Multiplicity of Solutions for a Class of Elliptic Problem of p-Laplacian Type with a p-Gradient Term
We consider the following problem: -Δpu=c(x)|u|q-1u+μ|∇u|p+h(x) in Ω, u=0 on ∂Ω, where Ω is a bounded set in RN (N≥3) with a smooth boundary, 1<p<N, q>0, μ∈R⁎, and c and h belong to Lk(Ω) for some k>N/p. In this paper, we assume that c≩0 a.e. in Ω and h without sign condition and th...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2019-01-01
|
| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2019/6824502 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | We consider the following problem: -Δpu=c(x)|u|q-1u+μ|∇u|p+h(x) in Ω, u=0 on ∂Ω, where Ω is a bounded set in RN (N≥3) with a smooth boundary, 1<p<N, q>0, μ∈R⁎, and c and h belong to Lk(Ω) for some k>N/p. In this paper, we assume that c≩0 a.e. in Ω and h without sign condition and then we prove the existence of at least two bounded solutions under the condition that ck and hk are suitably small. For this purpose, we use the Mountain Pass theorem, on an equivalent problem to (P) with variational structure. Here, the main difficulty is that the nonlinearity term considered does not satisfy Ambrosetti and Rabinowitz condition. The key idea is to replace the former condition by the nonquadraticity condition at infinity. |
|---|---|
| ISSN: | 0161-1712 1687-0425 |