information

The Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation:

The Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: Oct. 1 The Dirichlet s P rinciple In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: 1. Dirichlet s Principle. u = in, u = g on. ( 1 ) If we multiply

More information

Regularity and nonexistence results for anisotropic quasilinear elliptic equations in convex domains

Regularity and nonexistence results for anisotropic quasilinear elliptic equations in convex domains Regularity and nonexistence results for anisotropic quasilinear elliptic equations in convex domains Ilaria FRAGALÀ Filippo GAZZOLA Dipartimento di Matematica del Politecnico - Piazza L. da Vinci - 20133

More information

Minimization problems on the Hardy-Sobolev inequality

Minimization problems on the Hardy-Sobolev inequality manuscript No. (will be inserted by the editor) Minimization problems on the Hardy-Sobolev inequality Masato Hashizume Received: date / Accepted: date Abstract We study minimization problems on Hardy-Sobolev

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

Global unbounded solutions of the Fujita equation in the intermediate range

Global unbounded solutions of the Fujita equation in the intermediate range Global unbounded solutions of the Fujita equation in the intermediate range Peter Poláčik School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA Eiji Yanagida Department of Mathematics,

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

A Dirichlet problem in the strip

A Dirichlet problem in the strip Electronic Journal of Differential Equations, Vol. 1996(1996), No. 10, pp. 1 9. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp (login: ftp) 147.26.103.110 or 129.120.3.113

More information

ON A CONJECTURE OF P. PUCCI AND J. SERRIN

ON A CONJECTURE OF P. PUCCI AND J. SERRIN ON A CONJECTURE OF P. PUCCI AND J. SERRIN Hans-Christoph Grunau Received: AMS-Classification 1991): 35J65, 35J40 We are interested in the critical behaviour of certain dimensions in the semilinear polyharmonic

More information

REGULARITY AND COMPARISON PRINCIPLES FOR p-laplace EQUATIONS WITH VANISHING SOURCE TERM. Contents

REGULARITY AND COMPARISON PRINCIPLES FOR p-laplace EQUATIONS WITH VANISHING SOURCE TERM. Contents REGULARITY AND COMPARISON PRINCIPLES FOR p-laplace EQUATIONS WITH VANISHING SOURCE TERM BERARDINO SCIUNZI Abstract. We prove some sharp estimates on the summability properties of the second derivatives

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

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

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

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

New exact multiplicity results with an application to a population model

New exact multiplicity results with an application to a population model New exact multiplicity results with an application to a population model Philip Korman Department of Mathematical Sciences University of Cincinnati Cincinnati, Ohio 45221-0025 Junping Shi Department of

More information

A LOWER BOUND FOR THE GRADIENT OF -HARMONIC FUNCTIONS Edi Rosset. 1. Introduction. u xi u xj u xi x j

A LOWER BOUND FOR THE GRADIENT OF -HARMONIC FUNCTIONS Edi Rosset. 1. Introduction. u xi u xj u xi x j Electronic Journal of Differential Equations, Vol. 1996(1996) No. 0, pp. 1 7. ISSN 107-6691. URL: http://ejde.math.swt.edu (147.6.103.110) telnet (login: ejde), ftp, and gopher access: ejde.math.swt.edu

More information

Symmetry breaking for a problem in optimal insulation

Symmetry breaking for a problem in optimal insulation Symmetry breaking for a problem in optimal insulation Giuseppe Buttazzo Dipartimento di Matematica Università di Pisa buttazzo@dm.unipi.it http://cvgmt.sns.it Geometric and Analytic Inequalities Banff,

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

TOPICS IN NONLINEAR ANALYSIS AND APPLICATIONS. Dipartimento di Matematica e Applicazioni Università di Milano Bicocca March 15-16, 2017

TOPICS IN NONLINEAR ANALYSIS AND APPLICATIONS. Dipartimento di Matematica e Applicazioni Università di Milano Bicocca March 15-16, 2017 TOPICS IN NONLINEAR ANALYSIS AND APPLICATIONS Dipartimento di Matematica e Applicazioni Università di Milano Bicocca March 15-16, 2017 Abstracts of the talks Spectral stability under removal of small capacity

More information

Symmetry and monotonicity of least energy solutions

Symmetry and monotonicity of least energy solutions Symmetry and monotonicity of least energy solutions Jaeyoung BYEO, Louis JEAJEA and Mihai MARIŞ Abstract We give a simple proof of the fact that for a large class of quasilinear elliptic equations and

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

Symmetry and non-uniformly elliptic operators

Symmetry and non-uniformly elliptic operators Symmetry and non-uniformly elliptic operators Jean Dolbeault Ceremade, UMR no. 7534 CNRS, Université Paris IX-Dauphine, Place de Lattre de Tassigny, 75775 Paris Cédex 16, France E-mail: dolbeaul@ceremade.dauphine.fr

More information

Some nonlinear elliptic equations in R N

Some nonlinear elliptic equations in R N Nonlinear Analysis 39 000) 837 860 www.elsevier.nl/locate/na Some nonlinear elliptic equations in Monica Musso, Donato Passaseo Dipartimento di Matematica, Universita di Pisa, Via Buonarroti,, 5617 Pisa,

More information

Heat equations with singular potentials: Hardy & Carleman inequalities, well-posedness & control

Heat equations with singular potentials: Hardy & Carleman inequalities, well-posedness & control Outline Heat equations with singular potentials: Hardy & Carleman inequalities, well-posedness & control IMDEA-Matemáticas & Universidad Autónoma de Madrid Spain enrique.zuazua@uam.es Analysis and control

More information

A nodal solution of the scalar field equation at the second minimax level

A nodal solution of the scalar field equation at the second minimax level Bull. London Math. Soc. 46 (2014) 1218 1225 C 2014 London Mathematical Society doi:10.1112/blms/bdu075 A nodal solution of the scalar field equation at the second minimax level Kanishka Perera and Cyril

More information

Global Solutions for a Nonlinear Wave Equation with the p-laplacian Operator

Global Solutions for a Nonlinear Wave Equation with the p-laplacian Operator Global Solutions for a Nonlinear Wave Equation with the p-laplacian Operator Hongjun Gao Institute of Applied Physics and Computational Mathematics 188 Beijing, China To Fu Ma Departamento de Matemática

More information

A DEGREE THEORY FRAMEWORK FOR SEMILINEAR ELLIPTIC SYSTEMS

A DEGREE THEORY FRAMEWORK FOR SEMILINEAR ELLIPTIC SYSTEMS A DEGREE THEORY FRAMEWORK FOR SEMILINEAR ELLIPTIC SYSTEMS CONGMING LI AND JOHN VILLAVERT Abstract. This paper establishes the existence of positive entire solutions to some systems of semilinear elliptic

More information

EXISTENCE AND REGULARITY RESULTS FOR SOME NONLINEAR PARABOLIC EQUATIONS

EXISTENCE AND REGULARITY RESULTS FOR SOME NONLINEAR PARABOLIC EQUATIONS EXISTECE AD REGULARITY RESULTS FOR SOME OLIEAR PARABOLIC EUATIOS Lucio BOCCARDO 1 Andrea DALL AGLIO 2 Thierry GALLOUËT3 Luigi ORSIA 1 Abstract We prove summability results for the solutions of nonlinear

More information

Hardy Rellich inequalities with boundary remainder terms and applications

Hardy Rellich inequalities with boundary remainder terms and applications manuscripta mathematica manuscript No. (will be inserted by the editor) Elvise Berchio Daniele Cassani Filippo Gazzola Hardy Rellich inequalities with boundary remainder terms and applications Received:

More information

SYMMETRY AND REGULARITY OF AN OPTIMIZATION PROBLEM RELATED TO A NONLINEAR BVP

SYMMETRY AND REGULARITY OF AN OPTIMIZATION PROBLEM RELATED TO A NONLINEAR BVP lectronic ournal of Differential quations, Vol. 2013 2013, No. 108, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SYMMTRY AND RGULARITY

More information

VANISHING-CONCENTRATION-COMPACTNESS ALTERNATIVE FOR THE TRUDINGER-MOSER INEQUALITY IN R N

VANISHING-CONCENTRATION-COMPACTNESS ALTERNATIVE FOR THE TRUDINGER-MOSER INEQUALITY IN R N VAISHIG-COCETRATIO-COMPACTESS ALTERATIVE FOR THE TRUDIGER-MOSER IEQUALITY I R Abstract. Let 2, a > 0 0 < b. Our aim is to clarify the influence of the constraint S a,b = { u W 1, (R ) u a + u b = 1 } on

More information

arxiv: v1 [math.ap] 16 Jan 2015

arxiv: v1 [math.ap] 16 Jan 2015 Three positive solutions of a nonlinear Dirichlet problem with competing power nonlinearities Vladimir Lubyshev January 19, 2015 arxiv:1501.03870v1 [math.ap] 16 Jan 2015 Abstract This paper studies a nonlinear

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

A NOTE ON RANDOM HOLOMORPHIC ITERATION IN CONVEX DOMAINS

A NOTE ON RANDOM HOLOMORPHIC ITERATION IN CONVEX DOMAINS Proceedings of the Edinburgh Mathematical Society (2008) 51, 297 304 c DOI:10.1017/S0013091506001131 Printed in the United Kingdom A NOTE ON RANDOM HOLOMORPHIC ITERATION IN CONVEX DOMAINS FILIPPO BRACCI

More information

On semilinear elliptic equations with nonlocal nonlinearity

On semilinear elliptic equations with nonlocal nonlinearity On semilinear elliptic equations with nonlocal nonlinearity Shinji Kawano Department of Mathematics Hokkaido University Sapporo 060-0810, Japan Abstract We consider the problem 8 < A A + A p ka A 2 dx

More information

ANNALES DE L I. H. P., SECTION C

ANNALES DE L I. H. P., SECTION C ANNALES DE L I. H. P., SECTION C KAISING TSO Remarks on critical exponents for hessian operators Annales de l I. H. P., section C, tome 7, n o 2 (1990), p. 113-122

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

OPTIMAL POTENTIALS FOR SCHRÖDINGER OPERATORS. 1. Introduction In this paper we consider optimization problems of the form. min F (V ) : V V, (1.

OPTIMAL POTENTIALS FOR SCHRÖDINGER OPERATORS. 1. Introduction In this paper we consider optimization problems of the form. min F (V ) : V V, (1. OPTIMAL POTENTIALS FOR SCHRÖDINGER OPERATORS G. BUTTAZZO, A. GEROLIN, B. RUFFINI, AND B. VELICHKOV Abstract. We consider the Schrödinger operator + V (x) on H 0 (), where is a given domain of R d. Our

More information

Multiple Solutions for Parametric Neumann Problems with Indefinite and Unbounded Potential

Multiple Solutions for Parametric Neumann Problems with Indefinite and Unbounded Potential Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 8, Number 2, pp. 281 293 (2013) http://campus.mst.edu/adsa Multiple Solutions for Parametric Neumann Problems with Indefinite and Unbounded

More information

DEGREE AND SOBOLEV SPACES. Haïm Brezis Yanyan Li Petru Mironescu Louis Nirenberg. Introduction

DEGREE AND SOBOLEV SPACES. Haïm Brezis Yanyan Li Petru Mironescu Louis Nirenberg. Introduction Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 13, 1999, 181 190 DEGREE AND SOBOLEV SPACES Haïm Brezis Yanyan Li Petru Mironescu Louis Nirenberg Dedicated to Jürgen

More information

arxiv: v1 [math.ap] 28 Mar 2014

arxiv: v1 [math.ap] 28 Mar 2014 GROUNDSTATES OF NONLINEAR CHOQUARD EQUATIONS: HARDY-LITTLEWOOD-SOBOLEV CRITICAL EXPONENT VITALY MOROZ AND JEAN VAN SCHAFTINGEN arxiv:1403.7414v1 [math.ap] 28 Mar 2014 Abstract. We consider nonlinear Choquard

More information

NONTRIVIAL SOLUTIONS FOR SUPERQUADRATIC NONAUTONOMOUS PERIODIC SYSTEMS. Shouchuan Hu Nikolas S. Papageorgiou. 1. Introduction

NONTRIVIAL SOLUTIONS FOR SUPERQUADRATIC NONAUTONOMOUS PERIODIC SYSTEMS. Shouchuan Hu Nikolas S. Papageorgiou. 1. Introduction Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 34, 29, 327 338 NONTRIVIAL SOLUTIONS FOR SUPERQUADRATIC NONAUTONOMOUS PERIODIC SYSTEMS Shouchuan Hu Nikolas S. Papageorgiou

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

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

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

Xiyou Cheng Zhitao Zhang. 1. Introduction

Xiyou Cheng Zhitao Zhang. 1. Introduction Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 34, 2009, 267 277 EXISTENCE OF POSITIVE SOLUTIONS TO SYSTEMS OF NONLINEAR INTEGRAL OR DIFFERENTIAL EQUATIONS Xiyou

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

Non-radial solutions to a bi-harmonic equation with negative exponent

Non-radial solutions to a bi-harmonic equation with negative exponent Non-radial solutions to a bi-harmonic equation with negative exponent Ali Hyder Department of Mathematics, University of British Columbia, Vancouver BC V6TZ2, Canada ali.hyder@math.ubc.ca Juncheng Wei

More information

2 A Model, Harmonic Map, Problem

2 A Model, Harmonic Map, Problem ELLIPTIC SYSTEMS JOHN E. HUTCHINSON Department of Mathematics School of Mathematical Sciences, A.N.U. 1 Introduction Elliptic equations model the behaviour of scalar quantities u, such as temperature or

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

UNIQUENESS OF POSITIVE SOLUTION TO SOME COUPLED COOPERATIVE VARIATIONAL ELLIPTIC SYSTEMS

UNIQUENESS OF POSITIVE SOLUTION TO SOME COUPLED COOPERATIVE VARIATIONAL ELLIPTIC SYSTEMS TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9947(XX)0000-0 UNIQUENESS OF POSITIVE SOLUTION TO SOME COUPLED COOPERATIVE VARIATIONAL ELLIPTIC SYSTEMS YULIAN

More information

GRAND SOBOLEV SPACES AND THEIR APPLICATIONS TO VARIATIONAL PROBLEMS

GRAND SOBOLEV SPACES AND THEIR APPLICATIONS TO VARIATIONAL PROBLEMS LE MATEMATICHE Vol. LI (1996) Fasc. II, pp. 335347 GRAND SOBOLEV SPACES AND THEIR APPLICATIONS TO VARIATIONAL PROBLEMS CARLO SBORDONE Dedicated to Professor Francesco Guglielmino on his 7th birthday W

More information

H u + u p = 0 in H n, (1.1) the only non negative, C2, cylindrical solution of

H u + u p = 0 in H n, (1.1) the only non negative, C2, cylindrical solution of NONLINEAR LIOUVILLE THEOREMS IN THE HEISENBERG GROUP VIA THE MOVING PLANE METHOD I. Birindelli Università La Sapienza - Dip. Matematica P.zza A.Moro 2-00185 Roma, Italy. J. Prajapat Tata Institute of Fundamental

More information

A Caffarelli-Kohn-Nirenberg type inequality with variable exponent and applications to PDE s

A Caffarelli-Kohn-Nirenberg type inequality with variable exponent and applications to PDE s A Caffarelli-Kohn-Nirenberg type ineuality with variable exponent and applications to PDE s Mihai Mihăilescu a,b Vicenţiu Rădulescu a,c Denisa Stancu-Dumitru a a Department of Mathematics, University of

More information

EXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN EQUATION WITH DEGENERATE MOBILITY

EXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN EQUATION WITH DEGENERATE MOBILITY Electronic Journal of Differential Equations, Vol. 216 216), No. 329, pp. 1 22. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN

More information

ASYMMETRIC SUPERLINEAR PROBLEMS UNDER STRONG RESONANCE CONDITIONS

ASYMMETRIC SUPERLINEAR PROBLEMS UNDER STRONG RESONANCE CONDITIONS Electronic Journal of Differential Equations, Vol. 07 (07), No. 49, pp. 7. ISSN: 07-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ASYMMETRIC SUPERLINEAR PROBLEMS UNDER STRONG RESONANCE

More information

Non-degeneracy of perturbed solutions of semilinear partial differential equations

Non-degeneracy of perturbed solutions of semilinear partial differential equations Non-degeneracy of perturbed solutions of semilinear partial differential equations Robert Magnus, Olivier Moschetta Abstract The equation u + F(V (εx, u = 0 is considered in R n. For small ε > 0 it is

More information

ON THE SCHRÖDINGER EQUATION INVOLVING A CRITICAL SOBOLEV EXPONENT AND MAGNETIC FIELD. Jan Chabrowski Andrzej Szulkin. 1.

ON THE SCHRÖDINGER EQUATION INVOLVING A CRITICAL SOBOLEV EXPONENT AND MAGNETIC FIELD. Jan Chabrowski Andrzej Szulkin. 1. Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 25, 2005, 3 21 ON THE SCHRÖDINGER EQUATION INVOLVING A CRITICAL SOBOLEV EXPONENT AND MAGNETIC FIELD Jan Chabrowski

More information

BLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED

BLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED BLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED TAOUFIK HMIDI AND SAHBI KERAANI Abstract. In this note we prove a refined version of compactness lemma adapted to the blowup analysis

More information

ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR PARABOLIC OPERATORS OF LERAY-LIONS TYPE AND MEASURE DATA

ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR PARABOLIC OPERATORS OF LERAY-LIONS TYPE AND MEASURE DATA ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR PARABOLIC OPERATORS OF LERAY-LIONS TYPE AND MEASURE DATA FRANCESCO PETITTA Abstract. Let R N a bounded open set, N 2, and let p > 1; we study the asymptotic behavior

More information

The critical dimension for a fourth order elliptic problem with singular nonlinearity

The critical dimension for a fourth order elliptic problem with singular nonlinearity The critical dimension for a fourth order elliptic problem with singular nonlinearity Craig COWAN Pierpaolo ESPOSITO Nassif GHOUSSOU September 8, 008 Abstract We study the regularity of the extremal solution

More information

The Existence of Nodal Solutions for the Half-Quasilinear p-laplacian Problems

The Existence of Nodal Solutions for the Half-Quasilinear p-laplacian Problems Journal of Mathematical Research with Applications Mar., 017, Vol. 37, No., pp. 4 5 DOI:13770/j.issn:095-651.017.0.013 Http://jmre.dlut.edu.cn The Existence of Nodal Solutions for the Half-Quasilinear

More information

ASYMPTOTIC BEHAVIOUR OF NONLINEAR ELLIPTIC SYSTEMS ON VARYING DOMAINS

ASYMPTOTIC BEHAVIOUR OF NONLINEAR ELLIPTIC SYSTEMS ON VARYING DOMAINS ASYMPTOTIC BEHAVIOUR OF NONLINEAR ELLIPTIC SYSTEMS ON VARYING DOMAINS Juan CASADO DIAZ ( 1 ) Adriana GARRONI ( 2 ) Abstract We consider a monotone operator of the form Au = div(a(x, Du)), with R N and

More information

A note on the moving hyperplane method

A note on the moving hyperplane method 001-Luminy conference on Quasilinear Elliptic and Parabolic Equations and Systems, Electronic Journal of Differential Equations, Conference 08, 00, pp 1 6. http://ejde.math.swt.edu or http://ejde.math.unt.edu

More information

Robustness for a Liouville type theorem in exterior domains

Robustness for a Liouville type theorem in exterior domains Robustness for a Liouville type theorem in exterior domains Juliette Bouhours 1 arxiv:1207.0329v3 [math.ap] 24 Oct 2014 1 UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris,

More information

Positive eigenfunctions for the p-laplace operator revisited

Positive eigenfunctions for the p-laplace operator revisited Positive eigenfunctions for the p-laplace operator revisited B. Kawohl & P. Lindqvist Sept. 2006 Abstract: We give a short proof that positive eigenfunctions for the p-laplacian are necessarily associated

More information

AN APPLICATION OF THE LERAY-SCHAUDER DEGREE THEORY TO THE VARIABLE COEFFICIENT SEMILINEAR BIHARMONIC PROBLEM. Q-Heung Choi and Tacksun Jung

AN APPLICATION OF THE LERAY-SCHAUDER DEGREE THEORY TO THE VARIABLE COEFFICIENT SEMILINEAR BIHARMONIC PROBLEM. Q-Heung Choi and Tacksun Jung Korean J. Math. 19 (2011), No. 1, pp. 65 75 AN APPLICATION OF THE LERAY-SCHAUDER DEGREE THEORY TO THE VARIABLE COEFFICIENT SEMILINEAR BIHARMONIC PROBLEM Q-Heung Choi and Tacksun Jung Abstract. We obtain

More information

Some Remarks About the Density of Smooth Functions in Weighted Sobolev Spaces

Some Remarks About the Density of Smooth Functions in Weighted Sobolev Spaces Journal of Convex nalysis Volume 1 (1994), No. 2, 135 142 Some Remarks bout the Density of Smooth Functions in Weighted Sobolev Spaces Valeria Chiadò Piat Dipartimento di Matematica, Politecnico di Torino,

More information

Localization phenomena in degenerate logistic equation

Localization phenomena in degenerate logistic equation Localization phenomena in degenerate logistic equation José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal 1,2 rpardo@mat.ucm.es 1 Universidad Complutense de Madrid, Madrid, Spain 2 Instituto de Ciencias

More information

SPECTRAL PROPERTIES AND NODAL SOLUTIONS FOR SECOND-ORDER, m-point, BOUNDARY VALUE PROBLEMS

SPECTRAL PROPERTIES AND NODAL SOLUTIONS FOR SECOND-ORDER, m-point, BOUNDARY VALUE PROBLEMS SPECTRAL PROPERTIES AND NODAL SOLUTIONS FOR SECOND-ORDER, m-point, BOUNDARY VALUE PROBLEMS BRYAN P. RYNNE Abstract. We consider the m-point boundary value problem consisting of the equation u = f(u), on

More information

Nonnegative solutions with a nontrivial nodal set for elliptic equations on smooth symmetric domains

Nonnegative solutions with a nontrivial nodal set for elliptic equations on smooth symmetric domains Nonnegative solutions with a nontrivial nodal set for elliptic equations on smooth symmetric domains P. Poláčik School of Mathematics, University of Minnesota Minneapolis, MN 55455, USA Susanna Terracini

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

ON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM

ON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM ON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM OLEG ZUBELEVICH DEPARTMENT OF MATHEMATICS THE BUDGET AND TREASURY ACADEMY OF THE MINISTRY OF FINANCE OF THE RUSSIAN FEDERATION 7, ZLATOUSTINSKY MALIY PER.,

More information

NONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS

NONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS Fixed Point Theory, Volume 9, No. 1, 28, 3-16 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html NONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS GIOVANNI ANELLO Department of Mathematics University

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

Saddle Solutions of the Balanced Bistable Diffusion Equation

Saddle Solutions of the Balanced Bistable Diffusion Equation Saddle Solutions of the Balanced Bistable Diffusion Equation JUNPING SHI College of William and Mary Harbin Normal University Abstract We prove that u + f (u) = 0 has a unique entire solution u(x, y) on

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

Tiziana Cardinali Francesco Portigiani Paola Rubbioni. 1. Introduction

Tiziana Cardinali Francesco Portigiani Paola Rubbioni. 1. Introduction Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 32, 2008, 247 259 LOCAL MILD SOLUTIONS AND IMPULSIVE MILD SOLUTIONS FOR SEMILINEAR CAUCHY PROBLEMS INVOLVING LOWER

More information

Note on the Chen-Lin Result with the Li-Zhang Method

Note on the Chen-Lin Result with the Li-Zhang Method J. Math. Sci. Univ. Tokyo 18 (2011), 429 439. Note on the Chen-Lin Result with the Li-Zhang Method By Samy Skander Bahoura Abstract. We give a new proof of the Chen-Lin result with the method of moving

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

STOKES PROBLEM WITH SEVERAL TYPES OF BOUNDARY CONDITIONS IN AN EXTERIOR DOMAIN

STOKES PROBLEM WITH SEVERAL TYPES OF BOUNDARY CONDITIONS IN AN EXTERIOR DOMAIN Electronic Journal of Differential Equations, Vol. 2013 2013, No. 196, pp. 1 28. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu STOKES PROBLEM

More information

EIGENVALUE QUESTIONS ON SOME QUASILINEAR ELLIPTIC PROBLEMS. lim u(x) = 0, (1.2)

EIGENVALUE QUESTIONS ON SOME QUASILINEAR ELLIPTIC PROBLEMS. lim u(x) = 0, (1.2) Proceedings of Equadiff-11 2005, pp. 455 458 Home Page Page 1 of 7 EIGENVALUE QUESTIONS ON SOME QUASILINEAR ELLIPTIC PROBLEMS M. N. POULOU AND N. M. STAVRAKAKIS Abstract. We present resent results on some

More information

ON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS

ON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS Bendikov, A. and Saloff-Coste, L. Osaka J. Math. 4 (5), 677 7 ON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS ALEXANDER BENDIKOV and LAURENT SALOFF-COSTE (Received March 4, 4)

More information

EXISTENCE AND NONEXISTENCE OF ENTIRE SOLUTIONS TO THE LOGISTIC DIFFERENTIAL EQUATION

EXISTENCE AND NONEXISTENCE OF ENTIRE SOLUTIONS TO THE LOGISTIC DIFFERENTIAL EQUATION EXISTENCE AND NONEXISTENCE OF ENTIRE SOLUTIONS TO THE LOGISTIC DIFFERENTIAL EQUATION MARIUS GHERGU AND VICENŢIURĂDULESCU Received 26 February 23 We consider the one-dimensional logistic problem r α A u

More information

On the Second Minimax Level of the Scalar Field Equation and Symmetry Breaking

On the Second Minimax Level of the Scalar Field Equation and Symmetry Breaking arxiv:128.1139v3 [math.ap] 2 May 213 On the Second Minimax Level of the Scalar Field Equation and Symmetry Breaking Kanishka Perera Department of Mathematical Sciences Florida Institute of Technology Melbourne,

More information

LOWER AND UPPER SOLUTIONS TO SEMILINEAR BOUNDARY VALUE PROBLEMS: AN ABSTRACT APPROACH. Alessandro Fonda Rodica Toader. 1.

LOWER AND UPPER SOLUTIONS TO SEMILINEAR BOUNDARY VALUE PROBLEMS: AN ABSTRACT APPROACH. Alessandro Fonda Rodica Toader. 1. Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder University Centre Volume 38, 2011, 59 93 LOWER AND UPPER SOLUTIONS TO SEMILINEAR BOUNDARY VALUE PROBLEMS: AN ABSTRACT APPROACH

More information

Equilibria with a nontrivial nodal set and the dynamics of parabolic equations on symmetric domains

Equilibria with a nontrivial nodal set and the dynamics of parabolic equations on symmetric domains Equilibria with a nontrivial nodal set and the dynamics of parabolic equations on symmetric domains J. Földes Department of Mathematics, Univerité Libre de Bruxelles 1050 Brussels, Belgium P. Poláčik School

More information

The Maximum Principles and Symmetry results for Viscosity Solutions of Fully Nonlinear Equations

The Maximum Principles and Symmetry results for Viscosity Solutions of Fully Nonlinear Equations The Maximum Principles and Symmetry results for Viscosity Solutions of Fully Nonlinear Equations Guozhen Lu and Jiuyi Zhu Abstract. This paper is concerned about maximum principles and radial symmetry

More information

Existence of at least two periodic solutions of the forced relativistic pendulum

Existence of at least two periodic solutions of the forced relativistic pendulum Existence of at least two periodic solutions of the forced relativistic pendulum Cristian Bereanu Institute of Mathematics Simion Stoilow, Romanian Academy 21, Calea Griviţei, RO-172-Bucharest, Sector

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

Lebesgue-Stieltjes measures and the play operator

Lebesgue-Stieltjes measures and the play operator Lebesgue-Stieltjes measures and the play operator Vincenzo Recupero Politecnico di Torino, Dipartimento di Matematica Corso Duca degli Abruzzi, 24, 10129 Torino - Italy E-mail: vincenzo.recupero@polito.it

More information