Multiple solutions for a class of nonhomogeneous semilinear equations with critical cone Sobolev exponent

2018 ◽  
Vol 147 (2) ◽  
pp. 597-608
Author(s):  
Morteza Koozehgar Kalleji ◽  
Mohsen Alimohammady ◽  
Ali Asghar Jafari
Keyword(s):  
Multiple Solutions ◽  
Semilinear Equations ◽  
Critical Cone ◽  
Sobolev Exponent

Multiplicity results for nonlinear elliptic equations involving critical Sobolev exponent

1986 ◽  
Vol 103 (3-4) ◽  
pp. 275-285 ◽  
Author(s):  
A. Capozzi ◽  
G. Palmieri
Keyword(s):  
Elliptic Equations ◽  
Multiple Solutions ◽  
Nonlinear Elliptic Equations ◽  
Critical Sobolev Exponent ◽  
Sobolev Embedding ◽  
Multiplicity Results ◽  
Variational Techniques ◽  
Nonlinear Elliptic ◽  
Sobolev Exponent ◽  
Image Position

SynopsisIn this paper we study the following boundary value problemwhere Ω is a bounded domain in Rn, n≧3, x ∈Rn, p* = 2n/(n – 2) is the critical exponent for the Sobolev embedding is a real parameter and f(x, t) increases, at infinity, more slowly than .By using variational techniques, we prove the existence of multiple solutions to the equations (0.1), in the case when λ belongs to a suitable left neighbourhood of an arbitrary eigenvalue of −Δ, and the existence of at least one solution for any λ sufficiently large.

On inhomogeneous biharmonic equations involving critical exponents

1999 ◽  
Vol 129 (5) ◽  
pp. 925-946 ◽  
Author(s):  
Yinbin Deng ◽  
Gengsheng Wang
Keyword(s):  
Variational Principle ◽  
Boundary Value Problem ◽  
Multiple Solutions ◽  
Critical Sobolev Exponent ◽  
Mountain Pass Lemma ◽  
Biharmonic Equations ◽  
Bounded Smooth Domain ◽  
Bounded Smooth ◽  
Sobolev Exponent ◽  
Image Position

In this paper, we consider the existence of multiple solutions of biharmonic equations boundary value problemwhere Ω is a bounded smooth domain in ℝN, N ≥ 5; λ ∈ ℝ1 is a given constant; p = 2N/(N − 4) is the critical Sobolev exponent for the embedding ; Δ2 = ΔΔ denotes iterated N-dimensional Laplacian; f(x) is a given function. Some results on the existence and non-existence of multiple solutions for the above problem have been obtained by Ekeland's variational principle and the mountain-pass lemma under some assumptions on f(x) and N.

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