Ytukyg

Let $Omega$ be a non-empty bounded open subset of $mathbb{R}^N$, $lambdain mathbb{R}$ be an eigenvalue of $-Delta$ on the Sobolev space $H^1_0(Omega)$ and $fin L^infty(Omegatimesmathbb{R})$ such that

• $forall xin Omega, tmapsto f(x,t)in C(mathbb{R});$


• $forall M>0, exists r>0, forall |s|>r, forall xin Omega, int_0^s f(x,t)operatorname{d}tge M;$


• $forall varepsilon>0, exists r>0, forall|s|>r, forall xinOmega,left| frac{1}{s}int_0^s f(x,t)operatorname{d}tright|levarepsilon.$

Define: $$I:H^1_0(Omega)tomathbb{R}, umapstofrac{1}{2}|u|^2_{H^1_0}-frac {lambda}{2}|u|_2^2-int_Omegaint_0^{u(x)}f(x,t)operatorname{d}toperatorname{d}x.$$
Then $Iin C^1(H^1_0(Omega),mathbb{R})$ and
$$forall u,vin H^1_0(Omega), operatorname{d}I(u)(v)=int_Omega nabla u(x)cdot nabla v(x) operatorname{d}x-lambdaint_Omega u(x)v(x) operatorname{d}x-int_Omega f(x,u(x))v(x)operatorname{d}x$$

Is it true that $I$ satisfies the Palais-Smale condition? I.e. is it true that for all $(u_n)_{ninmathbb{N}}subset H^1_0(Omega)$ such that $(I(u_n))_{ninmathbb{N}}$ is bounded and $|operatorname{d}I(u_n)|to0, nrightarrowinfty$ there exists a subsequence $(u_{n_k})_{kinmathbb{N}}$ that converges in $H^1_0(Omega)$ to some $bar u in H^1_0(Omega)$?

In my lecture notes on calculus of variations, it is claimed that every sequence $(u_n)_{ninmathbb{N}}$ that satisfies the previous condition is actually bounded in $H^1_0(Omega)$ and so, by the fact that there exists a subsequence that weakly converges in $H^1_0(Omega)$ to some $bar u in H^1_0(Omega)$, it easily follows (from the fact that the differential of $I$ can be expressed as a sum of a homeomorphism and a compact operator) that this subsequence also converges to $bar u$ in $H^1_0(Omega)$.

My problem is proving the fact that a sequence $(u_n)_{ninmathbb{N}}$ as before is actually bounded in $H^1_0(Omega)$.
In particular, what I have proved is the following.
First, decompose $H^1_0(Omega)$ as the orthogonal sum: $$H^1_0(Omega)=E_-oplus E_0oplus E_+$$
where $E_-$ is the vector space generated by the eigenfunctions relative to eigenvalues less than $lambda$, $E_0$ is the eigenspace relative to $lambda$ and $E_+$ is the closure of the vector space generated by the the eigenfunctions relative to eigenvalues greater than $lambda$. Define $P_-$ as the orthogonal projection of $H^1_0(Omega)$ onto $E_-$, define $P_0$ as the orthogonal projection of $H^1_0(Omega)$ onto $E_0$ and define $P_+$ as the orthogonal projection of $H^1_0(Omega)$ onto $E_+$.

Then, using the relations:
$$operatorname{d}I(u_n)(P_-u_n)ge -C|P_-u_n|_{H^1_0}$$
and
$$operatorname{d}I(u_n)(P_+u_n)le C|P_+u_n|_{H^1_0}$$
and the estimates of $|cdot|_2^2$ from below with respect to $|cdot|_{H^1_0}^2$ on $E_-$ and of $|cdot|_2^2$ from above with respect to $|cdot|_{H^1_0}^2$ on $E_+$, we obtain that $(P_-u_n)_{ninmathbb{N}}$ and $(P_+u_n)_{ninmathbb{N}}$ are bounded in $H^1_0(Omega)$.

It remains to show that $(P_0 u_n)_{ninmathbb{N}}$ is bounded in $H^1_0(Omega)$.

Thanks to the previous estimates, what I proved is that for some constant $B,C>0$ we have that:
$$Cge |I(u_n)|=left|frac{1}{2}|u_n|^2_{H^1_0}-frac{lambda}{2}|u_n|_2^2-int_Omegaint_0^{u_n(x)}f(x,t)operatorname{d}toperatorname{d}xright|ge left|int_Omegaint_0^{u_n(x)}f(x,t)operatorname{d}toperatorname{d}xright|-B$$
and so the sequence:
$$left(int_Omegaint_0^{u_n(x)}f(x,t)operatorname{d}toperatorname{d}xright)_{ninmathbb{N}}$$
is bounded in $mathbb{R}$.

Now, I suspect that I have to use the hypothesis $$forall M>0, exists r>0, forall |s|>r, forall xin Omega, int_0^s f(x,t)operatorname{d}tge M$$
with the boundedness of the sequences
$$left(int_Omegaint_0^{u_n(x)}f(x,t)operatorname{d}toperatorname{d}xright)_{ninmathbb{N}}, left(P_-u_nright)_{ninmathbb{N}}, left(P_+u_nright)_{ninmathbb{N}}$$
to conclude that actually $(P_0u_n)_{ninmathbb{N}}$ (or directly the sequence $(u_n)_{ninmathbb{N}}$) is bounded in $H^1_0(Omega)$, but I can't see how...

Any suggestion?