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On an asymptotically linear elliptic Dirichlet problem. (English) Zbl 1047.35047
From the introduction: We consider the existence of one-signed solutions for the following Dirichlet problem: $-\Delta u=f(x,u),\;x\in \Omega,\quad u=0,\;x\in \partial\Omega,\tag{1}$ where $$\Omega$$ is a bounded domain in $$\mathbb{R}^N$$ $$(N\geq 1)$$ with smooth boundary $$\partial \Omega$$. The conditions imposed on $$f(x,t)$$ are as follows:
$$(f_1)$$ $$f\in C(\Omega\times\mathbb{R},\mathbb{R})$$; $$f(x,0)=0$$, for all $$x\in \Omega$$.
$$(f_2)$$ $$\lim_{| t|\to 0}(f(x,t)/t) =\mu$$, $$\lim_{| t|\to\infty} (f(x,t)/t)=\ell$$ uniformly in $$x\in\Omega$$.
Since we assume $$(f_2)$$, problem (1) is called asymptotically linear at both zero and infinity. Obviously, the constant function $$u=0$$ is a trivial solution of problem (1). Therefore, we are interested in finding nontrivial solutions. Let $$F(x,u)= \int^u_0f(x,s)\,ds$$. It follows from $$(f_1)$$ and $$(f_2)$$ that the functional $J(u)=\tfrac 12\int_\Omega|\nabla u|^2\,dx-\int_\Omega F(x,u) \,dx\tag{2}$ is of class $$C^1$$ on the Sobolev space $$H^1_0:=H^1_0(\Omega)$$ with norm $\| u\|:=\left(\int_\Omega |\nabla u|^2\right)^{1/2},\tag{3}$ and the critical points of $$J$$ are solutions of (1). Thus we will try to find critical points of $$J$$. In doing so, we have to prove that the functional $$J$$ satisfies the (PS) condition. We prove that $$(f_1)$$ and $$(f_2)$$ are sufficient to obtain a positive solution and a negative solution of problem (1). Our main result is the following. Theorem 1.1 Assume that $$f$$ satisfies $$(f_1)$$ and $$(f_2)$$. If $$\mu<\lambda_1<\ell$$, the problem (1) has at least two nontrivial solutions, one is positive, the other is negative. Note that in Theorem 1.1, even in the resonant case, we do not need to assume any additional conditions to ensure that $$J$$ satisfy the (PS) conditions.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35B34 Resonance in context of PDEs 35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
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