O. Jokhadze

A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

NONEXISTENCE OF POSITIVE SOLUTIONS FOR SOME CLASSES OF NONLINEAR ELLIPTIC INEQUALITES IN ℝ^{N}

The report deals with the investigation of the nonexistence of positive solutions for a class of nonlinear elliptic inequalites

− div (A(x, u, Du)Du) ≥ a(x)u^{q}, u ≥ 0, u ≢ 0 in ℝ^{ℕ},  (1)

where div := ∑_{i=1}^{N} \frac{∂}{∂x_{i}}, D := (\frac{∂}{∂x_{1}} , \frac{∂}{∂x_{2}} , …, \frac{∂}{∂x_{N}} ); ℝ := ] − ∞, ∞[ , ℝ^{ℕ} :=

\underbrace{ℝ × ⋯ × ℝ}_{ℕ−times}; D_{i}u, i = 1, …, N are understood in the sense of distributions.

Bellow we consider two particular cases of the function A:

(i) A(x, u, Du) := |x|^{α_{1}}|u|^{q_{1}}|Du|^{p−2}, where | · | denotes the Euclidean norm in the space ℝ^{ℕ};

(ii) A(x, u, Du) := A(|Du|) and the continuous function A : ℝ_{+} → ℝ_{+} satysfies the condition of uniform boundedness from above: 0 < A(t) ≤ C, t ∈ ℝ_{+} := [⊬, ∞[ , C := const. The latter case includes two important examples:

(a) A(t) := \frac{1}{\sqrt{1+t^{2}}} , t ∈ ℝ_{+} (the operator of mean curvature),

(b) A(t) := \frac{1}{(1+|t|^{k})^{s}}, t ∈ ℝ_{+}, k, s > 0 (the generalised operator of mean curvature).

The continuous function a : ℝ^{ℕ} → ℝ_{+} is such that

a(x) ≥ C_{0}|x|^{γ}, ∀x : |x| ≥ R_{0} > 0, γ ∈ ℝ, ℝ_{⊬}, ℂ_{⊬} := const . (2)

There is hold the following

Theorem. Let N + α_{1} > p > 1 − q_{1} (N > 2). It is also assumed that the functions A and a satisfy the conditions (i) ((ii)) and (2), respectively. If

q_{1} + p − 1 < q ≤ \frac{(N + γ)(q_{1} + p − 1)}{N + α_{1} − p} ( 1 < q ≤ \frac{N + γ}{N − 2} ),

then the problem (1) has no solution in the functional class W_{a,α,α_{1},loc}^{1,p,q,q_{1}}(ℝ^{ℕ}):= {≊ : ℝ^{ℕ} → ℝ_{+}, ⅁(↶)≊^{∥+α}, |↶|^{α_{⊮}}|𝔻≊|^{∣}≊^{∥_{⊮}+α−⊮} ∈ 𝕃_{⋖⋊}^{⊮}(ℝ^{ℕ})} (W_{a,α,0,loc}^{1,2,q,0}(ℝ^{ℕ})) for sufficiently small α < 0, respectively.

These problems were posed in the paper [1] by E. Mitidieri and S. I. Pohozaev.

References

1. E. Mitidieri, S. I. Pohozaev, Nonexistence of positive solutions for quasilinear elliptic problems on ℝ^{ℕ}. Proc. Steklov Math. Inst. 227(1999), 192–222.