An Application of a Mountain Pass Theorem

Abstract

We are concerned with the following Dirichlet problem:

$$ \begin{array}{*{20}c} {{ - \Delta u{\left( x \right)} = f{\left( {x,u} \right)},}} & {{x \in \Omega ,}} & {{u \in H^{1}_{0} {\left( \Omega \right)},}} \\ \end{array} $$

((P))

where f(x, t) ∈C (${\ifmmode\expandafter\bar\else\expandafter\=\fi{\Omega }}$×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L ^∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0,

$$ 0 < \theta F{\left( {x,s} \right)} \leqslant f{\left( {x,s} \right)}s,\;{\text{for}}\;{\text{all}}\;{\left| s \right|} \geqslant M\;{\text{and}}\;x \in \Omega , $$

((AR))

is no longer true, where $F{\left( {x,s} \right)} = {\int_0^s {f{\left( {x,t} \right)}dt} }.$ As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞.