Multiple positive solutions for indefinite semilinear elliptic problems involving a critical Sobolev exponent

Ching-Yu, Chen; Wu, Tsung‐fang

doi:10.1017/s0308210512000133

Cited by 15 publications

(19 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Set

{\overset{w}{true}}_{ϵ, z} false(x false) = η false(x - z false) w_{ϵ} false(x - z false)

. By He and Yang,, Lemma 4.2

false‖ {\overset{w}{true}}_{ϵ, z} ‖^{2} = {(‖‖, w_{ϵ})}_{D^{1, 2} false(R^{4} false)}^{2} + O false(ϵ^{2} false) = \frac{b S_{4}^{2}}{1 - a S_{4}^{2}} + O false(ϵ^{2} false),

\int_{Ω} false| {\overset{w}{true}}_{ϵ, z} |^{4} d x = \int_{R^{4}} {(||, w_{ϵ})}^{4} d x + O false(ϵ^{4} false) = \frac{{((), b S_{4})}^{2}}{{((), 1 - a S_{4}^{2})}^{2}} + O false(ϵ^{4} false) .

Similar to Chen and Wu,, Lemma 3.1

\int_{Ω} h {(||, {\overset{w}{true}}_{ϵ, z})}^{4} d x = \frac{{((), b S_{4})}^{2}}{{((), 1 - a S_{4}^{2})}^{2}} + o false(ϵ false), \int_{Ω} {\overset{w}{true}}_{ϵ, z}^{q} d x = o false(ϵ false) uniformly for z \in M .

…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…The existence and multiplicity of positive solutions of Equation have been researched widely. For example, when g = h ≡1, Ambrosetti et al obtained the result that there exists λ 0 >0, such that Equation has at least 2 positive solutions for 0< λ < λ 0 , a positive solution for λ = λ 0 , and no positive solution for λ > λ 0 ; Furthermore, in Chen and Wu, when λ g ( x ) is replaced by

λ g^{+} + g^{-} ((), g^{\pm} = \max ({}, \pm g, 0)),

the authors proved that existence and multiplicity of positive solutions. Besides, when g and h are sign changing, similar problems have been discussed; see Hsu and Lin and Wu .…”

Section: Introductionmentioning

confidence: 99%

“…By using the Talenti function of the semilinear elliptic problem, Chen and Wu obtained the existence and multiplicity of positive solutions of Equation . Inspired by Chen and Wu, in this paper, we consider the existence and multiplicity of positive solutions of the Kirchhoff problem in

R^{4}

. Similar to Naimen, the Talenti function multiplied by an appropriate constant, we give an affirmative answer to the existence and multiplicity of positive solutions for the Kirchhoff problem involving critical Sobolev exponent.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Kirchhoff‐type equation involving critical exponent and sign‐changing weight functions in dimension four

Lou

2018

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…Set

{\overset{w}{true}}_{ϵ, z} false(x false) = η false(x - z false) w_{ϵ} false(x - z false)

. By He and Yang,, Lemma 4.2

false‖ {\overset{w}{true}}_{ϵ, z} ‖^{2} = {(‖‖, w_{ϵ})}_{D^{1, 2} false(R^{4} false)}^{2} + O false(ϵ^{2} false) = \frac{b S_{4}^{2}}{1 - a S_{4}^{2}} + O false(ϵ^{2} false),

\int_{Ω} false| {\overset{w}{true}}_{ϵ, z} |^{4} d x = \int_{R^{4}} {(||, w_{ϵ})}^{4} d x + O false(ϵ^{4} false) = \frac{{((), b S_{4})}^{2}}{{((), 1 - a S_{4}^{2})}^{2}} + O false(ϵ^{4} false) .

Similar to Chen and Wu,, Lemma 3.1

\int_{Ω} h {(||, {\overset{w}{true}}_{ϵ, z})}^{4} d x = \frac{{((), b S_{4})}^{2}}{{((), 1 - a S_{4}^{2})}^{2}} + o false(ϵ false), \int_{Ω} {\overset{w}{true}}_{ϵ, z}^{q} d x = o false(ϵ false) uniformly for z \in M .

…”

Section: Proof Of Theoremmentioning

confidence: 99%

λ g^{+} + g^{-} ((), g^{\pm} = \max ({}, \pm g, 0)),

the authors proved that existence and multiplicity of positive solutions. Besides, when g and h are sign changing, similar problems have been discussed; see Hsu and Lin and Wu .…”

Section: Introductionmentioning

confidence: 99%

R^{4}

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Kirchhoff‐type equation involving critical exponent and sign‐changing weight functions in dimension four

Lou

2018

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…Furthermore, by means of the tool of Nehari manifold, Zhang et al [36] established the existence theorem of ground states for generalized Choquard equation when the nonlinear term is concave-convex. On the other hand, there are a great deal of results on the existence of elliptic boundary value problems with sign-changing weights [37][38][39][40][41][42][43][44][45]. We should point out that Hsu and Lin [38] showed the existence of positive solutions for elliptic equations with concave-convex nonlinearities and sign-changing weights.…”

Section: Introductionmentioning

confidence: 97%

“…In view of the same method, Chen and Wu [41] obtained the existence of positive solutions for a class of critical semilinear problem. Chen et al [42] established multiplicity .…”

Section: Introductionmentioning

confidence: 99%

The Choquard Equation with Weighted Terms and Sobolev-Hardy Exponent

Sang

Luo

Wang

2018

Journal of Function Spaces

View full text Add to dashboard Cite

show abstract

Positive solutions for a p−q‐Laplacian system with critical nonlinearities

Yin

Han

2017

Math Methods in App Sciences

View full text Add to dashboard Cite

In this paper, our main purpose is to establish the existence results of positive solutions for a p − q-Laplacian system involving concave-convex nonlinearities:where Ω is a bounded domain in R N , , > 0 and 1 < r < q < p < N. We assume 1 < , and + = p * = Np N−p is the critical Sobolev exponent and △ s · = div(|∇ · | s−2 ∇·) is the s-Laplacian operator. The main results are obtained by variational methods.

show abstract

Multiple positive solutions for indefinite semilinear elliptic problems involving a critical Sobolev exponent

Cited by 15 publications

References 20 publications

A Kirchhoff‐type equation involving critical exponent and sign‐changing weight functions in dimension four

A Kirchhoff‐type equation involving critical exponent and sign‐changing weight functions in dimension four

The Choquard Equation with Weighted Terms and Sobolev-Hardy Exponent

Positive solutions for a p−q‐Laplacian system with critical nonlinearities

Contact Info

Product

Resources

About