Multiplicity for semilinear elliptic equations involving singular nonlinearity

J. Hernández, J. Karátson, L. P. Simon

Research output: Contribution to journalArticle

18 Citations (Scopus)

Abstract

We investigate the number of positive solutions of the semilinear boundary value problem{A formula is presented}where BR ⊂ Rn is the ball centered at the origin with radius R, and f : ( 0, ∞ ) → ( 0, ∞ ) is a locally Lipschitz continuous function satisfying the following conditions:(H1) f ( u ) {greater than or slanted equal to} m > 0 ( u > 0 ) ;(H2)there exists {A formula is presented}where 2* = frac(n + 2, n - 2) if n > 2 and 2* = ∞ if n {less-than or slanted equal to} 2.(H3){A formula is presented}The motivating example, involving singular nonlinearity, is the problem{A formula is presented}with some constants {A formula is presented}This problem can be reduced to (1) with f ( s ) = s- α + sp using a suitable transformation. Problems with singular nonlinearities have been studied extensively recently involving more general nonlinearities which may also depend on x. For the case of singularities in x, see [10,13] and references therein. We are mainly interested here in singularities in u. For nonlinearities where the term up does not appear, existence and uniqueness has been proved in [3,5,9,11,12,17]. For equations of the form Δ u + frac(λ, uα) - up, existence and uniqueness for p > 0 (and also for p = - β <0, β <α) can be found in [12]. (For p = - β <0, α <β <1 non-existence is expected [12].). In the presence of up the problem exhibits a different behavior according to the sign of λ. If λ <0, then there may be no positive solution; existence and non-existence results for the concave case p ∈ [ 0, 1 ) can be found in [6,21,24] and generalized for x-dependent nonlinearities in [4,12]. In the latter case non-existence of positive solutions under the critical value λ* still allows existence of non-negative solutions with a free boundary (see [4]). Related results can be found in [20] for the nonlinearity λ ( up - u- α ). When the parameter λ multiplies the term up, then existence and uniqueness holds for all λ if u- α has a negative coefficient [12]. Here non-existence can also occur (namely for λ {less-than or slanted equal to} over(λ, -) with some over(λ, -) > 0) when u- α has a changing-sign coefficient or when the term up is replaced by a bounded non-negative function f ( x ) (see [6,12]). Conversely, a unique positive solution exists exactly for λ {less-than or slanted equal to} over(λ, -) with some over(λ, -) > 0 in two related cases when the equation contains three terms: Δ u + λ u + f ( x ) equals u- α or - u- α. For p = 0 there are also some multiplicity results in [1]. For λ > 0, existence is easier to prove on a ball and has also been obtained on smooth bounded domains. For p ∈ ( 0, 1 ) there is always a positive solution, which is also unique [12,21]. On the other hand, for p > 1 existence only holds for small λ [2,12]. Existence of a second positive solution has been proved for associated non-singular problems by using variational arguments. Some related results for different concave and/or convex nonlinearities can be found in [14-16,19]. For the singular case we prove here the existence of a second positive solution in the case of a ball and, moreover, that there are exactly two radial positive solutions for n = 1. At the best of our knowledge, the celebrated result by Gidas et al. [8] has not been extended to the case of singular nonlinearities. Hence we cannot be sure that our problems have only radial solutions in the case of a ball, and consequently our results for the case n > 1 refer only to positive radial solutions. The main results of our paper are formulated as follows. {A formulation is presented}. In order to have exact multiplicity for n = 1, f is assumed to satisfy a further condition:(H4) f is a strictly convex C2 function(i.e. f {greater than or slanted equal to} 0 and f does not vanish identically on any interval).{A formulation is presented}. Concerning the main example (2), Theorems 1 and 2 yield the following result using the transformation {A formula is presented}{A formulation is presented}. The proof of our results relies on the shooting method. Theorem 1 is proved in Section 3 via characterizing the shape of the time-map. The exact multiplicity is dealt with in Section 4 in a little more general setting than in Theorem 2, namely, we assume a certain admissibility condition (H5), which is seen to be always satisfied in the case n = 1.

Original languageEnglish
Pages (from-to)265-283
Number of pages19
JournalNonlinear Analysis
Volume65
Issue number2
DOIs
Publication statusPublished - Jul 15 2006

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Singular Nonlinearity
Semilinear Elliptic Equations
Multiplicity
Formulation
Positive Solution
Ball
Exact multiplicity
Positive Radial Solutions
Nonlinearity
Less than or equal to
Existence and Uniqueness
Term
Theorem
Time Map
Boundary value problems
Singularity
Radial Solutions
Shooting Method
Multiplicity Results
Lipschitz Function

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics
  • Mathematics(all)

Cite this

Multiplicity for semilinear elliptic equations involving singular nonlinearity. / Hernández, J.; Karátson, J.; Simon, L. P.

In: Nonlinear Analysis, Vol. 65, No. 2, 15.07.2006, p. 265-283.

Research output: Contribution to journalArticle

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abstract = "We investigate the number of positive solutions of the semilinear boundary value problem{A formula is presented}where BR ⊂ Rn is the ball centered at the origin with radius R, and f : ( 0, ∞ ) → ( 0, ∞ ) is a locally Lipschitz continuous function satisfying the following conditions:(H1) f ( u ) {greater than or slanted equal to} m > 0 ( u > 0 ) ;(H2)there exists {A formula is presented}where 2* = frac(n + 2, n - 2) if n > 2 and 2* = ∞ if n {less-than or slanted equal to} 2.(H3){A formula is presented}The motivating example, involving singular nonlinearity, is the problem{A formula is presented}with some constants {A formula is presented}This problem can be reduced to (1) with f ( s ) = s- α + sp using a suitable transformation. Problems with singular nonlinearities have been studied extensively recently involving more general nonlinearities which may also depend on x. For the case of singularities in x, see [10,13] and references therein. We are mainly interested here in singularities in u. For nonlinearities where the term up does not appear, existence and uniqueness has been proved in [3,5,9,11,12,17]. For equations of the form Δ u + frac(λ, uα) - up, existence and uniqueness for p > 0 (and also for p = - β <0, β <α) can be found in [12]. (For p = - β <0, α <β <1 non-existence is expected [12].). In the presence of up the problem exhibits a different behavior according to the sign of λ. If λ <0, then there may be no positive solution; existence and non-existence results for the concave case p ∈ [ 0, 1 ) can be found in [6,21,24] and generalized for x-dependent nonlinearities in [4,12]. In the latter case non-existence of positive solutions under the critical value λ* still allows existence of non-negative solutions with a free boundary (see [4]). Related results can be found in [20] for the nonlinearity λ ( up - u- α ). When the parameter λ multiplies the term up, then existence and uniqueness holds for all λ if u- α has a negative coefficient [12]. Here non-existence can also occur (namely for λ {less-than or slanted equal to} over(λ, -) with some over(λ, -) > 0) when u- α has a changing-sign coefficient or when the term up is replaced by a bounded non-negative function f ( x ) (see [6,12]). Conversely, a unique positive solution exists exactly for λ {less-than or slanted equal to} over(λ, -) with some over(λ, -) > 0 in two related cases when the equation contains three terms: Δ u + λ u + f ( x ) equals u- α or - u- α. For p = 0 there are also some multiplicity results in [1]. For λ > 0, existence is easier to prove on a ball and has also been obtained on smooth bounded domains. For p ∈ ( 0, 1 ) there is always a positive solution, which is also unique [12,21]. On the other hand, for p > 1 existence only holds for small λ [2,12]. Existence of a second positive solution has been proved for associated non-singular problems by using variational arguments. Some related results for different concave and/or convex nonlinearities can be found in [14-16,19]. For the singular case we prove here the existence of a second positive solution in the case of a ball and, moreover, that there are exactly two radial positive solutions for n = 1. At the best of our knowledge, the celebrated result by Gidas et al. [8] has not been extended to the case of singular nonlinearities. Hence we cannot be sure that our problems have only radial solutions in the case of a ball, and consequently our results for the case n > 1 refer only to positive radial solutions. The main results of our paper are formulated as follows. {A formulation is presented}. In order to have exact multiplicity for n = 1, f is assumed to satisfy a further condition:(H4) f is a strictly convex C2 function(i.e. f″ {greater than or slanted equal to} 0 and f″ does not vanish identically on any interval).{A formulation is presented}. Concerning the main example (2), Theorems 1 and 2 yield the following result using the transformation {A formula is presented}{A formulation is presented}. The proof of our results relies on the shooting method. Theorem 1 is proved in Section 3 via characterizing the shape of the time-map. The exact multiplicity is dealt with in Section 4 in a little more general setting than in Theorem 2, namely, we assume a certain admissibility condition (H5), which is seen to be always satisfied in the case n = 1.",
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T1 - Multiplicity for semilinear elliptic equations involving singular nonlinearity

AU - Hernández, J.

AU - Karátson, J.

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N2 - We investigate the number of positive solutions of the semilinear boundary value problem{A formula is presented}where BR ⊂ Rn is the ball centered at the origin with radius R, and f : ( 0, ∞ ) → ( 0, ∞ ) is a locally Lipschitz continuous function satisfying the following conditions:(H1) f ( u ) {greater than or slanted equal to} m > 0 ( u > 0 ) ;(H2)there exists {A formula is presented}where 2* = frac(n + 2, n - 2) if n > 2 and 2* = ∞ if n {less-than or slanted equal to} 2.(H3){A formula is presented}The motivating example, involving singular nonlinearity, is the problem{A formula is presented}with some constants {A formula is presented}This problem can be reduced to (1) with f ( s ) = s- α + sp using a suitable transformation. Problems with singular nonlinearities have been studied extensively recently involving more general nonlinearities which may also depend on x. For the case of singularities in x, see [10,13] and references therein. We are mainly interested here in singularities in u. For nonlinearities where the term up does not appear, existence and uniqueness has been proved in [3,5,9,11,12,17]. For equations of the form Δ u + frac(λ, uα) - up, existence and uniqueness for p > 0 (and also for p = - β <0, β <α) can be found in [12]. (For p = - β <0, α <β <1 non-existence is expected [12].). In the presence of up the problem exhibits a different behavior according to the sign of λ. If λ <0, then there may be no positive solution; existence and non-existence results for the concave case p ∈ [ 0, 1 ) can be found in [6,21,24] and generalized for x-dependent nonlinearities in [4,12]. In the latter case non-existence of positive solutions under the critical value λ* still allows existence of non-negative solutions with a free boundary (see [4]). Related results can be found in [20] for the nonlinearity λ ( up - u- α ). When the parameter λ multiplies the term up, then existence and uniqueness holds for all λ if u- α has a negative coefficient [12]. Here non-existence can also occur (namely for λ {less-than or slanted equal to} over(λ, -) with some over(λ, -) > 0) when u- α has a changing-sign coefficient or when the term up is replaced by a bounded non-negative function f ( x ) (see [6,12]). Conversely, a unique positive solution exists exactly for λ {less-than or slanted equal to} over(λ, -) with some over(λ, -) > 0 in two related cases when the equation contains three terms: Δ u + λ u + f ( x ) equals u- α or - u- α. For p = 0 there are also some multiplicity results in [1]. For λ > 0, existence is easier to prove on a ball and has also been obtained on smooth bounded domains. For p ∈ ( 0, 1 ) there is always a positive solution, which is also unique [12,21]. On the other hand, for p > 1 existence only holds for small λ [2,12]. Existence of a second positive solution has been proved for associated non-singular problems by using variational arguments. Some related results for different concave and/or convex nonlinearities can be found in [14-16,19]. For the singular case we prove here the existence of a second positive solution in the case of a ball and, moreover, that there are exactly two radial positive solutions for n = 1. At the best of our knowledge, the celebrated result by Gidas et al. [8] has not been extended to the case of singular nonlinearities. Hence we cannot be sure that our problems have only radial solutions in the case of a ball, and consequently our results for the case n > 1 refer only to positive radial solutions. The main results of our paper are formulated as follows. {A formulation is presented}. In order to have exact multiplicity for n = 1, f is assumed to satisfy a further condition:(H4) f is a strictly convex C2 function(i.e. f″ {greater than or slanted equal to} 0 and f″ does not vanish identically on any interval).{A formulation is presented}. Concerning the main example (2), Theorems 1 and 2 yield the following result using the transformation {A formula is presented}{A formulation is presented}. The proof of our results relies on the shooting method. Theorem 1 is proved in Section 3 via characterizing the shape of the time-map. The exact multiplicity is dealt with in Section 4 in a little more general setting than in Theorem 2, namely, we assume a certain admissibility condition (H5), which is seen to be always satisfied in the case n = 1.

AB - We investigate the number of positive solutions of the semilinear boundary value problem{A formula is presented}where BR ⊂ Rn is the ball centered at the origin with radius R, and f : ( 0, ∞ ) → ( 0, ∞ ) is a locally Lipschitz continuous function satisfying the following conditions:(H1) f ( u ) {greater than or slanted equal to} m > 0 ( u > 0 ) ;(H2)there exists {A formula is presented}where 2* = frac(n + 2, n - 2) if n > 2 and 2* = ∞ if n {less-than or slanted equal to} 2.(H3){A formula is presented}The motivating example, involving singular nonlinearity, is the problem{A formula is presented}with some constants {A formula is presented}This problem can be reduced to (1) with f ( s ) = s- α + sp using a suitable transformation. Problems with singular nonlinearities have been studied extensively recently involving more general nonlinearities which may also depend on x. For the case of singularities in x, see [10,13] and references therein. We are mainly interested here in singularities in u. For nonlinearities where the term up does not appear, existence and uniqueness has been proved in [3,5,9,11,12,17]. For equations of the form Δ u + frac(λ, uα) - up, existence and uniqueness for p > 0 (and also for p = - β <0, β <α) can be found in [12]. (For p = - β <0, α <β <1 non-existence is expected [12].). In the presence of up the problem exhibits a different behavior according to the sign of λ. If λ <0, then there may be no positive solution; existence and non-existence results for the concave case p ∈ [ 0, 1 ) can be found in [6,21,24] and generalized for x-dependent nonlinearities in [4,12]. In the latter case non-existence of positive solutions under the critical value λ* still allows existence of non-negative solutions with a free boundary (see [4]). Related results can be found in [20] for the nonlinearity λ ( up - u- α ). When the parameter λ multiplies the term up, then existence and uniqueness holds for all λ if u- α has a negative coefficient [12]. Here non-existence can also occur (namely for λ {less-than or slanted equal to} over(λ, -) with some over(λ, -) > 0) when u- α has a changing-sign coefficient or when the term up is replaced by a bounded non-negative function f ( x ) (see [6,12]). Conversely, a unique positive solution exists exactly for λ {less-than or slanted equal to} over(λ, -) with some over(λ, -) > 0 in two related cases when the equation contains three terms: Δ u + λ u + f ( x ) equals u- α or - u- α. For p = 0 there are also some multiplicity results in [1]. For λ > 0, existence is easier to prove on a ball and has also been obtained on smooth bounded domains. For p ∈ ( 0, 1 ) there is always a positive solution, which is also unique [12,21]. On the other hand, for p > 1 existence only holds for small λ [2,12]. Existence of a second positive solution has been proved for associated non-singular problems by using variational arguments. Some related results for different concave and/or convex nonlinearities can be found in [14-16,19]. For the singular case we prove here the existence of a second positive solution in the case of a ball and, moreover, that there are exactly two radial positive solutions for n = 1. At the best of our knowledge, the celebrated result by Gidas et al. [8] has not been extended to the case of singular nonlinearities. Hence we cannot be sure that our problems have only radial solutions in the case of a ball, and consequently our results for the case n > 1 refer only to positive radial solutions. The main results of our paper are formulated as follows. {A formulation is presented}. In order to have exact multiplicity for n = 1, f is assumed to satisfy a further condition:(H4) f is a strictly convex C2 function(i.e. f″ {greater than or slanted equal to} 0 and f″ does not vanish identically on any interval).{A formulation is presented}. Concerning the main example (2), Theorems 1 and 2 yield the following result using the transformation {A formula is presented}{A formulation is presented}. The proof of our results relies on the shooting method. Theorem 1 is proved in Section 3 via characterizing the shape of the time-map. The exact multiplicity is dealt with in Section 4 in a little more general setting than in Theorem 2, namely, we assume a certain admissibility condition (H5), which is seen to be always satisfied in the case n = 1.

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